﻿ 微積分統一教學網-考題分析

# 微積分統一教學網

## 考題分析

### 手機版選單

#### 字級

 Calculating Limits with Limit Laws: 1-1(a),   1-3(a),   1-4(c),   1-6,   1-8(a),   1-9(a),   1-10(a)(b),   1-11(a),   1-12(a),   1-13(a),   1-15 Special Limits ($\lim_{x\to 0} \frac{\sin x}{x}=1$): 1-1(b),   1-2,   1-4(a)(b),   1-5(a),   1-8(c),   1-9(b),   1-11(b),   1-12(b),   1-14(a)(b) Calculating Limits by l'Hospital's Rule: 1-3(b)(c),   1-5(b),   1-7,   1-8(b)(d),   1-10(c),   1-11(c),   1-12(c),   1-14(c),   1-16,   1-17,   1-18,   1-19,   1-20,   1-21,   1-22

1-1

0.287

0.608

7.232

12

0.751

0.465

Evaluate the following limits.
(a) $\lim\limits_{x\rightarrow 2^-}\frac{x-\sqrt{2x}}{|x^2-4|}$.
(b) $\lim\limits_{x\rightarrow 0}\frac{1-\sin^2(ax)-\cos(ax)}{1+\sin^2(bx)-\cos(bx)}$, where $a$ and $b$ are non-zero real numbers.

1-2

0.527

0.629

9.040

15

0.893

0.366

Find $\lim_{x\rightarrow 0^{-}}$ and $\lim_{x\rightarrow 0^{+}}$ of the following functions: $$(a)\;\frac{\sin(|x|)}{x},\;\; (b)\;\frac{\cos x - 1}{\sin( x \sin x)},\;\; (c)\;\frac{\cos(\sin x) - 1}{\tan^{2} x}$$

1-3

0.535

0.398

7.580

20

0.665

0.130

Find the limits.
(a) $\lim_{x\rightarrow\infty} x^{\frac{3}{2}}\left(\sqrt{x+2}-2\sqrt{x+1}+\sqrt{x}\right)$.
(b) $\lim_{x\rightarrow\infty} \left[x+x^2\ln\left(1-\frac{2}{x}\right)\right]$.
(c) $\lim_{x\rightarrow 0} \left(\cosh 3x\right)^{\csc^2x}$, where $\cosh x=\frac{e^x+e^{-x}}{2}$

1-4

0.509

0.608

9.260

15

0.863

0.354

Find the following limits.
(a) $\lim\limits_{x\to0}\sqrt{x}\sin\left(\sin\left(\frac{1}{|x|}\right)\right)$.
(b) $\lim\limits_{x\to\infty}\sqrt{x}\sin\left(\sin\left(\frac{1}{|x|}\right)\right)$.
(c) $\lim\limits_{x\to-\infty}\left(\sqrt{x^2+2x-1}+x\right)$.

1-5

0.393

0.402

5.390

14

0.598

0.205

Evaluate the limit, if it exists.
(a) $\lim\limits_{x\rightarrow\infty}x\left(\sqrt{x^6-3x^5+1}-x^3\right)\tan\frac{1}{x^3}$,
(b) $\lim\limits_{x\rightarrow \infty}\left(1-\frac{1}{x}\right)^{x^2}\cdot e^x$.

1-6

0.365

0.393

3.970

10

0.575

0.211

Find all horizontal, vertical and slant asymptotes, if any, of $f(x)=\frac{[ x^3-\sqrt{x^6}]}{x^2}$ where $[ x ]$ denotes the greatest integer function.

1-7

0.585

0.643

6.850

10

0.935

0.350

Find the limit $\lim_{x\rightarrow 0}\left(\frac{a + x}{a - x}\right)^{1/x}, a > 0.$

1-8

0.364

0.501

16.690

32

0.683

0.319

Evaluate the following limits.
(a) $\lim\limits_{x\rightarrow -\infty}(\sqrt{x^2+x}+x)$
(b) $\lim\limits_{x\rightarrow 0}\left(\frac{\sin x}{x}\right)^{\frac{1}{1-\cos x}}$
(c) $\lim\limits_{x\rightarrow \infty}\frac{\sin\left(\frac{1}{\sqrt{x^2+1}}\right)}{\sqrt{x^2+2}-\sqrt{x^2-1}}$
(d) $\lim\limits_{x\rightarrow \infty}\left[\left(\frac{x}{1+x}\right)^x-\frac{1}{e}\right]x$

1-9

0.347

0.774

8.040

10

0.948

0.601

Find the following limits if they exist.
(a) $\lim\limits_{t\rightarrow 0}\left(\frac{1}{t\sqrt{1+|t|}}-\frac{1}{t}\right)$
(b) $\lim\limits_{x\rightarrow\infty}\sqrt{2x^2+1}\sin\left(\frac{1}{x}\right)$

1-10

0.414

0.677

8.340

12

0.884

0.470

Find the following limits.
(a) $\lim\limits_{x\rightarrow 0^+}\sqrt{x}e^{\sin(\pi/x)}$
(b) $\lim\limits_{x\rightarrow 0}\frac{(x - 1)^{1/3} + (x + 1)^{1/3}}{x}$
(c) $\lim\limits_{x\rightarrow 0}\left(\cos (x^2)\right)^{\frac{1}{x^4}}$

1-11

0.386

0.566

11.560

20

0.759

0.373

Compute each of the following limits if it exists or explain why it doesn't exist.
(a) $\lim\limits_{x\rightarrow 0} \sin\left(\frac{1}{x^2}\right) \sin x$.
(b) $\lim\limits_{x\rightarrow 0}\frac{\tan x}{\sqrt{1-\cos 3x}}$.
(c) $\lim\limits_{x\rightarrow 0} (\cos x)^{\frac{2}{x^2}}$.
(d) $\lim\limits_{x\rightarrow +\infty}\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}$.

1-12

0.279

0.649

9.660

15

0.789

0.509

Compute each of the following limits if it exists or explain why it doesn't exist.
(a) $\lim\limits_{x\rightarrow 0} \sin\left(\frac{1}{x^2}\right) \sin x$.
(b) $\lim\limits_{x\rightarrow 0}\frac{\tan x}{\sqrt{1-\cos 3x}}$.
(c) $\lim\limits_{x\rightarrow 0} (\cos x)^{\frac{2}{x^2}}$.

1-13

0.445

0.714

7.660

10

0.936

0.491

(a) Compute the limit if it exists or explain why it doesn't exist. $\lim\limits_{x\rightarrow +\infty}\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}.$ (b) Determine for what values of $a$, $0<a </a 1-14 0.404 0.706 10.16 14 0.908 0.504 Find the limit, if it exists. (a)$\lim\limits_{x\rightarrow 0}\frac{\sin(x^2)}{1-\cos x}$. (b)$\lim\limits_{x\rightarrow 0}\frac{|\tan x|}{1-\sqrt{1+2x}}$. (c)$\lim\limits_{x\rightarrow \infty}\left(1+\frac{2}{x}\right)^{[x]}$. (Hint:$[x]$is the greatest integer function,$x-1< [x] \leq x$and$\lim\limits_{x\rightarrow\infty}\left(1+\frac{1}{x}\right)^x=e.)$1-15 0.460 0.747 9.55 12 0.977 0.517 Find the following limits. (a)$\displaystyle{\lim_{x\rightarrow \infty}} \frac{\sqrt{2x^8+4x+1}+5x^3}{x^4+1}$. (b)$\lim\limits_{x\rightarrow 1}\tan^{-1}\left(\dfrac{4\sqrt{x}-4}{x^2-1}\right)$. 1-16 0.406 0.503 7.621 15 0.706 0.299 Compute the following limits (a)$\lim\limits_{x\rightarrow 0^+}\left(x^{-1}+x^{-2}\right)^{\frac{1}{\ln x}}$(b)$\lim\limits_{n\rightarrow \infty}\frac{\left[\ln\left(1-\frac{1}{n}\right)\right]^3}{\tan\left(\frac{1}{n}\right)- \sin\left(\frac{1}{n}\right)}$(c)$\lim\limits_{n\rightarrow\infty}\left(1+\frac{1}{\ln n}\right)^{n^{\alpha}}$, where$\alpha>0$1-17 0.538 0.452 3.690 8 0.721 0.183 Find (a)$\lim\limits_{n\rightarrow\infty}\left(\sin\frac{1}{n}\right)^{\frac{1}{n}}$and (b)$\lim\limits_{n\rightarrow\infty}\left(1+\frac{1}{n^2}\right)^n$. 1-18 0.315 0.788 7.363 9 0.945 0.630 (a) Find$\lim_{t\rightarrow 0^+}\frac{t-\ln(1+t)}{t^2}$(b) Use (a) to find$\lim_{t\rightarrow 0^+}\frac{\sqrt{t-\ln(1+t)}}{t}$1-19 0.548 0.666 5.876 9 0.940 0.392 Evaluate$\lim_{x\rightarrow\infty}\left(2^x+3^x+5^x\right)^{\frac{1}{x}}$. 1-20 0.426 0.762 8.060 10 0.975 0.549 Find the limit$\lim\limits_{x\to 0}\frac{\ln(\cos(ax))}{\ln(\cos(bx))}$, where$a$,$b$are constant. 1-21 0.423 0.733 7.750 10 0.945 0.522 Find the following limits if they exist. (a)$\lim\limits_{x\rightarrow 0^+}(1-\cos x )^{\frac{1}{\ln x}}$(b)$\lim_{x\to1}\left(\frac{1}{\ln x}-\frac{1}{x-1}\right)$1-22 0.691 0.535 6.49 12 0.880 0.189 Find the following limits. (a)$\displaystyle\lim_{x\to0} \frac{\displaystyle\tan^{-1} ( a\,x)}{\displaystyle\tan^{-1} (b\,x)}$, where$a$and$b\ne 0$are constants. (b)$\lim\limits_{x\rightarrow 0}(1+\sin 2x)^{\frac{1}{3x}}$. 主題分類表：  Definition of Continuity: 2-1(a), 2-2(a), 2-3(a), 2-4(a)(b), 2-7(a), 2-8(a), 2-9(a), 2-10(a), 2-11(a) 編號 鑑別度 難易度 平均 總分 高分群答對率 低分群答對率 題目 2-1 0.486 0.538 6.457 12 0.781 0.295 Suppose that$f(x)$is differentiable on$(-1,1)$with$\lim\limits_{x\rightarrow 0}\frac{f(x)}{x^2}=L$, where$L$is a constant. Define$g(x)=\begin{cases}f(x)\sin (\frac{1}{x})&\mbox{, for }0<|x|<1;\\ A&\mbox{, for } x=0.\end{cases}$(a) Find$f(0)$and$f'(0)$. (b) If$g$is continuous at$x=0$, find the value of$A$and compute$g'(0)$. (c) Write down a formula of$g'(x)$in terms of$f(x)$and$f'(x)$for$0<|x|<1$. (d) Suppose that$f'(x)$and$g'(x)$are both continuous at$0$. Find the value of$L$. 2-2 0.518 0.394 4.439 12 0.653 0.135 Let$a<b$. A function$f$in said to be a$contraction$on$[a,b]$if there exists$K$,$0<K<1$, such that for all$x_1,x_2\in [a,b]$we have$\left|f(x_1)-f(x_2)\right|\leq K\left|x_1-x_2\right|$. (a) Show by the$\epsilon-\delta$definition that if$f$is a contraction on$[a,b]$, then$f$is continuous on$[a,b]$. (b) Suppose that$f$is continuous on$[a,b]$and differentiable on$(a,b)$with$\left|f'(x)\right|\leq q$,$0<q<1$, for all$x\in (a,b)$. Show that$f$is a contraction on$[a,b]$and has at most one fixed point on$[a,b]$. (A point$c\in [a,b]$is called a fixed point of$f$if$f(c)=c$.) 2-3 0.552 0.620 10.010 15 0.896 0.344 Let$f(x)=\begin{cases} |x|^x &\mbox{ if }x\ne 0\\ 1 &\mbox{ if }x=0. \end{cases}$(a) Determine whether$f(x)$is continuous at$0$. (b) Determine whether$f(x)$is differentiable at$0$. 2-4 0.462 0.597 9.500 15 0.828 0.366 Let$f(x)=\begin{cases} (1+x)^{\frac{1}{x}}, &\text{for }x\ne 0,\ x>-1,\\ a, &\text{for }x=0. \end{cases}$(a) Find the value of$a$such that$f(x)$is continuous at$x=0$. (b) Find$\lim\limits_{x\rightarrow\infty}f(x)$. (c) Compute$f'(x)$, for$x\ne 0$. (d) Is$f(x)$differentiable at$x=0$? If$f(x)$is differentiable at$x=0$, then find$f'(0)$. 2-5 0.508 0.517 5.980 12 0.771 0.264 Suppose that$f(x)$is twice differentiable. Let$L(x)$be the linearization of$f(x)$at$x=a$. Define$g(x)=\begin{cases} \frac{f(x)-L(x)}{x-a}, &\text{for }x\ne a.\\ 0, &\text{for }x=a. \end{cases}$(a) Show that$g(x)$is continuous at$x=a$. (b) Show that$g(x)$is differentiable at$x=a$and compute$g'(a)$. Write down the linearization,$L_g(x)$, of$g(x)$at$x=a$. (c) Because$f(x)=L(x)+g(x)(x-a)$, we can use$L(x)+L_g(x)(x-a)$as a better approximation of$f(x)$. Use this better approximation to estimate$\tan^{-1}(1.01)$. 2-6 0.368 0.493 5.050 10 0.677 0.309 Let$f(x)=\begin{cases} ae^{x}+bx & x<0\\ m & x=0\\ e^{-\frac{1}{x^2}}+\sqrt{x+5} & x>0. \end{cases}$Find the constants$m$,$a$,$b$, such that$f(x)$is differentiable everywhere. 2-7 0.221 0.703 7.190 10 0.813 0.592 Define$f(x)=\begin{cases} x^2\sin\frac{1}{x}\ &\mbox{if }x\ne 0,\\ 0\ &\mbox{if }x=0. \end{cases}$(a) Show that$f(x)$is continuous at$x=0$. (b) Calculate$f'(x)$when$x\neq0$. (c) Find$f'(0)$if it exists. 2-8 0.504 0.464 5.130 12 0.716 0.212 Let$f(x)=\begin{cases} x^\alpha\sin\left(\frac{1}{x^\beta}\right), &x>0\\ 0, &x=0\\ \frac{\sin(x^\beta)}{1-\cos x}, &x<0. \end{cases}$(a) For what values of$\alpha$and$\beta$will$f(x)$be continuous at$x=0$? (b) For what values of$\alpha$and$\beta$will$f(x)$be differentiable at$x=0$? 2-9 0.336 0.774 9.750 12 0.942 0.606 Suppose that$f(x)=\begin{cases} \sin x+b\ln(x+1)+c\ {\rm if}\ x\geq0 \\ e^{x^2}\ {\rm if}\ x<0 \end{cases}.$(a) Find$b,\ c$such that$f(x)$is continuous. (b) Find$b,\ c$such that$f(x)$is differentiable. (c) For$b,\ c$in (b), is$f'(x)$continuous? 2-10 0.407 0.658 7.910 12 0.861 0.454 Let$f(x)=\begin{cases} x^{\frac{4}{3}}\cos\left(\frac{1}{x}\right), &\text{for }x\ne 0\\ 0, &\text{for }x=0. \end{cases}$(a) Is$f(x)$continuous at$x=0$? (b) Compute$f'(x)$for$x\ne 0$and$f'(0)$. (c) Is$f'(x)$continuous at$x=0$? 2-11 0.396 0.315 2.83 9 0.513 0.117 Let$f(x)$be a continuous function. It is given that $\lim_{h\to 0}\dfrac{f(h)}{h}=2020.$ (a) Compute$f(0)$. Then, prove that$f$is differentiable at$x=0$and compute$f'(0)$. (b) Suppose in addition that$f$is twice differentiable and that$f''(x)\geq 2$for all$x>0$. Using Mean Value Theorem, or otherwise, prove that $f(x)\geq 2020x+x^2\mbox{ for all } x\geq 0.$ 主題分類表：  Definition: 3-a-1, 3-a-2(b), 3-a-3, 3-a-4(b), 3-a-5, 3-a-6, 3-a-7, 3-a-8, 3-a-9, 3-a-10(a), 3-a-11, 3-a-12, 3-a-13, 3-a-14(a), 3-a-15 Techniques: 3-a-10(b), 3-b-1, 3-b-2, 3-b-3(a)(b)(c), 3-b-4, 3-b-5, 3-b-6, 3-b-7(a), 3-b-8, 3-b-9, 3-b-10, 3-b-11, 3-b-12 Implicit Differentiation: 3-b-3(d), 3-b-7(b), 3-c-1, 3-c-2, 3-c-3, 3-c-4, 3-c-5, 3-c-6, 3-c-7 Tangent lines: 3-d-1, 3-d-2, 3-d-3, 3-d-4, 3-d-5, 3-d-6 Related Rates: 3-e全部 編號 副主題 鑑別度 難易度 平均 總分 高分群答對率 低分群答對率 題目 3-a-1 a. Definition 0.486 0.538 6.457 12 0.781 0.295 Suppose that$f(x)$is differentiable on$(-1,1)$with$\lim\limits_{x\rightarrow 0}\frac{f(x)}{x^2}=L$, where$L$is a constant. Define$g(x)=\begin{cases}f(x)\sin (\frac{1}{x})&\mbox{, for }0<|x|<1;\\ A&\mbox{, for } x=0.\end{cases}$(a) Find$f(0)$and$f'(0)$. (b) If$g$is continuous at$x=0$, find the value of$A$and compute$g'(0)$. (c) Write down a formula of$g'(x)$in terms of$f(x)$and$f'(x)$for$0<|x|<1$. (d) Suppose that$f'(x)$and$g'(x)$are both continuous at$0$. Find the value of$L$. 3-a-2 a. Definition 0.552 0.620 10.010 15 0.896 0.344 Let$f(x)=\begin{cases} |x|^x &\mbox{ if }x\ne 0\\ 1 &\mbox{ if }x=0. \end{cases}$(a) Determine whether$f(x)$is continuous at$0$. (b) Determine whether$f(x)$is differentiable at$0$. 3-a-3 a. Definition 0.462 0.597 9.500 15 0.828 0.366 Let$f(x)=\begin{cases} (1+x)^{\frac{1}{x}}, &\text{for }x\ne 0,\ x>-1,\\ a, &\text{for }x=0. \end{cases}$(a) Find the value of$a$such that$f(x)$is continuous at$x=0$. (b) Find$\lim\limits_{x\rightarrow\infty}f(x)$. (c) Compute$f'(x)$, for$x\ne 0$. (d) Is$f(x)$differentiable at$x=0$? If$f(x)$is differentiable at$x=0$, then find$f'(0)$. 3-a-4 a. Definition 0.508 0.517 5.980 12 0.771 0.264 Suppose that$f(x)$is twice differentiable. Let$L(x)$be the linearization of$f(x)$at$x=a$. Define$g(x)=\begin{cases} \frac{f(x)-L(x)}{x-a}, &\text{for }x\ne a.\\ 0, &\text{for }x=a. \end{cases}$(a) Show that$g(x)$is continuous at$x=a$. (b) Show that$g(x)$is differentiable at$x=a$and compute$g'(a)$. Write down the linearization,$L_g(x)$, of$g(x)$at$x=a$. (c) Because$f(x)=L(x)+g(x)(x-a)$, we can use$L(x)+L_g(x)(x-a)$as a better approximation of$f(x)$. Use this better approximation to estimate$\tan^{-1}(1.01)$. 3-a-5 a. Definition 0.368 0.493 5.050 10 0.677 0.309 Let$f(x)=\begin{cases} ae^{x}+bx & x<0\\ m & x=0\\ e^{-\frac{1}{x^2}}+\sqrt{x+5} & x>0. \end{cases}$Find the constants$m$,$a$,$b$, such that$f(x)$is differentiable everywhere. 3-a-6 a. Definition 0.221 0.703 7.190 10 0.813 0.592 Define$f(x)=\begin{cases} x^2\sin\frac{1}{x}\ &\mbox{if }x\ne 0,\\ 0\ &\mbox{if }x=0. \end{cases}$(a) Show that$f(x)$is continuous at$x=0$. (b) Calculate$f'(x)$when$x\neq0$. (c) Find$f'(0)$if it exists. 3-a-7 a. Definition 0.504 0.464 5.130 12 0.716 0.212 Let$f(x)=\begin{cases} x^\alpha\sin\left(\frac{1}{x^\beta}\right), &x>0\\ 0, &x=0\\ \frac{\sin(x^\beta)}{1-\cos x}, &x<0. \end{cases}$(a) For what values of$\alpha$and$\beta$will$f(x)$be continuous at$x=0$? (b) For what values of$\alpha$and$\beta$will$f(x)$be differentiable at$x=0? 3-a-8 a. Definition 0.334 0.627 6.590 10 0.794 0.460 Student A used some mathematical software to plot a dolphin-like curve as Figure. On the dolphin's back, he considered two functions: \begin{align*} f(x)&=-\frac18\left(x-\frac72\right)^2+\frac{65}{32},\quad 0.7<x\leq 4 \\ g(x)&=\frac{a}{x-7}+b, 4<x<6.3. \end{align*} Find constantsa$and$b$such that the union of two functions$f(x)$and$g(x)$is differentiable on$(0.7,6.3). 3-a-9 a. Definition 0.306 0.727 8.990 12 0.880 0.574 Student A used some mathematical software to plot a Lapras-like curve (乘龍) as Figure. (a) On Lapras' body, he considered two functions: \begin{align*} f(x)&=-\frac3{10}(x-4)^2-\frac{17}{10}, 0.3<x\leq 5\2mm] g(x)&=\frac{a}{x-4}+bx-\frac{22}5, 5<x<8. \end{align*} Find constants a and b such that the union of f(x) and g(x) is differentiable at x=5. (b) The upper part of Lapras' back is depicted by h(x). Which of the following three functions can be a good candidate for h(x)? Give reasons why other two functions are not good candidates. \[ \frac32\left(\cos^{-1}x-\frac\pi2\right),\qquad\qquad -2x^{\frac13},\qquad\qquad \sinh x. 3-a-10 a. Definition 0.542 0.559 6.530 12 0.830 0.288 Suppose thatf(x)$is twice differentiable,$\lim\limits_{x\rightarrow 1}\frac{\left(f(x)\right)^3-8}{x-1}=18$, and$\lim\limits_{t\rightarrow 0}\frac{f'(1+t)-f'(1-3t)}{t}=1$. (a) Find$f(1)$,$f'(1)$and$f''(1)$. (b) Suppose that$g(x)=f(e^{2x})$is a one-to-one function and$h(x)=g^{-1}(x)$, the inverse function of$g(x)$. Find$h(2)$,$h'(2)$and$h''(2)$. 3-a-11 a. Definition 0.336 0.774 9.750 12 0.942 0.606 Suppose that$f(x)=\begin{cases} \sin x+b\ln(x+1)+c\ {\rm if}\ x\geq0 \\ e^{x^2}\ {\rm if}\ x<0 \end{cases}.$(a) Find$b,\ c$such that$f(x)$is continuous. (b) Find$b,\ c$such that$f(x)$is differentiable. (c) For$b,\ c$in (b), is$f'(x)$continuous? 3-a-12 a. Definition 0.407 0.658 7.910 12 0.861 0.454 Let$f(x)=\begin{cases} x^{\frac{4}{3}}\cos\left(\frac{1}{x}\right), &\text{for }x\ne 0\\ 0, &\text{for }x=0. \end{cases}$(a) Is$f(x)$continuous at$x=0$? (b) Compute$f'(x)$for$x\ne 0$and$f'(0)$. (c) Is$f'(x)$continuous at$x=0$? 3-a-13 a. Definition 0.632 0.500 5.9 12 0.817 0.184 (a) Suppose that a function$f$has the property:$|f(x_1)-f(x_2)|\leq |x_1-x_2|^2$for any real number$x_1$,$x_2$. (i) Show that$f$is differentiable everywhere. (ii) Determine$f$explicitly. (b) Suppose now that another function$g$has the property:$g(3x)=2(g(x)+x)$for any real number$x$, and$g$is differentiable at$x=0$}. Find$g(0)$and$g'(0)$. 3-a-14 a. Definition 0.396 0.315 2.83 9 0.513 0.117 Let$f(x)$be a continuous function. It is given that $\lim_{h\to 0}\dfrac{f(h)}{h}=2020.$ (a) Compute$f(0)$. Then, prove that$f$is differentiable at$x=0$and compute$f'(0)$. (b) Suppose in addition that$f$is twice differentiable and that$f''(x)\geq 2$for all$x>0$. Using Mean Value Theorem, or otherwise, prove that $f(x)\geq 2020x+x^2\mbox{ for all } x\geq 0.$ 3-a-15 a. Definition 0.550 0.611 10.48 16 0.886 0.337 Consider the function $$f(x)=\begin{cases}x^3\sin\dfrac{1}{x}+2x&\text{ if }x>0,\\ ax+b&\text{ if }x\leq 0.\end{cases}$$ Suppose that$f$is differentiable everywhere. (a) Find the values of$a$and$b$. (b) Write down the linear approximation of$f(x)$, at$x=0$. Use the linear approximation to estimate$f(0.01)$. 3-b-1 b. Techniques 0.271 0.837 13.287 15 0.973 0.701 Find the derivative of the functions. (You need not simplify your answer.) (a)$f(x)=\log_2 \left(3^x+x^4+5^6\right)$(b)$f(x)=\sin^{-1}\left(\cos^2\left(\tan x^3\right)\right)$3-b-2 b. Techniques 0.480 0.750 7.223 9 0.989 0.510 Find$\frac{d}{dx}\left(\sec x\right)^x$,$-\frac{\pi}{2}<x<\frac{\pi}{2}$. 3-b-3 b. Techniques 0.359 0.700 14.410 20 0.879 0.521 Find the first-order derivative of the following functions. (a)$f(x)=\frac{\tan^{-1}x}{1+xe^x}$(b)$f(x)=\ln\left(\frac{\sqrt{\sin x\cos x}}{1+2\ln x}\right)$(c)$f(x)=x\tan(\sin^{-1}x)$(d)$x^y=y^x+y$. Find$\frac{dy}{dx}$at$(2,1)$. 3-b-4 b. Techniques 0.514 0.525 15.110 28 0.782 0.268 (a) Let$f(x)=\frac{\tan 2x\cdot\cos^{-1}x+\ln(1+x)}{3\sec^3x+x^3\sin^{-1}x}$. Find$f'(0)$. (b) Let$f(x)=\sin^{-1}(\tanh x)+\tan^{-1}(\sinh x)$. Find$f'(x)$. [Make your answer as simple as possible.] (c) Let$f(x)=a^{a^{\sqrt{x}}}+a^{x^{\sqrt{x}}}$, where$a>0$is a constant. Find$f'(x)$. (d) Let$f(x)=\log_{2^{x}}\left(\log_{x^2}e\right)$. Find$f'(x)$. 3-b-5 b. Techniques 0.515 0.675 7.410 10 0.932 0.417 If$y=f(u)$and$u=g(x)$, where$f$and$g$are twice differentiable functions, with$g(0)=1$,$f(1)=2$,$g'(0)=2$,$f'(1)=-1$,$g''(0)=1$and$f''(1)=3$. Find$\frac{d^2y}{dx^2}\Big|_{x=0}$. 3-b-6 b. Techniques 0.366 0.434 3.290 8 0.617 0.251 Find the$n$th derivative of the function$f(x)=\frac{x^n}{1-x}$. 3-b-7 b. Techniques 0.303 0.797 8.280 10 0.948 0.645 (a) Find$\frac{dy}{dx}$at$(x,y)=(\pi, 0)$, where$\tan (x- y)= \frac{y}{1+ x^2}$. (b) Find$\frac{dy}{dx}$where$x^{y^2}=y^{x^2}$,$x>0$,$y>0$. 3-b-8 b. Techniques 0.474 0.703 8.680 12 0.940 0.466 Find the derivative of the following functions. (a)$y= \left(\tan^{-1} x\right)^{\sin x}$,$x>0$. (b)$y= \log_{e^x}( \tan x)$,$0<x<\frac{\pi}{2}$. (c)$y= \frac{ (2x+1)^5(x^2+1)^3}{(3x-2)^6(x^3+1)^4}$, find$y'(0)$. 3-b-9 b. Techniques 0.285 0.827 12.850 15 0.970 0.685 Differentiate the following functions. (a)$f(x) = \frac{\sin x}{1+ \cos x}$. (b)$f(x)=\log_2\sqrt{x}+\tan^{-1}(x^3)$. (c)$f(x) = x^{\cos x}$. 3-b-10 b. Techniques 0.307 0.834 12.890 15 0.987 0.680 Differentiate the following functions. (a)$f(x)=\frac{\sec^{-1}(e^x)}{1+x^e}$. (b)$f(x)=\log_2\sqrt{x}+\tan^{-1}(x^3)$(c)$f(x) = x^{\cos x}$. 3-b-11 b. Techniques 0.258 0.807 11.7 14 0.936 0.678 (a)$f(x)=e^{x^2-x}$. Find$f'(x)$and$f''(x)$. (b)$f(x)=\sin^{-1}(\sqrt{1-x^2})$. Find$f'(x)$. (c)$f(x)=(\cos x)^{\log_2x}+x^2\cdot\sec x$. Find$f'(x)$. 3-b-12 b. Techniques 0.328 0.776 8.12 10 0.940 0.612 Find the derivative of$f(x)$. (a)$f(x)=\displaystyle x^{\frac{4}{3}}+x\cdot 2^{(x^2+1)}$. (b)$f(x)=\displaystyle x \cdot \sec ^{-1}x-\frac{1}{2} \ln\left(x^2+1\right)$. 3-c-1 c. Implicit differentiation 0.546 0.632 9.793 15 0.905 0.358 (a) Show that the function$f(x)=x^3+3x+1$is strictly increasing on$\mathbb{R}$. (b) If$g(x)$is the inverse function to the function$f(x)$of part (a). Find$g'(5)$and$g''(5)$. 3-c-2 c. Implicit differentiation 0.445 0.743 7.865 10 0.965 0.521 Let$y=f(x)$satisfy$x^3+2xy+y^3=13$. Find$y'$and$y''$at the point$x=1$,$y=2$. 3-c-3 c. Implicit differentiation 0.751 0.480 3.480 8 0.855 0.104 Find the length of the loop of the curve$3ay^2=x(a-x)^2$,$a>0$. 3-c-4 c. Implicit differentiation 0.352 0.798 10.130 12 0.974 0.622 If$xy+e^{y}=e$, (a) find$\frac{dy}{dx}$. (b) find the values of$y$,$y'$and$y''$at the point where$x=0$. 3-c-5 c. Implicit differentiation 0.538 0.651 5.660 8 0.920 0.382 Let$f(x)$be a twice differentiable one-to-one function. Suppose that$f(2)=1$,$f'(2)=3$,$f''(2)=e$. Find$\frac{d}{dx}f^{-1}(1)$and$\frac{d^2}{dx^2}f^{-1}(1)$. 3-c-6 c. Implicit differentiation 0.367 0.730 9.060 12 0.913 0.546 Let the curve$x^2y^2 + 2xy = 8$be given. (a) Express$y'$in terms of$x$and$y$. (b) Find points on the curve with$y=2$and the tangent lines at these points. (c) Find$y''$at the points in (b). 3-c-7 c. Implicit differentiation 0.671 0.496 6.2 12 0.832 0.161 Let$f(x)=\tan^{-1}x+2x$. (a) Show that$f(x)$is one to one. Therefore$f(x)$has inverse function. (b) Find$f^{-1}\left(\frac{\pi}{4}+2\right)$and$\frac{d}{dx}f^{-1}\Big|_{\frac{\pi}{4}+2}$. (c) Write down the linear approximation of$f^{-1}(x)$at$x=\frac{\pi}{4}+2$. Use the linear approximation to estimate$f^{-1}\left(\frac{\pi}{4}+1.95\right)$. 3-c-8 c. Implicit differentiation 0.514 0.724 9.56 12 0.981 0.468 (a)$f(x)=(x^2+1)^{\cos x}$. Find$f'(x)$. (b) Find the equation of the tangent line to the curve satisfying$x^{\frac{2}{3}}+y^{\frac{2}{3}}+y=6$at$(8,1)$. 3-d-1 d. Tangent Lines 0.542 0.566 4.640 8 0.837 0.295 Find the value of the number$c$such that the families of curves$y=(x+\alpha)^{-1}$and$y=c(x+\beta)^{1/3}$are orthogonal trajectories, that is, every curve in one family is orthogonal to every curve in the other family. 3-d-2 d. Tangent Lines 0.477 0.407 3.240 8 0.645 0.169 Suppose that three points on the parabola$y=x^2$have the property that their normal lines intersect at a common point. Show that the sum of their$x$-coordinates is 0. 3-d-3 d. Tangent Lines 0.444 0.751 7.950 10 0.973 0.529 Figure shows a circle with radius$1$inscribed in the parabola$y=2x^2$. Find the center of the circle$C$, and find the tangent points$P$and$Q$. 3-d-4 d. Tangent Lines 0.407 0.753 8.060 10 0.956 0.549 Consider any point$(x_0,y_0)$, where$x_0>0$and$y_0>0$, on the hyperbola$xy=k$, where$k>0$is a constant. (a) Find the equation of the tangent line at$(x_0,y_0)$. (b) Let$A$and$B$denote respectively the$x$-intercept and the$y$-intercept of the tangent line at$(x_0,y_0)$. Find the area of the triangle enclosed by the origin$O$and$A, B$. (c) Is the area of$\triangle OAB$a constant? That is, is the area of$\triangle OAB$independent of$x_0$and$y_0? 3-d-5 d. Tangent Lines 0.423 0.610 7.790 12 0.822 0.399 An elliptic billiard table (橢圓形撞球桌) is shaped by the equation \begin{align*} \frac{x^2}{4}+y^2=1. \end{align*} A billiard ball is located at theQ(1,0)$. Find all points$P$on the boundary of the elliptic billiard table such that the billiard ball will roll from$Q$to$P$and bounce back to$Q$again. (We assume that the angle of incidence is equal to the angle of bouncing back.) 3-d-6 d. Tangent Lines 0.564 0.651 6.680 10 0.933 0.369 The figure shows a lamp located 4 units to the right of the$y-$axis and a shadow created by the elliptical region$x^2+5y^2\leq 6$. If the point$(-6,0)$is on the edge of the shadow, how far above the$x-$axis is the lamp located? 3-d-7 d. Tangent Lines 0.510 0.617 7.670 12 0.872 0.362 A {\it terminator} (晨昏圈) is a circle that separates the illuminated day side and the dark night side of the Earth. The terminator curve can be characterized by the function$\phi=\tan^{-1}(\cot{\phi_0}\cdot\sin{\theta})$on the world map with spherical coordinates$(\theta,\phi)\in[-\pi,\pi)\times\left[-\frac\pi2,\frac\pi2\right]$, where$\theta$is the modified longitude (經度),$\phi$is the latitude (緯度), and$\phi_0$is a constant called the declination (赤緯). Now we focus on the time 4AM, GMT+0, December 2, then$\cot{\phi_0}=\sqrt{6}$and the terminator curve will be $\phi=f(\theta)=\tan^{-1}\left(\sqrt{6}\sin{\theta}\right).$ The curve is shown in Figure. The gray part is at night. (a) City$P$is located at$P\left(\frac\pi4,\frac\pi3\right)$. Find the equation of the tangent line$L$to the terminator curve at$P$. (b) Compute$f''(\theta)$. (c) City$Q$is located at$Q\left(\frac\pi3,\left(\frac{\sqrt{3}}{48}+\frac13\right)\pi\right)$. Does City$Q$lie above, or on, or below the tangent line$L$? (d) Is City$Q$in the daytime or night-time at 4AM, GMT+0, December 2? Explain your answer. (You may use the results of (a),(b), and (c).) (e) Explain that there must be some place with 9-hours daytime on December 2. (You may observe the regions near the North Pole and South Pole first.) 3-e-1 e. Related rates 0.318 0.814 8.639 10 0.973 0.655 A rhombus (菱形) has sides 10in. long. Two of its opposite vertices are pulled apart at a rate of 2 in. per second. How fast is the area changing when the vertices being pulled are 16 in apart? 3-e-2 e. Related rates 0.554 0.443 4.544 10 0.720 0.167 The minute hand on a watch is 13 mm long and the hour hand is 11 mm long. How fast is the distance between the tips of the hands changing at two o'clock? 3-e-3 e. Related rates 0.638 0.578 9.004 15 0.897 0.259 The minute hand (分針) on a clock is 8cm long and the hour hand (時針) is 4cm long. How fast is the distance between the tips of the hands changing at two o'clock? Give your answer in the unit cm/hour. 3-e-4 e. Related rates 0.469 0.683 8.780 12 0.918 0.449 A baseball diamond is a square with side 27 m. A player runs from the first base to the second base at a rate of$5$m/s. (a) At what rate is the player's distance from the third base changing when the player is 9m from the first base? (b) At what rate is the angle$\theta$changing at the moment in part (a)? (c) The player slides into the second base at a rate of$4.5$m/s. At what rate is the angle$\theta$changing as the player touches the second base? 3-e-5 e. Related rates 0.444 0.443 4.490 10 0.665 0.221 The lengths of line segments$\overline{AB}$and$\overline{AC}$are fixed but the angle$\theta$between them decreases with time$t$so that the area of the triangle$\triangle ABC$decays exponentially. Suppose that$T_0$is the time required for half of the area to decay and at time$t=0$, the angle$\theta$is$\frac{\pi}{3}$. How fast is$\theta$decreasing when$t=2T_0$? 3-e-6 e. Related rates 0.652 0.359 3.250 12 0.685 0.033 A container in the shape of an inverted cone has height 16 cm and radius 5 cm at the top. It is partially filled with a liquid that oozes(滲出) through the sides at a rate proportional to the area of the container that is in contact with the liquid. If we pour the liquid into the container at a rate of 2 cm$^3$/min, then the height of the liquid decreases at a rate of 0.3 cm/min when the height is 10 cm. If our goal is to keep the liquid at a constant height of 10 cm, at what rate should we pour the liquid into the container? You may need this: The surface area of a cone is$\pi rl$, where$r$is the radius and$l$is the slant height(斜高). 3-e-7 e. Related rates 0.597 0.619 6.590 10 0.917 0.321 The top of a ladder slides down a vertical wall at a rate of$0.1$m/s. At the moment when the bottom of the ladder is 4 m from the wall, it slides away from the wall at a rate of 0.2 m/s. (a) How long is the ladder? (b) At that moment, how fast is the angle between the ladder and the ground changing? 3-e-8 e. Related rates 0.612 0.651 7.080 10 0.957 0.345 A plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is$\pi/3$, this angle is decreasing at a rate of$\pi/6$rad$/$min. How fast is the plane travelling at that time? 3-e-9 e. Related rates 0.654 0.469 5.070 10 0.796 0.142 In the engine, a 7-inch connecting rod is fastened to a crank of radius 3 inches. The crankshaft(機軸) rotates counterclockwise at a constant rate of 100 revolutions per minute. Find the velocity of the piston(活塞) when$\theta=\frac{\pi}{3}$. (Reminder: the angular velocity of a circular motion at a constant speed of 1 revolution per minute is 2$\pi$rad/min.) 圖 1 圖 2 3-e-10 e. Related rates 0.534 0.498 6.91 13 0.765 0.231 The minute hand on a clock is 6 cm long and the hour hand is 3 cm long. Let$\theta(t)$be the angle between the minute hand and the hour hand at time$t$. Let$d(t)$be the distance between the tips of the hands at time$t$. (a) Find$\left|\frac{d\theta}{dt}\right|$, ($\text{rad}/\text{hour}$). (b) Find$d'(t)$in terms of$\theta$. (c) Find the maximum value of$d'(t)$. 3-e-11 e. Related rates 0.480 0.655 6.730 10 0.895 0.415 A rectangle has vertices$(-x,0), (x,0), (x,y), (-x,y)$, where$y\geq 0$and where$x^2+y^2=1$. Suppose that$x$is changing with the time$t$in the way$x(t)=t^2, -1<t<1$. (a) Find the rate of change of$y(t)$with respect to$t$. (b) Find the rate of change of the area of the rectangle with respect to$t$. 主題分類表：  Linear approximations: 4-a-1 到 4-a-8 The Mean Value Theorem: 4-a-3(c), 4-a-6(b), 4-c-1, 4-c-2, 4-c-3, 4-c-4(b), 4-c-8 Sketching a curve: 4-d全部 (4-d-11 只有 (b) 小題) Optimization Problems: 4-b-1, 4-e全部 編號 副主題 鑑別度 難易度 平均 總分 高分群答對率 低分群答對率 題目 4-a-1 a. Linear approximations 0.394 0.767 7.989 10 0.964 0.570 Let$f(x)=\frac{1+\cos x}{1+\sin x}$. Use a differential to estimate$f(44^{\circ})$. 4-a-2 a. Linear approximations 0.452 0.704 6.767 9 0.930 0.478 Find the linear approximation of the function $g(x)=\sin^{-1}\left(\frac{x-1}{x+1}\right)-\tan^{-1}\left(\sqrt{x}\right)\mbox{ at the point }x=3$ 4-a-3 a. Linear approximations 0.297 0.514 6.640 12 0.662 0.365 (a) Find the linearization of$f(x)=\sin^{-1} x$at$x=0.5$. Denote the linearization by$L(x)$. (b) Use linear approximation to estimate$\sin^{-1}(0.49)$. (c) Let$g(x)=\sin^{-1}x-L(x)$. Use the Mean Value Theorem twice to estimate$|g(0.49)-g(0.5)|$and get an upper bound for the quantity. 4-a-4 a. Linear approximations 0.508 0.517 5.980 12 0.771 0.264 Suppose that$f(x)$is twice differentiable. Let$L(x)$be the linearization of$f(x)$at$x=a$. Define$g(x)=\begin{cases} \frac{f(x)-L(x)}{x-a}, &\text{for }x\ne a.\\ 0, &\text{for }x=a. \end{cases}$(a) Show that$g(x)$is continuous at$x=a$. (b) Show that$g(x)$is differentiable at$x=a$and compute$g'(a)$. Write down the linearization,$L_g(x)$, of$g(x)$at$x=a$. (c) Because$f(x)=L(x)+g(x)(x-a)$, we can use$L(x)+L_g(x)(x-a)$as a better approximation of$f(x)$. Use this better approximation to estimate$\tan^{-1}(1.01)$. 4-a-5 a. Linear approximations 0.713 0.600 6.490 10 0.957 0.243 (a) Find the linear approximation of$\tan^{-1}x$at the point$p$. (b) Use (a) to approximate$\tan^{-1}\frac{3}{5}$with$p=\tan\left(\frac{\pi}{6}\right)$. 4-a-6 a. Linear approximations 0.397 0.450 4.460 10 0.649 0.252 (a) Find the linearization of$f(x) = \sin x$at$\frac{\pi}{6}$. (b) Explain why$f$satisfies the hypotheses of the Mean Value Theorem on$[\frac{\pi}{6} , \frac{\pi}{2}]$. Use the theorem to prove that $\sin x < \frac{1}{2} + \frac{\sqrt{3}}{2}(x - \frac{\pi}{6}) \ \ \text{ for }x\in( \frac{\pi}{6}, \frac{\pi}{2}]$ 4-a-7 a. Linear approximations 0.671 0.496 6.2 12 0.832 0.161 Let$f(x)=\tan^{-1}x+2x$. (a) Show that$f(x)$is one to one. Therefore$f(x)$has inverse function. (b) Find$f^{-1}\left(\frac{\pi}{4}+2\right)$and$\frac{d}{dx}f^{-1}\Big|_{\frac{\pi}{4}+2}$. (c) Write down the linear approximation of$f^{-1}(x)$at$x=\frac{\pi}{4}+2$. Use the linear approximation to estimate$f^{-1}\left(\frac{\pi}{4}+1.95\right)$. 4-a-8 a. Linear approximations 0.550 0.611 10.48 16 0.886 0.337 Consider the function $$f(x)=\begin{cases}x^3\sin\dfrac{1}{x}+2x&\text{ if }x>0,\\ ax+b&\text{ if }x\leq 0.\end{cases}$$ Suppose that$f$is differentiable everywhere. (a) Find the values of$a$and$b$. (b) Write down the linear approximation of$f(x)$, at$x=0$. Use the linear approximation to estimate$f(0.01)$. 4-b-1 b. Fermat's Theorem 0.405 0.279 2.700 12 0.482 0.077 (a) Suppose that$f(x)$and$g(x)$are differentiable on open interval containing$[a,b]$and$f(a)>g(a)$,$f(b)>g(b)$. Show that if the equation$f(x)=g(x)$has exactly one solution on$[a,b]$then at the solution$x_0\in[a,b]$,$f(x)$and$g(x)$have the same tangent line. (Hint: Consider$h(x)=f(x)-g(x)$. Show that$h(x)\geq 0$for all$x\in[a,b]$.) (b) For$\alpha>0$, if the equation$e^x=kx^{\alpha}$has exact one solution on$[0,\infty)$, solve$k$in terms of$\alpha$. 4-b-2 b. Fermat's Theorem 0.303 0.387 3.490 10 0.539 0.236 Suppose that$f$is a differentiable function. If$f'(a)>0$and$f'(b)<0$, explain that there exists$c\in(a,b)$such that$f'(c)=0$. (Note that$f'$may not be continuous.) 4-c-1 c. The Mean Value Theorem 0.518 0.394 4.439 12 0.653 0.135 Let$a<b$. A function$f$in said to be a$contraction$on$[a,b]$if there exists$K$,$0<K<1$, such that for all$x_1,x_2\in [a,b]$we have$\left|f(x_1)-f(x_2)\right|\leq K\left|x_1-x_2\right|$. (a) Show by the$\epsilon-\delta$definition that if$f$is a contraction on$[a,b]$, then$f$is continuous on$[a,b]$. (b) Suppose that$f$is continuous on$[a,b]$and differentiable on$(a,b)$with$\left|f'(x)\right|\leq q$,$0<q<1$, for all$x\in (a,b)$. Show that$f$is a contraction on$[a,b]$and has at most one fixed point on$[a,b]$. (A point$c\in [a,b]$is called a fixed point of$f$if$f(c)=c$.) 4-c-2 c. The Mean Value Theorem 0.627 0.658 7.270 10 0.971 0.344 Show that$|\tan{\frac{x}{2}}-\tan{\frac{y}{2}}|\geq\frac{|x-y|}{2}$for$x,y\in(-\pi,\pi)$. 4-c-3 c. The Mean Value Theorem 0.215 0.291 2.780 10 0.399 0.184 Suppose that the function$f(x)$is continuous on$[a,b]$and differentiable on$(a,b)$, and$0<a <b$. If$f(a)=ka$,$f(b)=kb$for some$k$, show that there exists$c\in(a,b)$such that the tangent line of$y=f(x)$at$c$passes through the origin. [Apply Rolle's Theorem.] 4-c-4 c. The Mean Value Theorem 0.396 0.315 2.83 9 0.513 0.117 Let$f(x)$be a continuous function. It is given that $\lim_{h\to 0}\dfrac{f(h)}{h}=2020.$ (a) Compute$f(0)$. Then, prove that$f$is differentiable at$x=0$and compute$f'(0)$. (b) Suppose in addition that$f$is twice differentiable and that$f''(x)\geq 2$for all$x>0$. Using Mean Value Theorem, or otherwise, prove that $f(x)\geq 2020x+x^2\mbox{ for all } x\geq 0.$ 4-c-5 c. The Mean Value Theorem 0.417 0.484 4.940 10 0.693 0.276$f(x)$is a differentiable function defined on$\mathbb{R}$. Let$g(x)=f(x)\cdot |f(x)|$. (a) Find the domain of$g'(x)$and compute$g'(x)$. (Hint: To compute$g'(x_0)$you may need to discuss the cases$f(x_0)>0$,$f(x_0)<0$, and$f(x_0)=0$seperately.) (b) Suppose that$f'(x)>0$on the interval$(a,b)$. Show that$g(x)$has at most one critical point on$(a,b)$. (c) Suppose that$f'(x)>0$on the interval$(a,b)$. Show that$g(x_1)<g(x_2)$for all$a\leq x_1<x_2\leq b$. 4-d-1 d. Sketching a curve 0.391 0.555 8.951 16 0.751 0.360 Let$f(x)=3\frac{x^{\frac{2}{3}}}{x-1}$. (a) Find all critical points. (b) Find the intervals of increasing and intervals of decreasing. (c) Find the intervals on which$f$is concave up and intervals on which$f$is concave down. (d) Find the points of inflection. (e) Determine whether$f$has any vertical tangent or vertical cusps. (f) Find all vertical or horizontal asymptotes. (g) Draw the graph of$f(x)$. 4-d-2 d. Sketching a curve 0.265 0.816 21.078 25 0.949 0.683 Let$f(x)=\frac{(x+1)^2}{x^2+1}$. (a) Find$f'$and$f''$. (b) Find the intervals on which$f$increases and the intervals on which$f$decreases. Indicate local extreme values and absolute extreme values. (c) Find the intervals on which the graph of$f$is concave up and the intervals on which the graph of$f$is concave down. Indicate points of inflection. (d) Find vertical and horizontal asymptotes if any. Sketch the graph of$f$. 4-d-3 d. Sketching a curve 0.349 0.470 9.157 20 0.644 0.295 Let$f(x)=(x-1)^{\frac{5}{3}}(x^2-1)^{-\frac{1}{3}}$(a) What is the domain of$f(x)$? (b) Does$f(x)$have any vertical or horizontal asymptote? (c) Calculate$\lim_{x\rightarrow\pm\infty}\left(f(x)-x\right)$and find the slant asymptote of$f(x)$. (d) Find the intervals of increase or decrease. (e) Find the intervals of concavity and the inflection points. (f) Find the local maximum and minimum values. (g) Sketch the graph of$f(x)$. 4-d-4 d. Sketching a curve 0.295 0.842 17.930 20 0.990 0.695 Let$y=f(x)=x-\frac{x^2}{6}-\frac{2\ln x}{3}$,$x>0$. Answer the following questions and give your reasons (including computations). Put$\textbf{None}$in the blank if the item asked does$\textbf{not}$exist. (a) Find the interval(s) on which$f$is increasing. (b) Find the local maximal point(s) and minimal point(s) of$f$, if any. (c) Find the interval(s) on which$f$is concave up. (d) Find the inflection point(s) if any. (e) Sketch the graph of$f$. Indicate all information in (a)-(d). 4-d-5 d. Sketching a curve 0.530 0.611 10.200 16 0.876 0.346 Consider the function$f: (1, \infty) \rightarrow \mathbb{R}$,$f(x) = \ln x - \ln (\ln x)$. (a) Compute$f'(x)$. Find the intervals in the domain of$f$on which$f$is increasing and those on which$f$is decreasing. What are the extreme values of$f$? (b) Compute$f''(x)$. Find the intervals on which the graph of$f$is concave upward and those on which the graph of$f$is concave downward. Is there any inflection point of the curve$y=f(x)$? (c) Find the vertical and horizontal asymptotes of the curve$y = f(x)$if any. Find$\lim\limits_{x\rightarrow \infty}\frac{f(x)}{x}$. Does the curve$y=f(x)$have any slant asymptote? (d) Draw the graph of$f(x)$. 4-d-6 d. Sketching a curve 0.365 0.393 3.970 10 0.575 0.211 Find all horizontal, vertical and slant asymptotes, if any, of$f(x)=\frac{[ x^3-\sqrt{x^6}]}{x^2}$where$[ x ]$denotes the greatest integer function. 4-d-7 d. Sketching a curve 0.421 0.528 9.240 18 0.738 0.317 Let$y=3\cos^2 x+\sin x$,$x\in[-\pi,\pi]$. (a) Find the intervals of increase or decrease. (b) Find the intervals of concavity. (c) Find the local maximum and minimum values. (d) Find the global maximum and minimum values. (e) Find the inflection points. (f) Sketch the graph of$y=f(x)$. 4-d-8 d. Sketching a curve 0.442 0.733 18.720 24 0.954 0.512 Let$f(x)=\frac{x^4}{(1+x)^3}$. Answer the following questions by filling each blank below. Show your work (computations and reasoning) in the space following. Put None in the blank if the item asked does not exist. (a) The function is increasing on the interval(s) and decreasing on the The local maximal point(s)$(x,y)=$The local minimal point(s)$(x,y)=$(b) The function is concave upward on the interval(s) and concave downward on the interval(s) The inflection point(s)$(x,y)=$(c) The vertical asymptote line(s) of the function is(are) The horizontal asymptote line(s) is(are) (d) Sketch the graph of the function. Indicate, if any, where it is increasing/decreasing, where it concaves upward/downward, all relative maxima/minima, inflection points and asymptotic line(s) (if any). 4-d-9 d. Sketching a curve 0.432 0.552 11.370 20 0.768 0.336 Let$f(x)=(x^3+x^2)^{1/3}$. (a) Find all asymptotes of$f(x)$. (b) Find the intervals of increase or decrease. (c) Find the intervals of concavity. (d) Find the local maximum and minimum values. (e) Find the inflection points. (f) Sketch the graph of$y=f(x)$. 4-d-10 d. Sketching a curve 0.340 0.796 20.110 24 0.967 0.626 Let$f(x)=\frac{-x^2+3x-1}{x^2+1}$. Answer the following questions by \underline{filling each blank below}. Show your work (computations and reasoning) in the space following. Put {\bf None} in the blank if the item asked does {\bf not} exist. (a) The function is increasing on the interval(s) and decreasing on the interval(s) The local maximal point(s)$(x,y)=$. The local minimal point(s)$(x,y)=$. (b) The function is concave upward on the interval(s) and concave downward on the interval(s). The inflection point(s)$(x,y)=. (c) The vertical asymptotes lines of the function are____. The horizontal asymptotes lines are____. (d) Sketch the graph of the function. Indicate, if any, where it is increasing/decreasing, where it concaves upward/downward, all relative maxima/minima, inflection points and asymptotic line(s) (if any). 4-d-11 d. Sketching a curve 0.306 0.727 8.990 12 0.880 0.574 Student A used some mathematical software to plot a Lapras-like curve (乘龍) as Figure. (a) On Lapras' body, he considered two functions: \begin{align*} f(x)&=-\frac3{10}(x-4)^2-\frac{17}{10}, 0.3<x\leq 5\2mm] g(x)&=\frac{a}{x-4}+bx-\frac{22}5, 5<x<8. \end{align*} Find constants a and b such that the union of f(x) and g(x) is differentiable at x=5. (b) The upper part of Lapras' back is depicted by h(x). Which of the following three functions can be a good candidate for h(x)? Give reasons why other two functions are not good candidates. \[ \frac32\left(\cos^{-1}x-\frac\pi2\right),\qquad\qquad -2x^{\frac13},\qquad\qquad \sinh x. 4-d-12 d. Sketching a curve 0.339 0.782 11.340 14 0.952 0.612 Consider the function $f(x)=\frac{x^3}{(x+1)^2}.$ Answer the following questions by filling each blank below. Show your work (computations and reasoning) in the space following. Put {\bf None} in the blank if the item asked does {\it not} exist. (a) The horizontal asymptote off(x)$is: The vertical asymptote of$f(x)$is: The slant asymptote of$f(x)$is: (b)$f(x)$is increasing on the interval(s)$f(x)$is decreasing on the interval(s) Local maximum point(s) of$f(x)$: Local minimum point(s) of$f(x)$: (c)$f(x)$is concave upward on the interval(s)$f(x)$is concave downward on the interval(s) The inflection point(s)$(x,y)=$. (d) Sketch the graph of$y=f(x)$. Indicate, if any, asymptotes, intervals of increase or decrease, concavity, local extreme values, and points of inflection. 4-d-13 d. Sketching a curve 0.258 0.803 13.980 17 0.932 0.674 Let$h(x)=x^{1/3}(x-4)$. Then$h'(x)=\frac{4(x-1)}{3x^{2/3}}$and$h''(x)=\frac{4(x+2)}{9x^{5/3}}$. Answer the following questions by \underline{filling each blank below}. Show your work (computations and reasoning) in the space following. Put {\bf None} in the blank if the item asked does {\bf not} exist, each blank is worth 2 pts. (a) The function is increasing on the interval(s) and decreasing on the interval(s) The local maximal point(s)$(x,y)=$and The local minimal point(s)$(x,y)=$. (b) The function is concave upward on the interval(s) and concave downward on the interval(s) . The inflection point(s)$(x,y)=$. (c) Sketch the graph of the function. Indicate, if any, where it is increasing/decreasing, where it concaves upward/downward, all relative maxima/minima, inflection points and asymptotic line(s) (if any). 4-d-14 d. Sketching a curve 0.500 0.642 11.880 18 0.892 0.391 Let$f(x)=\frac{\ln |x|}{x}$,$x\neq0$. Answer the following questions by filling each blank below and give your reasons (including computations). Put {\bf None} in the blank if the item asked does {\it not} exist. (a) Find all asymptote(s) of the curve$y=f(x)$. Vertical asymptote(s), Horizontal asymptote(s), Slant saymptote(s). (b)$f(x)$is increasing on the interval(s),$f(x)$is decreasing on the interval(s) (c) Find all local extreme values of$f(x)$. Local maximum point(s), Local minimum point(s). (d)$f(x)$is concave upward on the interval(s),$f(x)$is concave downward on the interval(s) (e) List the inflection point(s) of the curve$y=f(x)$. (f) Sketch the graph of$f$, and indicate all asymptotes, extreme values, and inflection points. 4-d-15 d. Sketching a curve 0.289 0.743 9.95 13 0.887 0.598 Suppose that the \underline{derivative} of the function$f$is given, $\frac{d}{dx}f(x) = \sqrt{x(9+x)(9-x)}.$ (a) Find the critical numbers of$f$. (b) Find the intervals on which$f$is decreasing. (c) Find the intervals on which$f$is concave upward. (d) Sketch the curve$y=f(x)$assuming$f(0)=0$. The sketch just needs to capture increase/decrease and concavity of the function. 4-d-16 d. Sketching a curve 0.437 0.716 13.48 18 0.934 0.497$f(x)=\frac{3x^2-x+2}{x-2}$. (a) Compute$f'(x)$and find interval(s) of increase of$f(x)$and interval(s) of decrease of$f(x)$. Find local extreme values of$f(x)$. (b) Compute$f''(x)$and find concavity and inflection points of$y=f(x)$. (c) Find all vertical, horizontal and slant asymptotes of the curve$y=f(x)$. (d) Sketch the graph of$f(x)$. 4-e-1 e. Optimization problems 0.409 0.702 8.748 12 0.906 0.497 Suppose$ABC$is a triangle with vertices$A=(-5,0)$,$B=(0,10)$and$C=(5,0)$. Let$P$be a point on the line segment that join$B$to the origin. Find the position of$P$that minimizes the sum of distances between$P$and the three vertices of the triangle$ABC$. 4-e-2 e. Optimization problems 0.417 0.588 6.125 10 0.797 0.380 Consider all the rectangles with base on the line$y=-2$and with two upper vertices on the ellipse$x^2+y^2/4=1$and symmetric with respect to the y-axis. Find the maximal possible area for such a rectangle. 4-e-3 e. Optimization problems 0.428 0.497 7.299 15 0.711 0.283 Two vertical poles$PQ$and$ST$are secured by a rope$PRS$as shown in the picture. Given that$\overline{PQ}=1$m,$\overline{ST}=3$m and$\overline{QT}=2$m, we want to find the position of$R$such that (a) the length of the rope$PRS$is maximized. (b) the angle$\theta=\angle PRS$is maximized. 4-e-4 e. Optimization problems 0.512 0.624 7.900 12 0.880 0.368 Suppose that it costs a manufacturer$C(x)$dollars to produces$x$units of products, and$C(x)$is differentiable for$x>0$. When$x$units of products are produced, we call$C'(x)$the "marginal cost" and$\frac{C(x)}{x}$the "average cost". (a) Write down the profit function,$P(x)$, when the manufacturer produces$x$units and sells them at a fixed price$p_0$per unit. Show that when the profit obtains its maximum value at$x_1>0$, the marginal cost equals$p_0$. (b) Assume that the average cost obtains its minimum value at$x_2>0$. Show that at$x=x_2$, the marginal cost equals the average cost. (c) Suppose that the second derivative of the average cost is positive near$x_2$. Is the average cost greater than the marginal cost when$x$is near$x_2$and$x<x_2$? What if$x>x_2$? 4-e-5 e. Optimization problems 0.699 0.599 6.270 10 0.949 0.250 Consider an isosceles triangle whose legs (the equal sides) have length$\ell$and whose vertex angle is$\theta$. As$\ell$and$\theta$vary, the area of the triangle stays the same. At which$\theta$does$\ell$attain its extreme value? Is this extreme value the maximum length or minimum length? 4-e-6 e. Optimization problems 0.332 0.405 3.860 10 0.571 0.239 A woman at a point$A$on the shore of a circular lake with radius 3 km wants to arrive at the point$C$diametrically opposite$A$on the other side of the lake in the shortest possible time. She can walk at the rate of 6 km/h and row a boat at$v$km/h. How should she proceed? [Discuss the cases according to$v$.] 4-e-7 e. Optimization problems 0.563 0.593 6.920 12 0.875 0.312 Choose the point$P$on the line segment$AB$so as (a) to maximize the angle$\theta$; (b) to minimize the angle$\theta$. 4-e-8 e. Optimization problems 0.358 0.451 5.450 12 0.631 0.272 A cone-shaped paper drinking cup is to be made to hold 9 cm$^3$of water. Find the height and radius of the cup that will use the smallest amount of paper. 4-e-9 e. Optimization problems 0.495 0.638 9.220 14 0.885 0.390 A circular cone frustum-shaped lampcover (正圓錐台形狀的燈罩) is made from an annulus piece of paper by cutting out some part of it and joining the edges$\overline{AB}$and$\overline{CD}$as Figure. Find the maximum enclosed volume of such a lampcover. Remark that the volume of a circular cone frustum is$V=\frac{\pi h}{3}(r_1^2+r_1 r_2+r_2^2)$, where$h$is the height of the frustum, and$r_1$,$r_2$are the radii of the two bases. 4-e-10 e. Optimization problems 0.368 0.740 9.260 12 0.924 0.556 An object at rest with mass$m$is dragged along a horizontal plane by a force acting along a rope attached to the object so that the object remains at rest as Figure. If the rope makes an angle$\theta$with a plane, where$0\leq\theta\leq\frac\pi2$, then the magnitude of the force$Fwill satisfy the equation \begin{align*} \mu(mg-F\sin\theta)=F\cos\theta, \end{align*} where\mu$is a positive constant called the coefficient of static friction and$g$is the gravitational constant. For what value of$\theta$is$F$smallest? 4-e-11 e. Optimization problems 0.677 0.562 8.540 15 0.900 0.224 A steel pipe is carried down a hallway 16 meter wide. At the end of the hall there is a right angled turn into a narrower hallway 2 meter wide. What is the length of the longest pipe that can be carried horizontally around the corner? 4-e-12 e. Optimization problems 0.724 0.521 5.450 10 0.883 0.159 A right circular cone is inscribed in a larger right circular cone so that its vertex is at the center of the base of the larger one. Denote the height of the large cone by$H$and the height of the small one by$h$. When the large cone is fixed, find$h$that maximizes the volume of the small cone and find out this maximum volume in terms of the volume of the large cone. (Hint: The volume of a right circular cone with height$h$and base radius$r$is$\frac{1}{3}\pi r^2 h$.) 圖 1 4-e-13 e. Optimization problems 0.534 0.498 6.91 13 0.765 0.231 The minute hand on a clock is 6 cm long and the hour hand is 3 cm long. Let$\theta(t)$be the angle between the minute hand and the hour hand at time$t$. Let$d(t)$be the distance between the tips of the hands at time$t$. (a) Find$\left|\frac{d\theta}{dt}\right|$, ($\text{rad}/\text{hour}$). (b) Find$d'(t)$in terms of$\theta$. (c) Find the maximum value of$d'(t)$. 4-e-14 e. Optimization problems 0.465 0.582 7.62 13 0.815 0.350 Three trainers P, Q, R have spotted Snorlax(卡比獸) in a rectangular field (See figure below). Trainer P is standing in the middle of the edge$\overline{AB}$whereas Trainers Q and R are at the two corners of the field opposite to$\overline{AB}$. Snorlax is currently at position S which is at a distance of$x$meters in front of trainer P. (a) Let$L$be the total distances of the three trainers from Snorlax. Prove that $\dfrac{dL}{dx}=1-\dfrac{2(4-x)}{\sqrt{(4-x)^2+1}}.$ (b) Find the values of$x$at which$L$attains its greatest and least value respectively. (c) Snorlax is now asleep. To wake Snorlax up, each trainer is going to play a magical flute. The intensity of sound energy received by Snorlax from each flute varies inversely with the square of the distance from the flute. In other words, if$E$is the total sound energy received by Snorlax from the three flutes, then $E=\dfrac{k}{\overline{PS}^2}+\dfrac{k}{\overline{QS}^2}+\dfrac{k}{\overline{RS}^2}\quad\mbox{ for some constant } k>0.$ Trainer P claims that when$L$attains the least value,$E$will attain its greatest value. Do you agree with trainer P? Explain your answer. 4-e-15 e. Optimization problems 0.631 0.597 12.73 20 0.912 0.281 A firm finds that the total cost$C(x)$(in dollars) of manufacturing$x$tennis rackets$/$day is given by$C(x)=400+4x+0.0001x^2$. Each racket can be sold at a price of$p$dollars related to$x$by the equation$p(x)=12-0.0004x$. (a) Find the daily level of production,$x_1$, that minimizes the average cost$\frac{C(x)}{x}$. (You need to check that the value you find is indeed the minimum value.) (b) Show that the average cost,$\frac{C(x)}{x}$, equals the marginal cost$C'(x)$when$x=x_1$. (c) Find the daily level of production,$x_2$, that maximizes the profit$\Pi(x)=x\cdot p(x)-C(x)$. (You need to check that the value you find is indeed the maximum value.) (d) Find the inverse function of$p(x)=12-0.0004x$which is denoted by$x=F(p)$. Find the point elasticity$\epsilon=\frac{F'(p)\cdot p}{F(p)}$. In the interval$p\in(0,12)$, find values of$p$such that$-1<\epsilon<0$(inelastic) and values of$p$such that$\epsilon<-1$(elastic). 主題分類表：  Riemann Sums: 5-a-1, 5-a-2, 5-a-3, 5-a-4, 5-a-5(a) The Fundamental Theorem of Calculus: 5-a-5(b), 5-b全部 編號 副主題 鑑別度 難易度 平均 總分 高分群答對率 低分群答對率 題目 5-a-1 a. Riemann Sums 0.691 0.504 6.018 12 0.850 0.159 Express the following limit as a definite integral of certain function and then evaluate the integral: $\lim\limits_{n\rightarrow\infty}\frac{1}{\sqrt{n}}\sum_{k=1}^{n} \frac{1}{\sqrt{n+\sqrt{nk}}}.$ 5-a-2 a. Riemann Sums 0.541 0.590 9.116 15 0.860 0.320 (a) Evaluate$\displaystyle\int\ln x~dx$. (b) Show that the function$f(x)=\ln x$is increasing in$x>0$. (c) Consider the definite integrals of$f(x)=\ln x$on$[1,n]$and$[1,n+1]$. By comparing the upper sum and the lower sum for$f(x)=\ln x$with regular partition of length$\triangle x=1$, derive the inequalities $\int_1^n\ln x~dx<\ln 1+\ln 2+\cdots+\ln n<\int_1^{n+1}\ln x~dx.$ Sketch a graph if necessary. (d) Prove that$\left(\frac{n}{e}\right)^n<\frac{n!}{e}<\left(\frac{n+1}{e}\right)^{n+1}$. 5-a-3 a. Riemann Sums 0.657 0.432 3.790 10 0.761 0.104 (a) Evaluate the limit$\begin{align*} I_m&=\lim_{n\rightarrow\infty}\sum_{i=1}^{nm}\frac{i^2 n^3}{n^6+i^6}\\ &=\lim_{n\rightarrow\infty}\left(\frac{n^3}{n^6+1^6} +\cdots+\frac{(nm)^2n^3}{n^6+(nm)^6}\right)\end{align*}$where$m$is a positive integer. (b) Compute$\lim_{m\rightarrow\infty}I_m$. 5-a-4 a. Riemann Sums 0.549 0.363 5.220 16 0.638 0.089 Let$M_n=\sum\limits_{i=1}^{n}\frac{1}{n+\sqrt{n(i-\frac{1}{2})}}$. (a) Recognize$M_n$as the Midpoint approximation of a definite integral and compute$I=\lim\limits_{n\rightarrow\infty}M_n$. (b) Write down the Trapezoidal approximation of the integral,$T_n$, and the right point approximation,$R_n$. (c) For any value of$n$, list$M_n$,$T_n$,$R_n$, and$I$in increasing order. 5-a-5 a. Riemann Sums 0.674 0.512 6.140 12 0.849 0.174 Find the limits. (a)$\lim\limits_{n\rightarrow +\infty} \left( \frac{n}{n^2 + 4\cdot 1^2} + \frac{n}{n^2 +4\cdot 2^2} + \frac{n}{n^2 +4\cdot 3^2} + \ldots + \frac{n}{5n^2} \right)=\lim\limits_{n\rightarrow +\infty}\sum\limits_{i=1}^{n}\frac{n}{n^2+4i^2}$. (b)$\lim\limits_{h \rightarrow 0} \frac{1}{h} \int_{1-h}^{\sqrt{1+h}} \sqrt{1+t^3} dt$. 5-b-1 b. The fundamental theorem of Calculus 0.312 0.328 4.042 12 0.484 0.172 Suppose that$f(x)$is continuous and increasing on$[-1,2]$with$f(x)>0$. Let$F(x)=\int_0^x\left(t\int_1^tf(u)du\right)dt$. (a) Classify all critical points of$F(x)$in$(-1,2)$. (b) Show that$F''(x)$is increasing on$(0,1)$and there is a point of inflection of$F(x)$on$(\frac{1}{2},1)$. 5-b-2 b. The fundamental theorem of Calculus 0.496 0.528 6.775 12 0.776 0.280 From the equation$\sqrt{1+y}-\int_0^{x^2-1}\frac{dt}{1+t^2}+\tan(xy)=1$, a differentiable function$y=y(x)$can be determined around$(x,y)=(1,0)$. (a) Evaluate$y'$at$(x,y)=(1,0)$. (b) Evaluate$y''$at$(x,y)=(1,0)$and determine the concavity of$y=y(x)$around this point. 5-b-3 b. The fundamental theorem of Calculus 0.499 0.546 5.484 10 0.796 0.297 Find$f'(2)$given that$f(x)=\int_{2x}^{x^3-4}\frac{x}{1+\sqrt{t}}dt$. 5-b-4 b. The fundamental theorem of Calculus 0.446 0.229 1.545 10 0.451 0.006 Let$A=\int_{0}^{\infty} e^{-x^2} dx$. Compute the limit $\lim_{x\to \infty} xe^{x^2}\left(A-\int_{0}^{x} e^{-t^2}\; dt\right).$ 5-b-5 b. The fundamental theorem of Calculus 0.465 0.689 7.404 10 0.921 0.457 Evaluate$\lim_{x\rightarrow 0^+}\frac{\displaystyle \int_0^{x^2}e^t\sqrt{t}\sin\sqrt{t}dt+x\cos x-x}{\displaystyle x^3}$. 5-b-6 b. The fundamental theorem of Calculus 0.589 0.606 7.390 12 0.901 0.311 Let$R$be the region bounded by the curves$y=e^{x^2}$,$y=0$,$x=\sqrt{a}$, and$x=\sqrt{a+1}$, where$a>0$. Find the number$a$such that the area of$R$attains the minimum value. 5-b-7 b. The fundamental theorem of Calculus 0.650 0.478 3.940 8 0.803 0.153 Evaluate$\lim\limits_{x\rightarrow 0}\frac{\int_{x}^{\tan x}\sqrt{1+t^3}dt}{x^3}$. (You may use the Mean Value Theorem for Integrals.) 5-b-8 b. The fundamental theorem of Calculus 0.695 0.519 4.130 8 0.867 0.172 Let$F(x)=\int_0^{x}\left(\int_0^{u^3}f(t)dt\right)du$and$G(x)=\int_0^{x^3}f(u)(x-\sqrt{u})du$,$x\geq 0$. Show that$F(x)=G(x)$for$x\geq 0$. 5-b-9 b. The fundamental theorem of Calculus 0.716 0.490 4.760 10 0.849 0.132 Evaluate$\lim\limits_{x\rightarrow 0} \frac{1}{x}\int_0^x (1-\tan t)^{1/t}dt$. 5-b-10 b. The fundamental theorem of Calculus 0.460 0.327 2.490 10 0.557 0.096 Let$f(x)$be a differentiable and increasing function on$[a,b]$, where$a>0$. Find a horizontal line$y=L$that will minimize the function$\begin{align*} F(L)=\int_a^bx|f(x)-L|\,\mathrm{d}x \\ =\int_a^{f^{-1}(L)}x(L-f(x))\,\mathrm{d}x+\int_{f^{-1}(L)}^bx(f(x)-L)\,\mathrm{d}x\end{align*}$. 5-b-11 b. The fundamental theorem of Calculus 0.523 0.702 4.49 6 0.963 0.440 Find$f'\left(\frac{\pi}{4}\right)$if$f(x)=e^{g(x)}$and$g(x)=\int_1^{\tan x} \sqrt{1+t^3}\,\mathrm{d} t$. 5-b-12 b. The fundamental theorem of Calculus 0.622 0.521 3.16 6 0.832 0.210 Find$f'(2)$if$f(x)=e^{g(x)}$and$g(x)=\int_4^{x^2} \frac{t}{1+t^4}\,\mathrm{d} t$. 主題分類表：  Techniques of Integration : 6-a 全 Improper Integrals : 6-b全部 編號 副主題 鑑別度 難易度 平均 總分 高分群答對率 低分群答對率 題目 6-a-1 a. Techniques of Integration 0.358 0.787 8.369 10 0.966 0.608 Calculate$\int\frac{\csc^22x}{\sqrt{2+\cot2x}}dx$. 6-a-2 a. Techniques of Integration 0.397 0.727 11.572 15 0.926 0.529 Evaluate the integral. (a)$\int_0^1x^2\sqrt{1-x}dx$. (b)$\int x\sqrt{3-2x-x^2}dx$. 6-a-3 a. Techniques of Integration 0.363 0.671 10.830 16 0.852 0.489 Compute the following integrals. (a)$\int \frac{\cos\theta-2\sin\theta}{2\cos\theta+\sin\theta}d\theta$. (b)$\int\frac{\sqrt{1+x^2}}{x^2}dx$. 6-a-4 a. Techniques of Integration 0.440 0.715 9.280 12 0.935 0.495 Evaluate the following integrals. (a)$\displaystyle\int_0^3 \frac{x^2}{\sqrt{x+1}} dx $. (b)$\displaystyle\int\frac{3x^3-2x-2}{x^2(x^2+1)}dx$. 6-a-5 a. Techniques of Integration 0.339 0.789 9.57 12 0.958 0.620 (a) Evaluate the integral$\int \frac{3x^2-x-5}{(x-2)(x^2+1)}\,\mathrm{d} x$. (b) Using the substitution$u=\dfrac{1}{x}$, show that for any$a>1$,$\displaystyle\int_{1/a}^a\dfrac{\ln x}{1+x+x^2}\,\mathrm{d} x=0.$6-a-6 a. Techniques of Integration 0.525 0.529 5.210 10 0.791 0.266 Evaluate (a)$\int_0^1 x\tan^{-1}(x^2)~dx$and (b)$\int_0^1 x(\tan^{-1}x)^2~dx$. 6-a-7 a. Techniques of Integration 0.362 0.794 12.790 15 0.975 0.613 Evaluate the integrals: (a)$\int\tan^{\frac{1}{3}}x\sec^4x \,dx$(b)$\int\frac{4}{x(x^2+2x+2)}\,dx$6-a-8 a. Techniques of Integration 0.586 0.654 6.380 10 0.947 0.361 Find$\int\frac{1}{\sec x-1}dx$. 6-a-9 a. Techniques of Integration 0.347 0.795 8.290 10 0.968 0.621 Evaluate the integrals. (a)$\int \tan x\ln(\cos x)\:dx$. (b)$\int \frac{\sin x -1}{\sin x \cos x }\: dx$. 6-a-10 a. Techniques of Integration 0.335 0.719 12.200 16 0.887 0.552 Three of these six antiderivatives are elementary. Compute them. (a)$\int x\cos xdx$(b)$\int\frac{\cos x}{x}dx$(c)$\int\frac{x}{\ln x}dx$(d)$\int\frac{\ln (x^2)}{x}dx$(e)$\int\sqrt{x-1}\sqrt{x}\sqrt{x+1}dx$(f)$\int\sqrt{x-1}\sqrt{x+1}xdx$6-a-11 a. Techniques of Integration 0.513 0.498 8.290 16 0.755 0.242 Compute the following integrals. (a)$\int \sqrt{1-x^2}\sin^{-1}xdx$. (b)$\int\ln(\sqrt{x}+\sqrt{1+x})dx$. 6-a-12 a. Techniques of Integration 0.517 0.656 8.300 12 0.914 0.398 Evaluate the integrals. (a)$\int x\sqrt{8+2x-x^2}\mathrm{d}x$(b)$\int\frac{x^2-1}{(x^2+2x+2)^2}\mathrm{d}x$6-a-13 a. Techniques of Integration 0.553 0.676 10.18 14 0.953 0.400 Compute the following integrals. (a)$\int (\cos x + \sec^2 x) \ln (\tan x)\,\mathrm{d} x$. (b)$\int x\sqrt{2x-x^2}\,\mathrm{d} x$. 6-a-14 a. Techniques of Integration 0.608 0.514 8.167 15 0.818 0.210 Calculate$\displaystyle\int \frac{x^4+2x^3+1}{(x-1)(x^2+1)^2} dx$. 6-a-15 a. Techniques of Integration 0.743 0.605 5.651 9 0.976 0.233 Evaluate$\int_0^1\frac{(2+x)^2}{1+x^2}dx$. 6-a-16 a. Techniques of Integration 0.361 0.785 12.433 15 0.966 0.605 Let$P(x)=x^3+x^2+x+1$and$Q(x)=x^3-x^2+x+1$. Evaluate$\int \frac{Q(x)}{P(x)}dx$. Note that$P(-1)=0$. 6-a-17 a. Techniques of Integration 0.544 0.544 9.150 16 0.816 0.272 (a) Evaluate the integral$\int\frac{dx}{x\sqrt{x^6-1}}$. (b) Evaluate the integral$\int\frac{(e^{3x}+1)}{(e^{2x}+1)^2}dx$. 6-a-18 a. Techniques of Integration 0.484 0.722 7.420 10 0.964 0.480 Find$\int\frac{1}{x^2(x^2+x+1)}dx$. 6-a-19 a. Techniques of Integration 0.328 0.804 10.000 12 0.968 0.640 Find the following indefinite integrals: (a)$\int\,\frac{3 t^2 + t +4}{t^3 + t}\,dt$. (b)$\int\,\cos\sqrt{x}\,dx$6-a-20 a. Techniques of Integration 0.414 0.748 9.300 12 0.955 0.541 Evaluate the integrals. (a)$\int\frac{1}{\sin x\cos^2x}\mathrm{d}x$. (b)$\int\tan^{-1}(\sqrt{x})\mathrm{d}x$. 6-a-21 a. Techniques of Integration 0.315 0.793 13.250 16 0.951 0.636 Evaluate the integrals. (a)$\int \frac{x\: dx}{\sqrt{25 -8x +x^2}}$. (b)$\int \frac{e^{2x}}{16-8e^x+e^{2x}}\: dx$. 6-a-22 a. Techniques of Integration 0.690 0.604 18.55 28 0.949 0.259 Compute the following integrals. (a)$\int_0^1 \sin^{-1}(x)\,\mathrm{d} x$(b)$\int \sqrt{1+x^2}\,\mathrm{d} x$(c)$\int \frac{x^3 + 4x^2 + 4x + 2}{x^4 + 2x^3 + 2x^2}\,\mathrm{d} x ~.$6-b-1 b. Improper Integrals 0.431 0.241 3.032 15 0.456 0.025 The gamma function$\Gamma(x)$is defined by the improper integral $\Gamma(x)=\int_0^{\infty}t^{x-1}e^{-t}~dt.$ (a) Show that the integral converges for$x\geq 1$. (b) Show that the integral also converges for$0<x<1$. (c) Show that$\Gamma(x+1)=x\Gamma(x)$for$x>0$. 6-b-2 b. Improper Integrals 0.679 0.394 3.285 10 0.733 0.054 Evaluate the improper integral$\int_0^2 \frac{1}{x}\sqrt{x(2-x)}~dx$. 6-b-3 b. Improper Integrals 0.514 0.623 13.423 20 0.880 0.366 Evaluate the integral. (a)$\int\frac{\sin x\cos x}{\sin^4x+\cos^4x}dx$. (b)$\int_1^{\infty}\frac{x^2-3}{(x^2-2x+3)(x^2+2x+3)}dx$. 6-b-4 b. Improper Integrals 0.562 0.491 4.734 10 0.772 0.210 Determine the values of$\alpha>0$such that$\int^{\infty}_{1}\frac{\ln x}{x^{\alpha}}dx$is convergent. 6-b-5 b. Improper Integrals 0.601 0.525 8.120 15 0.825 0.225 (a) Determine whether$\int_1^{\infty}\frac{\tan^{-1}x}{x^2}dx$converges or diverges. Evaluate the value if it converges. (b) Determine whether$\int_{-1}^{1}\frac{\tan^{-1}x}{x^2}dx$converges or diverges. Evaluate the value if it converges. 6-b-6 b. Improper Integrals 0.400 0.319 2.490 8 0.519 0.119 Find the values of$s$such that$F(s)=\int_0^{\infty}\sin t\ e^{-st}dt$converges and evaluate the integrals. 6-b-7 b. Improper Integrals 0.665 0.589 6.320 10 0.921 0.256 Evaluate the improper integral$\int_0^{\infty}x^3e^{-x^2}dx$. 6-b-8 b. Improper Integrals 0.331 0.406 3.290 8 0.572 0.241 Suppose that$f(x)$is a polynomial whose coefficients are integers, and $\int_0^{\infty}\frac{f(x)}{(x+1)^2(4x^2+1)}dx=2\ln 2+1.$ Find$f(x)$. 6-b-9 b. Improper Integrals 0.472 0.629 8.990 14 0.865 0.393 (a) Find the indefinite integrals$\int\frac{1}{\sqrt{x^2+1}}dx$and$\int\frac{1}{x+2}dx$. (You may use the integral formula of$\int\sec\theta\ d\theta$.) (b) Find the value of the constant$a$for which the improper integral $\int_0^\infty \left(\frac{1}{\sqrt{x^2+1}}-\frac{a}{x+2}\right)dx\ \ \text{converges}.$ (c) Evaluate the improper integral for this$a$. 6-b-10 b. Improper Integrals 0.399 0.365 3.470 10 0.565 0.166 Find the reduction formula$I_n=\int(\ln x)^n\mathrm{d}x$, where$n$is a positive number.(i.e. write$I_n$in terms of$I_{n-1}$.) Evaluate the improper integral$\int_0^1(\ln x)^n\mathrm{d}x$or explain why it is divergent. 6-b-11 b. Improper Integrals 0.267 0.195 2.000 12 0.328 0.062 Find the values of$a$and$b$for which the improper integral $\int_0^{\infty}\frac{e^{-ax}}{x^b(1+x^2)}\mathrm{d}x=\int_0^1\frac{e^{-ax}}{x^b(1+x^2)}\mathrm{d}x +\int_1^{\infty}\frac{e^{-ax}}{x^b(1+x^2)}\mathrm{d}x$ converges. Hint: Discuss cases$a=0$and$a\ne 0$, respectively. 6-b-12 b. Improper Integrals 0.418 0.695 7.280 10 0.904 0.486 Evaluate the following improper integrals. (a)$\displaystyle\int_e^{\infty} \frac{1}{x (\ln x)^2} dx $. (b)$\displaystyle\int_0^{1} \frac{1}{x+\sqrt{x}} dx $. 6-b-13 b. Improper Integrals 0.441 0.365 3.350 10 0.586 0.145 Find the value of the constant$c$for which the integral $\int_0^{\infty}\frac{x^2+8}{x^3+8}-\frac{c}{\sqrt{x^2+1}}\: dx\ \text{converges.}$ Evaluate the integral for this value of$c$. 6-b-14 b. Improper Integrals 0.477 0.690 7.360 10 0.928 0.451 (a) Determine the values of the constant$t$such that$\int_1^e \frac{1}{(\ln x)^t x}dx$is convergent. Evaluate the integral for such values of$t$. (b) Determine the values of the constant$t$such that$\int_e^{\infty} \frac{1}{(\ln x)^t x}dx$is convergent. Evaluate the integral for such values of$t$. 6-b-15 b. Improper Integrals 0.424 0.466 8.600 18 0.678 0.254 Let$R$be the region bounded above by the curve$ y = \tan^2 x $, left by$x = 0$, below by$ y=0 $, and right by$ x = \pi / 4 $. Let$\tilde{R}$be the region bounded above by the curve$ y = \tan^p x $, left by$x = 0$, below by$ y=0 $, and right by$ x = \pi / 2 $, where$ p >0 $is a constant. (a) Rotate$R$about the$x$-axis. Find the resulting volume. (b) Rotate$R$about the$y$-axis. Find the resulting volume. (c) Rotate$\tilde{R}$about the$x$-axis. Find the values of$p$such that the resulting volume is finite. (Hint: You may use the inequality$(\frac{\pi}{2}-x)\cdot \tan x<2$, for$\frac{\pi}{4}\leq x<\frac{\pi}{2}$.) 6-b-16 b. Improper Integrals 0.366 0.745 9.57 12 0.928 0.561 Determine whether the improper integral converges. If it converges, compute the value. (a)$\int_0^1\dfrac{\mathrm{d} t}{\sqrt{t}(1+t)}$. (b)$\int_0^{\infty}\dfrac{\mathrm{d} t}{\sqrt{t}(1+t)}. 6-b-17 b. Improper Integrals 0.659 0.537 5.85 10 0.866 0.208 Determine whether the following improper integral is convergent or divergent. Evaluate it if it is convergent. \begin{align*} \int_0^{\infty} e^{-2x}\cos x\,\mathrm{d} x ~. \end{align*} 主題分類表：  Area between curves: 7-a-1, 7-a-2, 7-b-1(a), 7-e-1(a), 7-e-3(a) Volumes: 7-b-1(b), 7-b-2~7-b-7, 7-d-3(a), 7-e-2(a)(b), 7-e-3(b), 7-e-4(a)(b) Arc length: 7-c-1, 7-c-2, 7-c-3(a), 7-d-2(a), 7-d-5(a), 7-e-1(b) Area of surface of revolution: 7-c-3(b), 7-d-1(a), 7-d-2(b), 7-d-3(b), 7-d-4, 7-d-5(b), 7-e-1(c) Centroid: 7-e-1(d), 7-e-2(c)(d), 7-e-3(c), 7-e-4(c), 7-e-5 編號 副主題 鑑別度 難易度 平均 總分 高分群答對率 低分群答對率 題目 7-a-1 a. Area between curves 0.589 0.606 7.390 12 0.901 0.311 LetR$be the region bounded by the curves$y=e^{x^2}$,$y=0$,$x=\sqrt{a}$, and$x=\sqrt{a+1}$, where$a>0$. Find the number$a$such that the area of$R$attains the minimum value. 7-a-2 a. Area between curves 0.575 0.483 5.82 12 0.770 0.196 Given an increasing supply function$S(q)$and a decreasing demand function$D(q)$where$S(q)$and$D(q)$are continuous, we define the total surplus at quantity$q$as$TS(q)=\int_0^q D(t)-S(t)\,\mathrm{d} t$for$q\geq 0$. (a) Show that if$D(q^*)=S(q^*)$for some$q^*>0$then$TS(q)$obtains the absolute maximum value at$q=q^*$. (b) Suppose that$D(q)=6-\left(1+\frac{q}{2}\right)^{2/3}$and$S(q)=\left(1+\frac{q}{2}\right)^{1/3}$. Compute$TS(q)$and find the absolute maximum value of$TS(q)$. 7-b-1 b Volumes 0.440 0.722 9.220 12 0.942 0.502 (a) Show that the area of an ellipse with the semi-major axis of length$a$and the semi-minor axis of length$b$is$ab\pi$. See Figure(a). (b) A toothpaste tube is modeled in Figure(b). • One side is flat and is located at$-\pi\leq x\leq\pi, y=0, z=0$. • The other side is a circle with radius$2$, so the equation of the circle is$x^2+y^2=4, z=20$. • Each cross-section for$0<z < 20$is an ellipse with the semi-major axis of length$a$and the semi-minor axis of length$b$, where $a=\pi+(2-\pi)\frac{z}{20}\ \ \text{and }b=\frac{1}{10}z.$ Find the volume of the modeled toothpaste tube with$0 \leq z \leq 20$. 7-b-2 b Volumes 0.443 0.551 5.393 10 0.772 0.329 Consider the region bounded by the curves: $y = \sin(\pi x/2)\mbox{ and } y = 6/(x^2 + 3 x + 2)\mbox{ for }0\leq x\leq 1.$ Note that the two curves meet at$x=1$. Find the volume of revolving the region about (a)the$y$-axis, (b)the$x$-axis. 7-b-3 b Volumes 0.486 0.458 5.490 12 0.701 0.216 Consider the intersection of three circular cylinders with radius$R$and all axes of cylinders lie in a plane with polar equations$\displaystyle\theta=0, \theta=\frac\pi3$, and$\displaystyle\theta=\frac{2\pi}3$. The cross-section is a hexagon, and the shape of the solid looks like the union of two umbrellas. (a) Find the area of the cross section of the solid when the height is$y, y\in[-R,R]$. (b) Find the volume of the solid. 7-b-4 b Volumes 0.370 0.325 2.380 8 0.510 0.140 A sphere of radius 1 overlaps a smaller shpere of radius$r$($0<r<1$) in such a way that their intersection is a circle of radius$r$, i.e. a great circle of the small sphere. Find$r$so that the volume inside the small sphere and outside the large sphere is as large as possible. 7-b-5 b Volumes 0.355 0.724 9.330 12 0.902 0.547 Find the volume of Gulliver's Tunnel (格列佛隧道), which is half of the solids of revolution obtained by rotating the region bounded by$y=\frac1{1+\mathrm{e}^{3x}}$,$y=0$,$x=-\frac{2}{3}\ln 3$, and$x=\ln 2$, about the$x$-axis. 7-b-6 b Volumes 0.424 0.466 8.600 18 0.678 0.254 Let$R$be the region bounded above by the curve$ y = \tan^2 x $, left by$x = 0$, below by$ y=0 $, and right by$ x = \pi / 4 $. Let$\tilde{R}$be the region bounded above by the curve$ y = \tan^p x $, left by$x = 0$, below by$ y=0 $, and right by$ x = \pi / 2 $, where$ p >0 $is a constant. (a) Rotate$R$about the$x$-axis. Find the resulting volume. (b) Rotate$R$about the$y$-axis. Find the resulting volume. (c) Rotate$\tilde{R}$about the$x$-axis. Find the values of$p$such that the resulting volume is finite. (Hint: You may use the inequality$(\frac{\pi}{2}-x)\cdot \tan x<2$, for$\frac{\pi}{4}\leq x<\frac{\pi}{2}$.) 7-b-7 b Volumes 0.629 0.617 8.87 14 0.932 0.303 The Apprentice is a reality show on the BBC in which candidates need to compete in various business-related challenges. This week the candidates are required to create their own perfume. The following perfume is designed by one of them which consists of a stopper and a bottle (See Figure 1). (a) Consider the region bounded by the curve$y=x\cos^{-1}(x)$and the$x$-axis in the interval$0\leq x\leq 1$. The stopper of the perfume is obtained by revolving this region about the$y$-axis. Find the volume of the stopper. (b) The bottle of the perfume is designed such that • the curved edges of the bottle are arcs of a circle of radius$r$(See Figure 2). • each of its cross-section is a regular hexagon (see Figure 3), The side-view of the bottle is given in Figure 4. Suppose the height of the bottle equals to$2h$(with$h<r$). (i) Find the cross-sectional area of the bottle at a height$y$from its centre. (ii) Find the volume of the bottle. 7-b-8 b Volumes 0.561 0.396 3.58 10 0.677 0.116 Consider the crescent-shaped region (called a lune) bounded by arcs of circles with radii$r$and$R$, where$0<r < R$. Rotate the region about the$y$-axis. Find the resulting volume. 7-c-1 c. Arc length 0.473 0.438 6.004 15 0.674 0.202 Let$y=h(x)$be decreasing on$[0,\frac{\pi}{2})$and is continuously differentiable on$(0,\frac{\pi}{2})$with$h(0)=0$. Let$s(x)$denote the arc length of$y=h(x)$from$(0,0)$to$(x,h(x))$. (a) Write down the formula for$s(x)$. (b) Suppose that$s(x)$is also given by$s(x)=\int_0^xe^{-h(t)}dt$. Find the function$h(x)$explicitly. (c) Find the function$s(x)$explicitly. 7-c-2 c. Arc length 0.751 0.480 3.480 8 0.855 0.104 Find the length of the loop of the curve$3ay^2=x(a-x)^2$,$a>0$. 7-c-3 c. Arc length 0.488 0.718 8.930 12 0.962 0.474 (a) Find the length of the curve $y = \int_0^x \sqrt{\cos (2t) }\mbox{ } dt$ from$x = 0$to$x = \pi / 4$. (b) Rotate the curve about the$y$-axis. Find the resulting surface area. 7-d-1 d. Area of surface of revolution 0.484 0.518 5.495 10 0.760 0.276 (a) If the infinite curve$y=e^{-x},\ x\geq0$, is rotated about the$x$-axis, find the area of the resulting surface. (b) Find the arc length of the infinite curve with polar equation$r=\theta^{-1}$,$\theta\geq 1$. 7-d-2 d. Area of surface of revolution 0.477 0.718 11.045 15 0.956 0.479 Let$h(x)=\sqrt{x-x^2}+\sin^{-1}\left(\sqrt{x}\right)$,$0\leq x\leq 1$. (a) Find the length of the curve$y=h(x)$. (b) Find the area of the surface generated by rotating the curve$y=h(x)$about the$x$-axis. 7-d-3 d. Area of surface of revolution 0.540 0.619 10.340 16 0.889 0.349 (a) Let$R$be the region bounded by the curves$y=\frac{\sqrt{4-x^2}}{(1+\sqrt{4-x^2})^2}$,$y=0$,$x=0$, and$x=2$. Find the volume of the solid obtained by rotating$R$about the$y$-axis. (b) Find the area of the infinite surface generated by rotating the curve$y=e^{-x}$,$0\leq x <\infty$, about the$x$-axis if it is finite (show explicitly in your calculation that you are using the definition of improper integrals). 7-d-4 d. Area of surface of revolution 0.564 0.660 7.130 10 0.942 0.378 Compute the area of the surface generated by rotating the curve$y=\ln x$,$0\leq x\leq 1$about the$y$-axis. 7-d-5 d. Area of surface of revolution 0.448 0.755 9.64 12 0.980 0.531 Let$C$be the curve defined by$y=f(x) = \int_e^x \sqrt{(\ln t)^{2}-1}\, \mathrm{d} t$,$e \leq x \leq e^2.$(a) Find the arc length of the curve$C$. (b) Find the area of the surface generated by rotating$C$about the$y$-axis. 7-e-1 e. Centroid and others 0.340 0.531 10.760 20 0.701 0.361 Let$C$be the curve$y=\ln x$,$0< x\leq 1$and$R$be the region bounded by$x$-axis,$y$-axis and the curve$C$. (a) Compute the area of$R$if it is finite. (b) Compute the arc length of$C$if it is finite. (c) Rotate$C$about the$y$-axis. Compute the area of the generated surface if it is finite. (d) Rotate$R$about the line$y=x$. Compute the volume of the generated solid if it is finite. 7-e-2 e. Centroid and others 0.556 0.619 13.310 20 0.897 0.341 Let$R$be the region bounded by the$x$-axis,$x=e$and the curve$y=\ln x$. (a) Find the volume of the solid obtained by revolving$R$about the$x$-axis. (b) Find the volume of the solid obtained by revolving$R$about the$y$-axis. (c) Find the centroid of$R$. (d) Find the volume of the solid obtained by revolving$R$about$x+y=1$. 7-e-3 e. Centroid and others 0.500 0.409 7.684 20 0.659 0.159 The figure shows a curve$C$with the property that, for every point$P$on the middle curve$y=2x^2$, the area of$B$is twice the area of$A$. (a) Find an equation$x=g(y)$for$C$. (b) Let$R$be the region bounded by the curve$C$,$y=x^2$,$x=2$and$y=8$. Find the volume of the solid obtained by rotating$R$about the$x$-axis. (c) Find the$y$-coordinate of the centroid of$R$. 7-e-4 e. Centroid and others 0.481 0.711 11.632 15 0.951 0.470 Let$\Omega$be the region bounded by$y=\cos x$,$y=0$,$x=0$and$x=\frac{\pi}{2}$. (a) Find the volume of the solid obtained by revolving$\Omega$about$x$-axis. (b) Find the volume of the solid obtained by revolving$\Omega$about$y$-axis. (c) Find the centroid of$\Omega$. 7-e-5 e. Centroid and others 0.369 0.692 10.644 15 0.876 0.507 Sketch the region bounded by the curves$x^{1/3}+y^{1/3}=1$and$x+y=1$. Locate the centroid of the region and find the volume generated by revolving the region about each of the coordinate axes. 主題分類表：  Separable Equations: 8-a-1, 8-a-2, 8-a-3, 8-a-4, 8-a-5, 8-a-6, 8-a-7, 8-a-8, 8-a-9(a), 8-a-10, 8-a-11(b), 8-a-12(a), 8-a-13(a), 8-b-3(a) Linear Differential Equations: 8-a-9(b), 8-a-11(a), 8-a-12(b), 8-a-13(b), 8-b-1, 8-b-2, 8-b-3(b), 8-b-4, 8-b-5, 8-b-6, 8-b-7, 8-b-8 編號 副主題 鑑別度 難易度 平均 總分 高分群答對率 低分群答對率 題目 8-a-1 a. Separable equations 0.637 0.418 5.381 15 0.736 0.099 Solve the initial-value problem $$\begin{cases} &(\cos^2x)y'-(\sec x+\sin^2x)y^2=2\sec x-2\cos^2x+2,\\ &y(0)=\sqrt{2}. \end{cases}$$ 8-a-2 a. Separable equations 0.525 0.678 7.819 10 0.941 0.416 Solve the initial-value problem: $\begin{cases} &(xy^2+y^2+x+1)dx+(y-1)dy=0,\\ &y(2)=0. \end{cases}$ 8-a-3 a. Separable equations 0.422 0.338 3.321 10 0.549 0.127 A function$y=y(x)$satisfies the equation $y(x)=x+\int_0^{x^2}\left(x-y(\sqrt{t})-\sqrt{t}-\frac{1}{2\sqrt{t}}+1\right)e^t dt,\ x\geq 0.$ (a) Find a differential equation with initial condition for$y$. (b) Solve the differential equation. 8-a-4 a. Separable equations 0.473 0.438 6.004 15 0.674 0.202 Let$y=h(x)$be decreasing on$[0,\frac{\pi}{2})$and is continuously differentiable on$(0,\frac{\pi}{2})$with$h(0)=0$. Let$s(x)$denote the arc length of$y=h(x)$from$(0,0)$to$(x,h(x))$. (a) Write down the formula for$s(x)$. (b) Suppose that$s(x)$is also given by$s(x)=\int_0^xe^{-h(t)}dt$. Find the function$h(x)$explicitly. (c) Find the function$s(x)$explicitly. 8-a-5 a. Separable equations 0.441 0.293 2.690 10 0.514 0.073 An inverted cycloid is defined by the parametried equations $x(\theta)=r(\theta - \sin\theta),\ y(\theta) = -r (1- \cos\theta), \ 0 \leq \theta \leq 2 \pi.$ Consider the motion of a particle without friction and rolling down the inverted cycloid released from$(x(\alpha),y(\alpha))$, where$\alpha\in[0,\pi]$. By the conservation of energy, the velocity of the particle at$(x(\theta),y(\theta))$is given by $v= \frac{ds}{dt} = \sqrt{2gr(\cos{\alpha}-\cos{\theta})}\ \ \ \ \ \ \ \ (*)$ where$s$is the arc length function and$\alpha \leq \theta \leq \pi$. (a) Derive a separable differential equation for$\frac{d \theta}{dt}$from (*). (b) Compute the time$T=\int_{\theta=\alpha}^{\pi}dt$for the particle to get to the lowest point$(x(\pi), y(\pi)).$8-a-6 a. Separable equations 0.340 0.791 8.430 10 0.961 0.621 Solve the initial value problem:$\frac{dy}{dx}=\frac{e^y}{1+x^2}$,$y(0)=-1$. 8-a-7 a. Separable equations 0.427 0.250 3.250 16 0.464 0.037 A candle is located at the origin$O$, a bug,$P$, crawls on the plane so that the angle between its velocity and the vector$\overrightarrow{PO}$is always$\frac{\pi}{6}$. (a) Suppose that the bug crawls at a curve with polar equation$r=f(\theta)$. Derive the differential equation that$f(\theta)$satisfies. (b) If the Cartesian coordinates for the bug's initial position are$(1,0)$, solve for the curve$r=f(\theta)$. (c) Compute the arc length function$s(\theta)$and find$\lim\limits_{\theta\rightarrow\infty}s(\theta)$. 8-a-8 a. Separable equations 0.658 0.557 7.010 12 0.886 0.228 (a) Solve the initial-value problem:$\frac{dx}{dt} = (a - x)(b - x)$, where$a>b>0$, for$x=x(t)$with$x(0)=0$. (b) Find$\lim_{t\rightarrow\infty} x(t)$. 8-a-9 a. Separable equations 0.526 0.614 7.840 12 0.878 0.351 (a) Find the orthogonal trajectories of the family of curves$y=\sqrt{x^3+c}$, where$c$is an arbitrary constant. (b) Solve the initial-value problem $y'+(\tan x)y=\sec^3 x,\ y(0)=1.$ 8-a-10 a. Separable equations 0.509 0.413 5.350 13 0.668 0.159 Find the orthogonal trajectories of the family of curves$y=\tan^{-1}(kx)$, where$k$is an arbitrary constant. 8-a-11 a. Separable equations 0.515 0.619 7.520 12 0.876 0.362 (a) Solve the differential equation$x\,\frac{dy}{dx}\,-\,2 y \,=\,x^3\,\tan x\,\sec x$,$x > 0$, and$y(\pi/3)=0$. (b) Find the orthogonal trajectories of the family of curves$y=\frac{k}{x+1}$, where$k$is an arbitrary constant. 8-a-12 a. Separable equations 0.509 0.691 11.85 16 0.946 0.437 (a) Find the orthogonal trajectories of the family of curves$y=C\tan x$, where$C$is an arbitrary constant. (b) Solve the differential equation$(\cos x)\cdot y'+(\sin x)\cdot y=\tan x$,$y(0)=1$. 8-a-13 a. Separable equations 0.703 0.592 10.87 16 0.943 0.240 (a) Solve the initial-value problem:$\dfrac{du}{dt}=\dfrac{2\,t+\sec^2 t}{2\,u}$,$u(0)=-5$. (b) Solve the differential equation$(x^2+2)y'(x)+(4x)y=2x$with$y(0)=2$. 8-b-1 b. Linear equations 0.294 0.715 7.592 10 0.861 0.568 Solve the initial-value problem: $\begin{cases} &(\sec x) y'+y=(\tan x) e^{\cos x -\sin x}, \quad 0\leq x<\frac{\pi}{2},\\ &y(0)=0. \end{cases}$ 8-b-2 b. Linear equations 0.430 0.700 7.537 10 0.915 0.485 Solve$xy'-3y=5x^3$(a) with the initial condition$y(1)=2$. (b) with the initial condition$y(-1)=2$. 8-b-3 b. Linear equations 0.588 0.512 8.840 16 0.806 0.218 (a) Show that$y=0$is an orthogonal trajectory of the family of curves$x^2+\frac{y^2}{k}=1$, where$k>0$is an arbitrary constant. Find the orthogonal trajectories of the same family of curves when$y\neq 0$. (b) Find$u(t)$that satisfies the ordinary differential equation $u'(t)+\ln(t)u(t)=e^{-t\ln(t)}\text{, }t>0$ and the condition $\lim\limits_{t\rightarrow 0^{+}} u(t)=2.$ 8-b-4 b. Linear equations 0.681 0.557 4.500 8 0.897 0.216 Solve the differential equation$y'+\frac{2}{x}y=\frac{y^3}{x^2}$. (Hint: let$y^2=\frac{1}{u}.) 8-b-5 b. Linear equations 0.567 0.627 8.380 13 0.910 0.343 (a) Solve the initial value problem: \begin{align*} \left\{\begin{array}{l} 2x(x+3)y'+\left(4x+3\right)y=2x^{\frac12}(x+3)^{\frac12},\\ y(1)=\frac12,\quad x>0. \end{array}\right. \end{align*} (b) Find\lim\limits_{x\to\infty}y(x)$and$\lim\limits_{x\to0^+}y(x)$. 8-b-6 b. Linear equations 0.486 0.683 5.650 8 0.926 0.440 Solve the differential equation $x^2y'-y=2x^3e^{-\frac{1}{x}},\ y(1)=1.$ 8-b-7 b. Linear equations 0.531 0.672 8.600 12 0.938 0.407 (a) Solve the initial-value problem:$xy'-y=x^2\sin x$, with$y(\pi)=0$. (b) Find$\lim\limits_{x\rightarrow 0^+}\frac{y(x)}{x^2}. 8-b-8 b. Linear equations 0.585 0.659 8.850 12 0.951 0.367 Solve the initial value problem \begin{align*} \begin{cases} x^2y'-y=2xe^{-\frac{1}{x}}\ln x,\ x>0\\ y(1)=2 \end{cases} \end{align*} 主題分類表：  Calculus with parametric equations: 9-a全 Polar coordinates and polar curves: 9-b全 (除了9-b-4(a)) 編號 副主題 鑑別度 難易度 平均 總分 高分群答對率 低分群答對率 題目 9-a-1 a. Calculus with parametric equations 0.599 0.451 5.277 12 0.751 0.152 The cycloid is the curve parametrized byx(\theta)=R(\theta-\sin\theta)$,$y(\theta)=R(1-\cos\theta)$. (a) Find the arclength of the cycloid for$0\leq\theta\leq 2\pi$. (b) Find the area under the cycloid and above the$x$-axis for$0\leq\theta\leq 2\pi$. (c) Find the volume of the solid generated by revolving the region in (b) about the$x$-axis. 9-a-2 a. Calculus with parametric equations 0.473 0.702 11.258 15 0.939 0.465 Let the curve$C$defined by$\begin{cases} x=t^2\\ y=\frac{t^3}{3}-t \end{cases}$,$t\in\mathbb{R}$. (a) Find the point$P$where the curve intersects itself. (b) Find the equation of the tangent lines at the point$P$. (c) For which values of$t$is the curve increasing? (d) For which values of$t$is the curve concave upward? (e) Sketch the curve$C$. 9-a-3 a. Calculus with parametric equations 0.495 0.692 7.443 10 0.939 0.444 Find the arc length of the curve$x = t \sin 2 t,\;\; y= t \cos 2 t,\;\;0 \le t \le 1.$9-a-4 a. Calculus with parametric equations 0.638 0.565 8.190 14 0.884 0.246 A tractrix (the curve below) describes the path an obstinate dog takes when its master walks along the$y$-axis. One way to parameterize the curve is by means of $(x(t), y(t))=(\sin t, \cos t+\ln(\csc t-\cot t)), 0<t<\pi.$ (a) Find the slope of the tangent line at a point$P=(x(t_0),y(t_0))$on the curve, and show that the distance between$P$and the$y$-intercept of the tangent line at$P$is independent of$t_0\in (0,\pi)$. (b) Compute the arc length of the part of the tractrix from$t=\frac{\pi}{4}$to$t=\frac{\pi}{2}$. 9-a-5 a. Calculus with parametric equations 0.495 0.538 9.230 16 0.786 0.291 Let$a$and$b$be fixed numbers. (a) Find parametric equations for the curve$C$that consists of all possible positions of the point$P$in the figure using the angle$\theta$as the parameter. (b) What are the horizontal asymptotes of the curve$C$? (c) Let$R$be the region bounded by$C$and its horizontal asymptotes. Find the area of$R$if it is finite. (d) The region is$R$rotated about the$y$-axis. Find the volume of the resulting solid if it is finite. 9-a-6 a. Calculus with parametric equations 0.542 0.559 9.670 16 0.830 0.288 The front door of a school bus is designed as in Figure. It is a folding door with$\overline{AB}=\overline{BO}=\frac12$. The door is opened or closed by rotating$\overline{OB}$about$z$-axis, while$A$is moving along$y$-axis. The region swept out by the bus door on the$xy$-plane is enclosed by the two axes and the curve$\gamma(t)parametrized by \begin{align*} \gamma(t)=(x(t),y(t)) =\left\{ \begin{array}{ll} (\sin^3t,\cos^3t)& \text{if } 0\leq t\leq\frac\pi4\3mm] \left(\frac12\sin t,\frac12\cos t\right) & \text{if } \frac\pi4\leq t\leq\frac\pi2, \end{array}\right. \end{align*} where 0\leq t\leq\frac{\pi}{2} denotes the angle between the positive y-axis and \overline{OB}. (a) Find the area of this region. (b) Find the length of the curve \gamma(t). 9-a-7 a. Calculus with parametric equations 0.360 0.730 7.520 10 0.910 0.550 Find the arc length of the curve. x=\cos t+\ln(\tan \frac{1}{2}t), y=\sin t, \frac{\pi}{4}\leq t\leq \frac{3\pi}{4}. 9-a-8 a. Calculus with parametric equations 0.799 0.498 9.41 18 0.897 0.098 The eight-like curve has the following parametric equation: \begin{align*} x = 2\sqrt{2}\sin t ~,~~ y= \sin t\cos t ~~ \text{for }~ 0\leq t\leq 2\pi ~. \end{align*} (a) Find the tangent line at t=0. (b) Find its arc length. (c) Find the shaded area which is enclosed by the curve 0\leq t\leq\frac{\pi}{2} and the x-axis. 9-a-9 a. Calculus with parametric equations 0.628 0.636 14.550 21 0.950 0.322 (a) Find the length of the parametric curve C: x=\ln(\sec t +\tan t) - \sin t,\;y = \cos t,\;\;0 \le t \le \frac{\pi}{3}. (b) Rotate the curve C about x-axis. Find the surface area. (c) Find the volume of the solid bounded by the surface given in (b) and the planes x=0, x=\ln(2+\sqrt{3})-\frac{\sqrt{3}}{2}. 9-b-1 b. Polar coordinates and polar curves 0.507 0.435 6.984 15 0.688 0.181 Consider the curve given by the polar equation r=a+\cos\theta where 0<a<1. (a) Find the tangent lines of this curve at the origin. Find the constant a=a_0 such that the tangent lines at the origin are y=\frac{\sqrt{3}}{2}x and y=-\frac{\sqrt{3}}{2}x. (b) Draw the curve r=a_0+\cos\theta. (c) Compute the area of the region bounded by the inner loop of r=a_0+\cos\theta. (d) Find the area of the surface given by revolving the inner loop about the x-axis. 9-b-2 b. Polar coordinates and polar curves 0.444 0.629 6.434 10 0.851 0.407 Find the area both inside r^2=2\cos 2\theta and inside r=1. 9-b-3 b. Polar coordinates and polar curves 0.646 0.608 6.499 10 0.931 0.285 Find the area of the surface generated by revolving the curve r=\sin \theta, 0\le \theta\le \pi/2 about the x-axis. 9-b-4 b. Polar coordinates and polar curves 0.484 0.518 5.495 10 0.760 0.276 (a) If the infinite curve y=e^{-x},\ x\geq0, is rotated about the x-axis, find the area of the resulting surface. (b) Find the arc length of the infinite curve with polar equation r=\theta^{-1}, \theta\geq 1. 9-b-5 b. Polar coordinates and polar curves 0.341 0.280 2.700 10 0.451 0.110 (a) Find the points of intersection of the curves r=1+2\cos\theta and r^2=\cos\theta. (b) Find the area of the region in the second quadrant that lies inside r^2=\cos\theta and outside r=1+2\cos\theta. 9-b-6 b. Polar coordinates and polar curves 0.519 0.656 7.006 10 0.915 0.396 Find the area of the region that lies inside the curve r=2+\cos 2\theta but outside the curve r=2+\sin\theta. 9-b-7 b. Polar coordinates and polar curves 0.618 0.602 7.560 12 0.911 0.293 (a) Find f(\theta) such that r^2=f(\theta) is the polar equation of the curve given by x^2+y^2=(x^2-y^2)^2, (x,y)\ne (0,0). (b) Find the points of intersection of the curves x^2+y^2=(x^2-y^2)^2 and r=2 (express the intersection points in terms of polar coordinates). (c) Find the area of the region containing the origin and bounded by the curves x^2+y^2=(x^2-y^2)^2 and r=2 (the shaded region in the figure below). 9-b-8 b. Polar coordinates and polar curves 0.553 0.546 4.540 8 0.822 0.270 (a) Sketch the curve with polar equation r=1+2\cos 2\theta. (b) Find the area of the region inside both curves r=1 and r=1+2\cos 2\theta. 9-b-9 b. Polar coordinates and polar curves 0.497 0.727 10.770 14 0.976 0.479 (a) Find the length of the curve in polar coordinates: r=\sqrt{1+\sin 2\theta},\;\;0 \le \theta \le 2\pi. (b) Find the area enclosed by the curve given in (a). 9-b-10 b. Polar coordinates and polar curves 0.427 0.543 9.150 16 0.757 0.330 (a) Sketch the curve r=1+2\cos\theta. (b) Compute the area of the region that is inside the larger loop of the curve r=1+2\cos\theta and outside the smaller loop of the curve r=1+2\cos\theta. (c) Let C be the smaller loop of r=1+2\cos\theta, and rotate C about the x-axis. Find the area of the resulting surface. 9-b-11 b. Polar coordinates and polar curves 0.522 0.600 7.540 12 0.861 0.339 (a) Find all intersection points of the two curves r=\sqrt{2}\sin\theta and r^2=\cos2\theta in their polar equations. (b) Find the area of the shaded region in Figure. 9-b-12 b. Polar coordinates and polar curves 0.465 0.589 7.290 12 0.821 0.356 Consider the plum flower-like curve (梅花) as Figure. It is characterized by the polar equation \[r=\frac32+\cos\left(\frac52\theta\right). (a) Find the slopes of the tangent lines of the curve at the intersection pointP(r,\theta)=\left(\frac{3}{2},\frac{\pi}{5}\right)$. (b) Set up an integral that represents the length of the whole curve. You don't need to evaluate the integral. (c) Find the area of the shaded region. 9-b-13 b. Polar coordinates and polar curves 0.651 0.598 8.160 13 0.923 0.272 Find the area of the region that lies inside the curve$r = 1 + \cos\theta$but outside the curves$r = 2\cos\theta$and$r = -\cos\theta$. 9-b-14 b. Polar coordinates and polar curves 0.513 0.638 6.730 10 0.895 0.382 The curve$C:\ r=1+2\cos\theta$and its two tangent lines,$L_1$and$L_2$, at the pole are shown in the graph. (a) Find the area of the shaded region. (b) Now consider another curve$\tilde{C}:\ r=-1-2\cos(\theta-\frac{\pi}{6})$. How is the curve$\tilde{C}$related to the curve$C$? 9-b-15 b. Polar coordinates and polar curves 0.616 0.640 9.71 14 0.949 0.332 If you pour coffee mate into the coffee and stir it, the shape will be similar to a Fermat's spiral. The curve is given by$r^2=\theta$,$\theta>0$. Note that it can be realized as two curves$r=\sqrt{\theta}$(Solid curve) and$r=-\sqrt{\theta}$(Dotted curve). (a) On the solid curve$r=\sqrt{\theta}$, compute the slope of the tangent line at$\theta=\frac{\pi}{4}$. (b) Set up the integral, using the variable$\theta$only, that expresses the length of the curve connecting$(x,y)=(\sqrt{2\pi},0)$and$(x,y)=(-\sqrt{2\pi},0)$. DO NOT evaluate this integral! (c) If we fix$0<\theta_1<\theta_2<\frac{\pi}{2}$, among three regions R1, R2, R3(See Picture 3) cut by$\theta=\theta_1$,$\theta=\theta_2$and the Fermat's spiral, which region has the largest area? 9-b-16 b. Polar coordinates and polar curves 0.466 0.342 3.870 12 0.574 0.109 (a) There are two polar curves$C_1$and$C_2$with equations$r=r_1(\theta)$and$r=r_2(\theta)$. Suppose that they intersect orthogonally at the point with polar coordinates$(r_0,\theta_0)$. Show that$r'_1(\theta_0)\cdot r'_2(\theta_0)=-r_0^2$which means that$\left(\frac{1}{r_1}\frac{dr_1}{d\theta}\right)\cdot\left(\frac{1}{r_2}\frac{dr_2}{d\theta}\right) \Big|_{\theta=\theta_0}=-1$. (Hint: Compute the slopes of the tangent lines of$C_1$and$C_2$at the intersection.) (b) Consider the family of cardioids$r=r_c(\theta)=c\, (1+\cos\theta)$, where$c$is an arbitrary constant. Derive the differential equation that$r_c(\theta)$satisfies for all$c$. (c) Find the orthogonal trajectories of the family of cardioids$r=c\, (1+\cos\theta)$. 主題分類表：  Second order differential equations: 10-a全, 10-b-1(a), 10-b-4(a), 10-b-5(a) Laplace transform: 10-b-1(b)(c)(d), 10-b-2, 10-b-3, 10-b-4(b)(c), 10-b-5(b) 編號 副主題 鑑別度 難易度 平均 總分 高分群答對率 低分群答對率 題目 10-a-1 a. Second order differential equations 0.511 0.593 6.110 10 0.848 0.337 Solve the differential equation$y''+2y'+y=x^{-3}e^{-x}$with initial conditions$y(1)=y'(1)=0$. Find$\lim\limits_{x\rightarrow\infty}y(x)$. 10-a-2 a. Second order differential equations 0.552 0.531 5.610 10 0.806 0.255 Solve the differential equation $y''+y=x^2e^x+\tan x,\ \ x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right).$ 10-a-3 a. Second order differential equations 0.466 0.642 6.940 10 0.875 0.409 Solve the initial value problem $\begin{cases} y''+y=xe^{x}+\sec x,\ \ -\frac{\pi}{2}<x<\frac{\pi}{2}\\ y(0)=1,\ \ y'(0)=-3. \end{cases}$ 10-b-1 b. Laplace transform 0.445 0.714 11.880 16 0.936 0.491 Consider the differential equation$\begin{cases} y''(t)+2y'(t)+y(t)=f(t)\\ y(0)=1, y'(0)=0, \end{cases}$where$f(t)=\begin{cases} 2\cos t, &\text{for }0\leq t<\pi\\ 0, &\text{for }t\geq \pi. \end{cases}$(a) Find the general solution of the related homogeneous equation $y''(t)+2y'(t)+y(t)=0.$ (b) Write$f(t)$in terms of the unit step function $\mathcal{U}(t-\pi)=\begin{cases} 0,&0\leq t<\pi\\ 1, &t\geq \pi \end{cases}.$ Compute$\mathcal{L}\{f(t)\}$, the Laplace transform of$f(t)$. (c) Let$Y(s)$be the Laplace transform of the solution$y(t)$. Apply Laplace transform to the differential equation and solve for$Y(s)$. (d) Solve the differential equation. 10-b-2 b. Laplace transform 0.400 0.319 2.490 8 0.519 0.119 Find the values of$s$such that$F(s)=\int_0^{\infty}\sin t\ e^{-st}dt$converges and evaluate the integrals. 10-b-3 b. Laplace transform 0.567 0.462 5.240 12 0.745 0.179 (a) Suppose that$f(t)$is piecewise continuous on$[0,\infty)$and of exponential order. Let$F(s)=\mathfrak{L}\{f(t)\}$be the Laplace transform of$f(t)$. Show that$\mathfrak{L}\{f(t-a)\mathcal{U}(t-a)\}=e^{-as}F(s)$, where$\mathcal{U}(t-a)$is the unit step function defined as$\mathcal{U}(t-a)=\begin{cases} 0, &\text{if }0\leq t<a\\ 1, &\text{if }t\geq a \end{cases}$. (b) Express$g(t)=\left\{\begin{array}{ll} t, &\text{if }0\leq t<1\\ 1, &\text{if }t\geq 1\end{array}\right.$in terms of unit step functions. (c) Solve the differential equation$y''+4y=g(t)$, where$y(0)=1$and$y'(0)=0$. 10-b-4 b. Laplace transform 0.494 0.568 6.770 12 0.815 0.321 Consider the differential equation $y''(t)+2y'(t)+10y(t)=f(t),\ y(0)=0,\ y'(0)=1,$ where$f(t)=\begin{cases} 0, &0\leq t<\pi\\ 13\sin 2t, & t\geq \pi. \end{cases}$(a) Write down the general solution of its related homogeneous equation. (b) Derive the algebraic equation that$Y(s)$, the Laplace transform of the solution$y(t)$, satisfies. (c) Solve this differential equation. (Hint:$\mathfrak{L}\{f(t-a)\mathcal{U}(t-a)\}=e^{-as}\mathfrak{L}\{f(t)\}$, where$\mathcal{U}(t)$is the unit step function.) 10-b-5 b. Laplace transform 0.582 0.681 11.92 16 0.972 0.390 (a) Find the general solution of$y''(t)+y(t)=\sec^2t$, for$-\frac{\pi}{2} < t < \frac{\pi}{2}$. (b) Solve the differential equation$y''(t)+y(t)=f(t)$,$y(0)=0$,$y'(0)=1$, for$t\geq 0$, where$f(t)=t-5$for$5\leq t< 10$,$f(t)=0$, otherwise. (Hint:$ \mathcal{L} \{g(t-a) \mathcal{U}(t-a) \} = e^{-as} \mathcal{L} \{g(t)\} $) 編號 副主題 鑑別度 難易度 平均 總分 高分群答對率 低分群答對率 題目 11-a-1 a. Sequences 0.613 0.451 5.770 15 0.758 0.145 Let$\{f_n\}$be the Fibonacci sequence defined by$f_1=f_2=1$,$f_{n+1}=f_n+f_{n-1}$for$n\geq 2$. Define$a_n=\frac{f_{n+1}}{f_n}$,$n\geq 1$. (a) Show that$\{a_{2n}\}$is decreasing while$\{a_{2n+1}\}$is increasing and both$\lim_{n\rightarrow\infty}a_{2n}$and$\lim_{n\rightarrow\infty}a_{2n+1}$exist. Find the limits. (Hint.$\{a_n\}$satisfies the recursive relation$a_{n+1}=1+\frac{1}{a_n}$,$n\geq 1$. Express$a_{n+2}$in terms of$a_n$.) (b) Find the radius of convergence of the power series$f(x)=\sum_{n=1}^{\infty}f_nx^n$. 11-a-2 a. Sequences 0.605 0.517 5.040 10 0.820 0.214 Let$p\in(0,1)$. A sequence$\{x_n\}_{n=1}^\inftyis given by \begin{align*} x_1=\sqrt{p}\quad\text{and}\quad x_{n+1}=\sqrt{p+x_n},\ \text{for }n\geq 1. \end{align*} Determine whether the sequence is convergent or divergent with an argument. If it is convergent, find the limit. 11-b-1 b. Series and tests for series 0.535 0.585 6.930 12 0.853 0.318 Test the following two series\sum_{k=1}^\infty\frac{\ln k}{k^p}$, where$p=1$and$p=3/2$, for convergence. 11-b-2 b. Series and tests for series 0.236 0.849 10.357 12 0.967 0.731 Determine whether the series converge or diverge. (a)$\sum_{k=1}^\infty(\sqrt{k}-\sqrt{k-1})^{2k}$. (b)$\sum_{k=1}^\infty\frac{(k!)^2}{(5k)!}$. 11-b-3 b. Series and tests for series 0.434 0.473 6.240 15 0.690 0.255 (a) Let$\{b_n\}$be a sequence of nonzero numbers such that$\lim_{n\rightarrow\infty}b_n=\infty$. Determine whether the series$\sum_{k=1}^{\infty}(b_{k+1}-b_k)$and$\sum_{k=1}^{\infty}\left(\frac{1}{b_k}-\frac{1}{b_{k+1}}\right)$are convergent or divergent. Explain your answer. (b) Determine whether the series$\sum_{n=1}^{\infty}(-1)^n\left(n\sin\frac{1}{n}-1\right)$is absolutely convergent, conditionally convergent or divergent. (c) Find all values of$p$such that the series$\sum_{n=1}^{\infty}(-1)^n\frac{\ln n}{n^p}$converges conditionally. 11-b-4 b. Series and tests for series 0.203 0.539 5.400 10 0.640 0.437 Find the values of$\rho$for the convergence of the series below (a)$\sum_{n=0}^{\infty} e^{n(\rho^2-\rho-2)}$, (b)$\sum_{n=1}^{\infty} \frac{e^{\frac{1}{n}}-1}{n^{\rho}}$. 11-b-5 b. Series and tests for series 0.469 0.268 2.140 10 0.502 0.033 (a) Prove$\ln (n+1)<1+\frac{1}{2}+\cdots+\frac{1}{n}<1+\ln n$. (b) Test for convergence of$\sum_{n=1}^{\infty}a_n$, where$a_n=\frac{1}{1+\frac{1}{2}+\cdots+\frac{1}{n}}$. 11-b-6 b. Series and tests for series 0.501 0.625 7.800 12 0.875 0.374 Determine whether each of the following series is divergent, conditionally convergent, or absolutely convergent? (a)$\sum\limits_{n=1}^{\infty} \frac{(-1)^n}{n(1+\ln n)}$. (b)$\sum\limits_{n=1}^{\infty}(-1)^{n-1}\sin\left(\frac{1}{n^2}\right)$. (c)$\sum\limits_{n=1}^\infty (-1)^n\frac{\pi^nn!}{n^n}$. 11-b-7 b. Series and tests for series 0.515 0.608 11.340 18 0.865 0.351 Test the series for absolute convergence, conditional convergence or divergence. (a)$\sum\limits_{n=2}^{\infty}\frac{(-1)^n}{n(\ln n)^2}$. (b)$\sum\limits_{n=1}^{\infty}(-1)^n\tan\frac{1}{n}$. (c)$\sum\limits_{n=1}^{\infty} \frac{(-1)^n}{\frac{1}{1^2}+\frac{1}{2^2}+\cdots+\frac{1}{n^2}}$11-b-8 b. Series and tests for series 0.556 0.447 3.280 8 0.725 0.169 Compute the sum of the series$\begin{align*} S=(1)(\frac{1}{2})+(1-\frac{1}{3})(\frac{1}{2})^3+(1-\frac{1}{3}+\frac{1}{5}) (\frac{1}{2})^5\\ +(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7})(\frac{1}{2})^7 \\ +(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9})(\frac{1}{2})^9\\ +\cdots.\end{align*}$(Hint: imitate the method of deriving the sum of a geometric series.) 11-b-9 b. Series and tests for series 0.535 0.558 4.300 8 0.825 0.290 Determine whether the series$\sum\limits_{n=1}^{\infty}(-1)^n\sin\left(\frac{1}{\sqrt{n}}\right)\ln\left(1+ \frac{1}{\sqrt{n}}\right)$is divergent, conditionally convergent or absolutely convergent. 11-b-10 b. Series and tests for series 0.383 0.408 4.860 12 0.599 0.216 (a) Use a Riemann sum approximation of$\int_1^n\ln tdt$to show that$\ln(n!)\geq n\ln n-n+1$. (b) Find the interval of convergence of the power series$\sum\limits_{n=1}^{\infty}\frac{(2n)!}{n^{2n}}x^n$. 11-b-11 b. Series and tests for series 0.382 0.763 8.110 10 0.954 0.572 Find the sum of the series given below: (a)$\;\sum_{n=1}^{\infty}\;\frac{1}{2^n}\cos(\frac{n\pi}{2})$(b)$\;\sum_{n=1}^{\infty}\;\frac{1}{n(n+2)}$. 11-b-12 b. Series and tests for series 0.482 0.519 6.130 12 0.760 0.278 (a) Find the values of$p$for which the series$\sum\limits_{n=1}^\infty \frac{n}{(1+n^3)^p}$is convergent. (b) Determine whether the series$\sum\limits_{n=1}^\infty\frac{(-1)^n\tan^{-1}n}{n}$is absolutely convergent, conditionally convergent, or divergent. 11-b-13 b. Series and tests for series 0.483 0.437 6.180 15 0.679 0.195 Determine whether the series is absolutely convergent, conditionally convergent, or divergent. Please state the tests which you use. (a)$\sum\limits_{n=2}^{\infty}(-1)^n\frac{\ln(n!)}{n^3\ln n}$(b)$ \sum\limits_{n=2}^{\infty} (-1)^n \frac{\sqrt[n]{2}-1} {\ln n}$(c)$\sum\limits_{n=1}^{\infty} n^5 \frac{4^n-n^3}{(-5)^n+3^n}$11-b-14 b. Series and tests for series 0.421 0.392 5.390 15 0.602 0.181 Determine whether the series is absolutely convergent, conditionally convergent, or divergent. Please state the tests which you use. (a)$\sum\limits_{n=2}^{\infty}(-1)^n\frac{\ln(n!)}{n^3\ln n}$(b)$\sum\limits_{n=1}^{\infty} (-1)^n \left(\frac{1}{\sqrt{n}}-\sin\left(\frac{1}{\sqrt{n}}\right)\right)$(c)$\sum\limits_{n=1}^{\infty}\left(-1\right)^{\frac{n^3-n}{2}}\left(\frac{n+1}{n}\right)^{n^2}$11-b-15 b. Series and tests for series 0.536 0.420 3.780 10 0.688 0.152 (a) Find the constant$p$such that$\lim\limits_{n\rightarrow\infty}\frac{\frac{1}{\sqrt{1}}+ \frac{1}{\sqrt{2}}+\cdots+\frac{1}{\sqrt{n}}}{n^p}$is a finite nonzero constant. (b) Find the interval of convergence of the power series$\sum\limits_{n=1}^{\infty}\frac{(1-3x)^n}{\frac{1}{\sqrt{1}}+ \frac{1}{\sqrt{2}}+\cdots+\frac{1}{\sqrt{n}}}$. 11-c-1 c. Power series 0.639 0.480 7.099 15 0.799 0.160 Let$s_k=\sum\limits_{j=1}^{k}\frac{1}{j}$,$k=1,~2,~\cdots$, and$A(x)=\sum\limits_{k=1}^{\infty}s_kx^k$. (a) Find the interval of convergence of$A(x)$. (b) Express$A(x)$in terms of elementary functions by comparing$A(x)$and$xA(x)$. 11-c-2 c. Power series 0.705 0.477 5.847 12 0.830 0.125 Find the interval of convergence of the series$\sum_{k=1}^\infty\ln\Big(\frac{k+1}{k}\Big)x^{k}$. 11-c-3 c. Power series 0.468 0.534 5.290 10 0.768 0.300 (a) Expand the function$f(x)=(8+x)^{\frac{1}{3}}$as a power series centered at$x=0$. (You must write out the general terms.) Find the radius of convergence. (b) Find the sum of the series$\sum_{n=2}^{\infty}\frac{n^2+1}{n!}$. 11-c-4 c. Power series 0.550 0.624 6.670 10 0.899 0.350 (a) Find the radius of convergence and the interval of convergence of the power series$\sum_{n=0}^\infty\frac{(x-1)^n}{(-2)^n\sqrt{n}}$. (b) Let$f(x)=\sum_{n=0}^\infty\frac{(x-1)^n}{(-2)^n\sqrt{n}}$when the power series is convergent. Evaluate$f^{(3)}(1)$. 11-c-5 c. Power series 0.508 0.583 9.710 16 0.837 0.329 For each of the following power series, find the interval of convergence and the function represented by it. (a)$\sum\limits_{n=1}^\infty (-1)^n\frac{1}{n8^n}(x-8)^n.$(b)$\sum\limits_{n = 0}^\infty n^2 x^n$. (Hint: You can use the fact$\frac{1}{1-x}=\sum_{n=0}^{\infty} x^n$.) 11-c-6 c. Power series 0.570 0.628 5.050 8 0.913 0.343 Find the interval of convergence of the series$\sum\limits_{n=0}^{\infty}(n+3)x^n$, and compute the sum. 11-c-7 c. Power series 0.433 0.736 11.900 15 0.953 0.520 (a) Find the radius of convergence and the interval of convergence of the power series$f(x)=\sum\limits_{n=2}^{\infty} (-1)^n\,\frac{x^n}{4^n\,\ln n}$. (b) Evaluate$f^{(3)}(0)$. 11-c-8 c. Power series 0.611 0.411 3.060 8 0.717 0.106 Find the sum of the series$\sum\limits_{n=0}^{\infty}\frac{x^{4n}}{2n+1}$. 11-c-9 c. Power series 0.383 0.408 4.860 12 0.599 0.216 (a) Use a Riemann sum approximation of$\int_1^n\ln tdt$to show that$\ln(n!)\geq n\ln n-n+1$. (b) Find the interval of convergence of the power series$\sum\limits_{n=1}^{\infty}\frac{(2n)!}{n^{2n}}x^n$. 11-c-10 c. Power series 0.469 0.672 7.130 10 0.906 0.437 Consider the power series$\sum\limits_{n=1}^{\infty}n^{\sqrt{n}}\frac{x^n}{100^n}$. (a) Find the radius of convergence of the given series by the Root Test. (b) Find the interval of convergence of the given series. 11-c-11 c. Power series 0.397 0.682 8.330 12 0.881 0.483 Consider the power series$f(x)=\sum\limits_{n=2}^\infty\frac{1}{n(n-1)3^n}(x-2)^n$. (a) Find the interval of convergence for$f(x)$. (b) Write down the power series representation for$\frac{\mathrm{d}}{\mathrm{d}x}f(x)$and find its sum in the interior of the interval of convergence. 11-c-12 c. Power series 0.399 0.750 7.970 10 0.950 0.551 Consider the power series$\sum\limits_{n=1}^\infty\frac{(-1)^nx^n}{(n+1)\ln(n+1)}$. (a) Determine its radius of convergence. (b) Determine its interval of convergence. 11-c-13 c. Power series 0.536 0.420 3.780 10 0.688 0.152 (a) Find the constant$p$such that$\lim\limits_{n\rightarrow\infty}\frac{\frac{1}{\sqrt{1}}+ \frac{1}{\sqrt{2}}+\cdots+\frac{1}{\sqrt{n}}}{n^p}$is a finite nonzero constant. (b) Find the interval of convergence of the power series$\sum\limits_{n=1}^{\infty}\frac{(1-3x)^n}{\frac{1}{\sqrt{1}}+ \frac{1}{\sqrt{2}}+\cdots+\frac{1}{\sqrt{n}}}$. 11-c-14 c. Power series 0.471 0.597 5.830 10 0.833 0.362 Find the radius of convergence and the interval of convergence of the power series$\sum\limits_{n=2}^{\infty} \frac{(2x-1)^n}{n(\ln n)^{\frac 34}}$. 11-c-15 c. Power series 0.577 0.526 4.230 8 0.815 0.238 (a) Identify the power series$\sum\limits_{n=0}^{\infty}\frac{(-1)^n 2^{2n+1}} {2n+1} x^{2n+1}$as an elementary function. (b) Find the sum$\;\frac{1}{\sqrt{3}}\,-\,\frac{1}{9\sqrt{3}}\,+\,\frac{1}{45\sqrt{3}}\,-\,\frac{1}{189\sqrt{3}}\,+\,\cdots$11-d-1 d. Taylor series 0.752 0.537 7.775 15 0.912 0.161 (a) Expand$f(x)=(x-1)\ln(1+3x)$in powers of$x-1$. (b) For what values of$x$is the above expansion valid? (c) Find the sum$\sum\limits_{k=1}^{\infty}\frac{1}{k}\left(\frac{3}{4}\right)^k$. 11-d-2 d. Taylor series 0.559 0.597 7.448 12 0.877 0.318 (a) Find the Taylor series for$f(x)=(x^2+x+1)\sqrt{x+1}$at$x=0$up to the third power of$x$. (b) Let$f(x)=\ln\sqrt{\frac{1+x^2}{1-x^2}}$. Find$f^{(10)}(0)$. 11-d-3 d. Taylor series 0.443 0.554 8.150 15 0.776 0.332 (a) Write down the general terms the MacLaurin series of$\sin x$and$\sin^{-1}x$. (b) Find their radii of convergence. (c) Find$\lim_{x\rightarrow 0}\frac{\sin x\cdot\sin^{-1}x-x^2}{x^6}$. 11-d-4 d. Taylor series 0.537 0.430 6.310 14 0.699 0.162 Let$f(x)=\int_0^x \frac{1}{\sqrt{1+t^2}}\ dt$. (a) Write down the Maclaurin series for$f(x)$and find its radius of convergence. (b) Approximate$f(\frac{1}{2})$correct to within 0.01. 11-d-5 d. Taylor series 0.547 0.413 3.980 10 0.686 0.139 (a) Find the Maclaurin series for$\cos^{-1}x$. (Write down the general term explicitly.) (b) What is the radius of convergence of the series in (a). (c) Let$f(x)=\cos^{-1}(x^2)$. Find$f^{(10)}(0)$. 11-d-6 d. Taylor series 0.646 0.638 9.160 14 0.961 0.315 (a) Find the 4-th degree MacLaurin polynomials of$\sec x$and of$\left(1-x^2\right)^{-\frac{1}{2}}$(5\% each). (b) Find$\lim\limits_{x\rightarrow 0}\frac{\sec x-\left(1-x^2\right)^{-\frac{1}{2}}}{x^4}$. 11-d-7 d. Taylor series 0.514 0.623 5.120 8 0.880 0.365 Find the Maclaurin series expansion of the function$\ln(1+3x+2x^2)$. Write out the general terms. What is the radius of convergence? 11-d-8 d. Taylor series 0.768 0.419 4.610 12 0.803 0.035 (a) Find the Maclaurin series for$\sinh^{-1}x$. (b) Find$\lim\limits_{x\rightarrow 0}\frac{\sinh^{-1}(x^2)-x^2}{x^6}$. 11-d-9 d. Taylor series 0.572 0.425 4.600 12 0.711 0.139 (a) Find the Maclaurin series for$\cos^{-1}x$. (b) Find the radius of convergence of (a) by the Ratio Test. (c) Find$\lim\limits_{x\rightarrow 0^+}\frac{\frac{\pi}{2}-\cos^{-1}x-x}{x^3}$. 11-d-10 d. Taylor series 0.583 0.495 7.020 15 0.787 0.204 Let$\displaystyle f(x)=\int_{-1}^x\frac{1}{\sqrt{t^2+2t+2}}\,\mathrm{d}t$. (a) Find the Taylor series for$f(x)$centered at$a=-1$. (Hint: Complete the square first.) (b) Find$f^{(9)}(-1)$and$f^{(10)}(-1)$. (c) Write down the$3$rd-degree Taylor polynomial$T_3(x)$for$f(x)$centered at$a=-1$, and calculate$T_3\left(-\frac{1}{2}\right)$. Estimate the error$\left|f\left(-\frac{1}{2}\right)-T_3\left(-\frac{1}{2}\right)\right|$by some estimation theorem. 11-d-11 d. Taylor series 0.595 0.599 9.540 15 0.896 0.301 (a) Derive the MacLaurin series of$\tan^{-1}x$. (b) Find the value of$a\in \mathbb{R}$such that the limit$\lim\limits_{x\rightarrow 0}\frac{\sin(ax)-\sin x-\tan^{-1}x}{x^3}$is finite. (c) Evaluate the above limit. 11-d-12 d. Taylor series 0.599 0.530 5.680 10 0.829 0.230 Let$F(x) = \int_0^x \ln\left(1+\frac{t^2}{2}\right) dt.$(a) Find the Maclaurin series of$F(x)$and its radius of convergence. (b) Estimate$F(10^{-1})$up to an error within$10^{-7}$. 編號 鑑別度 難易度 平均 總分 高分群答對率 低分群答對率 題目 12-1 0.348 0.784 12.150 15 0.958 0.610 Let$\mathcal{C}$be the curve given by$\textbf{r}(t)=\frac{2}{3}(1+t)^{\frac{3}{2}}\textbf{i}+\frac{2}{3}(1-t)^{\frac{3}{2}}\textbf{j}+at\textbf{k}$,$t\in(-1,1)$,$a\in\mathbb{R}\setminus \{0\}$. (a) Find the length$s(b)$of the curve from$t=0$to$t=b\in(0,1)$. (b) Find the unit tangent, the principal normal, and the osculating plane of$\mathcal{C}$at$\textbf{r}(t)$. (c) Find the curvature$\kappa(t)$of$\mathcal{C}$at$\textbf{r}(t)$. 12-2 0.364 0.769 8.014 10 0.951 0.587 Find the curvature$\kappa(t)$of the curve$\textbf{r}(t)=e^t\textbf{i}+e^{-t}\textbf {j}+\sqrt{2}t\textbf{k}$. 12-3 0.355 0.803 16.970 20 0.980 0.625 A curve$C$is defined by${\bf r}(t)=\left\langle\cos t+t\sin t,\sin t-t\cos t,t^2\right\rangle$,$t\geq 0$. (a) Find the arc length function$s(t)$with the starting point$(1,0,0)$. (b) Find the unit tangent vector${\bf T}$, the unit normal vector${\bf N}$and the unit binormal vector${\bf B}$. (c) Find the curvature of$C$. 12-4 0.403 0.695 11.100 15 0.897 0.494 Let${\bf r}(t)=\left\langle \frac{t^4}{2}, t, \frac{4}{5}t^{\frac{5}{2}}\right\rangle$for$t\geq 0$. (a) Find the length of the arc$0\leq t\leq 2$of${\bf r}(t)$. (b) Find the curvature$\kappa(t)$. (c) Find${\bf T}(1)$,${\bf N}(1)$and${\bf B}(1)$, the principal unit normal vector and the binormal unit vector when$t=1$respectively. 12-5 0.284 0.491 5.420 11 0.633 0.349 Railway tracks cannot have large curvature at any point, otherwise the train might derail. Let$\mathbf{r}(t) = \langle 20t, a(t-t^2) , 5 \rangle$,$a\ne 0$be the vector function describing a track starting at$(0,0,5)$and ending at$(20,0,5)$. (a) Find the unit tangent vector$\vec{T}(t)$and the unit normal vector$\vec{N}(t)$. (b) If the curvature needs to be smaller than 0.001, what is the largest possible value of$a$? 12-6 0.375 0.797 9.950 12 0.984 0.609 (a) Parametrize the curve of intersection of the parabolic cylinders$x=y^2$and$z=x^2$by setting$t=y$. (b) Find the unit tangent \textbf{T} and the curvature$\kappa$at the point$(1,1,1)$. 12-7 0.412 0.613 7.860 12 0.819 0.407 Consider the curve$C:\ x=t^3,\ y=3t,\ z=t^4$. (a) Find the curvature of$C$at the point$(-1,-3,1)$. (b) Is there a point on the curve$C$where the osculating plane is parallel to the plane$x+y+z=1$? 12-8 0.324 0.799 10.040 12 0.960 0.637 Compute (a) the unit tangent$\mathbf{T}\left(\frac{\sqrt{2}}{2}\right)$, (b) the curvature$\kappa\left(\frac{\sqrt{2}}{2}\right)$and (c) the arc length from$t=0$to$t=\frac{\sqrt{2}}{2}$of a space curve$C$parameterized by $\mathbf{r}(t)=\left(\frac{\sqrt{2}}{2}t-\frac{1}{2}t^2\right)\,\mathbf{i}+\left(\frac{\sqrt{2}}{2}t+\frac12t^2\right)\,\mathbf{j}+\frac13t^3\,\mathbf{k}.$ 12-9 0.308 0.827 10.420 12 0.980 0.673 A plane curve$C$is parameterized by$\mathbf{r}(t)=(\cos t+t\sin t, \sin t-t\cos t), t>0$, as Figure. (a) Compute the unit tangent vector$\mathbf{T}(t)$, the unit normal vector$\mathbf{N}(t)$, and the curvature$\kappa(t)$. (b) Show that all centers of osculating circles,$\displaystyle\mathbf{r}(t)+\frac{1}{\kappa(t)}\mathbf{N}(t)$, lie on a circle. 12-10 0.280 0.826 17.090 20 0.966 0.686 Consider the space curve$\textbf{ r}(t)=t\textbf{i}+t^2\textbf{j}+\frac{2t^3}{3}\textbf{k}$. (a) Find the arc length of the curve from$t=0$to$t=a$. (b) Find the curvature$\kappa(0)$at$t=0$. (c) Find the unit tangent$\textbf{T}(0)$at$t=0$. (d) Find the unit normal$\textbf{N}(0)$at$t=0$. 12-11 0.475 0.683 8.710 12 0.921 0.446 Let${\bf r}(t)=(\sin t-t\cos t){\bf i}+(\cos t +t\sin t){\bf j}+t^2{\bf k}$,$0 \leq t \leq \pi $, be a vector function that parametrizes a curve in space. (a) Find the arc length of the curve. (b) At what point on the curve is the osculating plane parallel to the plane$ x+ \sqrt{3} y-z=0$? (c) Find the curvature of the curve. 12-12 0.418 0.343 2.650 8 0.551 0.134 A curve consists of two pieces of curves:$C_1:\ \mathbf{r}(t)=(t^2+3t)\mathbf{i}+(t^3-4t+1)\mathbf{j},\ t\leq 0$,$C_2:\ y=p(x),\ x>0,\ \text{where $p(x)$ is a polynomial of degree 2.}$Find the polynomial$p(x)$so that this curve is continuous and has continuous slope and continuous curvature. 編號 副主題 鑑別度 難易度 平均 總分 高分群答對率 低分群答對率 題目 13-a-1 a. Limit, continuity and differentiability of multi-variable functions 0.406 0.553 8.188 15 0.756 0.350 Let$f(x,y)=\begin{cases} \frac{x^2y}{x^2+y^2},&(x,y)\ne (0,0);\\ 0, &(x,y)=(0,0). \end{cases}$(a) Compute$f_x(0,0)$and$f_y(0,0)$. (b) Calculate$f_x(x,y)$and$f_y(x,y)$for$(x,y)\ne (0,0)$. (c) Are$f_x$and$f_y$continuous at$(0,0)$? (d) Determine$f_{xy}(0,0)$and$f_{yx}(0,0)$if they exist. If they do not exist, explain why. (e) Is$f(x,y)$differentiable at$(0,0)$? 13-a-2 a. Limit, continuity and differentiability of multi-variable functions 0.540 0.657 10.128 15 0.927 0.388 Let$f(x,y)=\begin{cases} \frac{x^2y}{x^2+y^2}&\mbox{ if }(x,y)\ne (0,0)\\ 0&\mbox{ if }(x,y)=(0,0) \end{cases}$(a) Find$\lim\limits_{(x,y)\rightarrow (0,0)}f(x,y)$. Is$f$continuous at$(0,0)$? (b) Find the partial derivative$\frac{\partial f}{\partial x}$at$(x,y)=(0,0)$and at$(x,y)\ne (0,0)$. (c) Is$\frac{\partial f}{\partial x}$continuous at$(0,0)$? 13-a-3 a. Limit, continuity and differentiability of multi-variable functions 0.484 0.648 6.630 10 0.890 0.406 Find the limit, if it exists, or show that it does not exist. (a)$\lim_{(x,y)\rightarrow(1,1)}\frac{xy-x-y+1}{x^2+y^2-2x-2y+2}$. (b)$\lim_{(x,y)\rightarrow(0,0)}\frac{x^3+y^3}{x^2+y^2}$. 13-a-4 a. Limit, continuity and differentiability of multi-variable functions 0.465 0.601 7.260 12 0.833 0.368 Let$f(x,y)=\begin{cases} \sin\frac{xy^2}{x^2+y^4}, &\text{if }(x,y)\ne (0,0)\\ 0, &\text{if }(x,y)=(0,0) \end{cases}$. (a) Compute$f_x$and$f_y$for all$(x,y)$including$(0,0)$. (b) Is$f(x,y)$continuous at$(0,0)$? Is$f(x,y)$differentiable at$(0,0)$? Justify your answers. 13-a-5 a. Limit, continuity and differentiability of multi-variable functions 0.294 0.416 4.910 12 0.563 0.269 Find the limit, if it exists, or show that the limit does not exist. (a)$\lim\limits_{(x,y)\rightarrow (0,0)}\frac{x^5+x^2y^3}{x^4+y^6}$. (b)$\lim\limits_{(x,y,z)\rightarrow (0,0,0)}\frac{e^{xyz}-1}{x^2+y^2+z^2}$. 13-a-6 a. Limit, continuity and differentiability of multi-variable functions 0.468 0.653 6.790 10 0.887 0.420 Let$f(x,y)=\begin{cases} \frac{x^3-y^3}{x^2+y^2}, &\text{for }(x,y)\ne (0,0)\\ 0, &\text{for }(x,y)=(0,0). \end{cases}$(a) Find$f_x(x,y)$and$f_y(x,y)$. (b) Are the functions$f_x$and$f_y$continuous at (0,0)? 13-a-7 a. Limit, continuity and differentiability of multi-variable functions 0.533 0.440 5.160 12 0.706 0.174 Let$f(x,y)=\begin{cases} (x^2+y^2)\sin\frac{1}{x^2+y^2} &\text{if }(x,y)\ne(0,0),\\ 0 &\text{if }(x,y)=(0,0). \end{cases}$(a) Is$f_x$continuous at$(0,0)$? (b) Write down the linear approximation$L(x,y)$of$f$at$(0,0)$. (c) Find the limit$\lim\limits_{(x,y)\rightarrow (0,0)}\frac{f(x,y)-L(x,y)}{\sqrt{x^2+y^2}}. 13-a-8 a. Limit, continuity and differentiability of multi-variable functions 0.461 0.632 6.460 10 0.863 0.402 Consider the function \begin{align*} f(x,y)=\left\{ \begin{array}{cl} \frac{x^2y}{x^4+2y^2} & \text{if } (x,y)\neq(0,0)\2mm] 0 & \text{if } (x,y)=(0,0).\\ \end{array} \right. \end{align*} (a) Find the limit \lim\limits_{(x,y)\to(0,0)}f(x,y) or explain why the limit does not exist. (b) Compute the directional derivative D_{\mathbf{u}}f(0,0), where \mathbf{u}=(\cos\theta,\sin\theta) is any direction. 13-b-1 b. Computing partial derivatives by the chain rule 0.697 0.585 7.645 12 0.933 0.237 Let u=u(x,y) be a function of rectangular coordinates x,y. Then u can be expressed in polar coordinates r,\theta with x=r\cos\theta, y=r\sin\theta. Express \frac{\partial u}{\partial x} and \frac{\partial u}{\partial y} in terms of r, \theta, \frac{\partial u}{\partial r} and \frac{\partial u}{\partial\theta}. 13-b-2 b. Computing partial derivatives by the chain rule 0.565 0.660 7.350 10 0.942 0.377 Suppose that z=f(x,y) is a smooth function and let x=uv, and y=v-u. Express \frac{\partial^2z}{\partial u\partial v} in terms of x,y,f_x,f_y,f_{xx},f_{xy}, and f_{yy}. 13-b-3 b. Computing partial derivatives by the chain rule 0.406 0.606 7.480 12 0.809 0.402 Suppose that x,\ y,\ z satisfy the relation x^2+2y^2+3z^2+xy-z=9. Find \frac{\partial^2 z}{\partial x^2}, \frac{\partial^2 z}{\partial x\partial y} and \frac{\partial^2 z}{\partial y^2}. 13-b-4 b. Computing partial derivatives by the chain rule 0.605 0.645 9.660 14 0.947 0.342 Consider a function z=z(x,y) satisfying the partial differential equation: \[ 6\,\frac{\partial^2 z}{\partial x^2}+\frac{\partial^2 z}{\partial x\partial y}-\frac{\partial^2 z}{\partial y^2}=0. Suppose thatu=x-2y$and$v=x+ay$will change the partial differential equation into$\frac{\partial^2 z}{\partial u\partial v}=0$. Find the constant$a$. 13-b-5 b. Computing partial derivatives by the chain rule 0.573 0.456 5.130 11 0.742 0.169 Let$z=f(x,y)$and$x=r\cos\theta$,$y=r\sin\theta$. (a) Express$\frac{\partial z}{\partial x}$in terms of$r$,$\theta$and partial derivatives with respect$r,\theta$. (b) Express$\frac{\partial^2z}{\partial x^2}$in terms of$r$,$\theta$and partial derivatives with respect$r,\theta$. 13-c-1 c. Tangent planes, linear approximation, directional derivatives, and the gradient vector 0.442 0.653 10.225 15 0.874 0.432 Let$f(x,y,z)=e^{xy}\ln z$. Find the directional derivatives of$f$at$P(1,0,e)$in the following directions. (a) In the direction in which$f$increases most rapidly at$P$. (b) In the directions parallel to the line in which the planes$x+y-z=2$and$4x-y-z=1$intersect. (c) In the direction of increasing$t$along the path $\textbf{r}(t)=\sqrt{1+t^2}\textbf{i}+\tan t\textbf{j}+e^{2t+1}\textbf{k}$ 13-c-2 c. Tangent planes, linear approximation, directional derivatives, and the gradient vector 0.433 0.756 12.324 15 0.973 0.540 Let$f(x,y)=xe^y+\cos(xy)$. (a) Find the direction (a unit vector$\textbf{u}$) in which$f(x,y)$increases most rapidly at$(2,0)$(that is,$f'_{\textbf{u}}(2,0)$is maximal). (b) Find the direction in which$f(x,y)$decreases most rapidly at$(2,0)$. (c) What are the directions of zero change in$f$at$(2,0)$. 13-c-3 c. Tangent planes, linear approximation, directional derivatives, and the gradient vector 0.801 0.564 13.300 20 0.965 0.164 Let$f(x, y)=\int_1^{2y-x^2}e^{t^2}dt$. (a) Find the rate of change of$f$at the point$P(1,1)$in the direction from$P$to$Q(6,13)$. (b) In what direction does$f$have the maximum rate of change? What is this rate of change. (c) Find the tangent plane and the normal line to the surface$S:\ z=f(x,y)$at the point$(1,1,0)$. (d) The sphere$x^2+y^2+z^2=2$intersects$S$in a curve$C$. Find the equations for the tangent line to$C$at the point$(1,1,0)$. 13-c-4 c. Tangent planes, linear approximation, directional derivatives, and the gradient vector 0.479 0.730 11.840 15 0.969 0.490 Let$f(x,y)=\sin(x-y)e^{-x^2-y^2}$,$P=(\sqrt{2},\sqrt{2})$. (a) Find the maximum rate of change of$f$at$P$. (b) Find the direction in which the maximum rate of change occurs. (c) Find the directional derivative$D_{{\bf u}}(P)$, where${\bf u}=\left(\tfrac{1}{2},\tfrac{\sqrt{3}}{2}\right)$. 13-c-5 c. Tangent planes, linear approximation, directional derivatives, and the gradient vector 0.647 0.559 7.130 12 0.882 0.235 Consider the level surface$z^2+z\tan^{-1}\frac{y}{x}=\frac{\pi}{4}+1$. (a) Find the tangent plane for the level surface at$(1,1,1)$. (b) The level surface defines$z$implicitly as a function of$x$and$y$,$z=f(x,y)$. Compute the directional derivative of$f(x,y)$at$(1,1)$in the direction of$-3\vec{i}+4\vec{j}$. (c) Use linear approximation to estimate the value of$f(0.98,1.04)$. 13-c-6 c. Tangent planes, linear approximation, directional derivatives, and the gradient vector 0.618 0.516 4.440 8 0.825 0.207 Find the tangent plane of the surface $\frac{4}{\pi}\arctan\frac{z}{2}=x^2+\int_{xy}^zxy\sqrt{1+t^3}dt$ at the point$(1,2,2)$. 13-c-7 c. Tangent planes, linear approximation, directional derivatives, and the gradient vector 0.583 0.651 5.310 8 0.942 0.359 Find all points at which the direction of fastest change of the function$f(x,y,z)=x^2+y^2+z^2-2x-4y-6z$is$\mathbf{i}+2\mathbf{j}+3\mathbf{k}$. 13-c-8 c. Tangent planes, linear approximation, directional derivatives, and the gradient vector 0.560 0.639 7.920 12 0.919 0.359 Let the unit vectors${\bf u}$and${\bf n}$be respectively the tangent direction and the normal direction (with positive$x$-components) of the circle$x^2+y^2-2x=0$at the point$\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$. Let$f(x,y)=\tan^{-1}\left(\frac{y}{x}\right)$. Find$\nabla f\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$,$D_{\textbf{u}}f\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$and$D_{\textbf{n}}f\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$. 13-c-9 c. Tangent planes, linear approximation, directional derivatives, and the gradient vector 0.347 0.802 8.410 10 0.976 0.629 Find the equation of the tangent plane to the elliptic paraboloid$\frac{z}{c}=\frac{x^2}{a^2}+\frac{y^2}{b^2}$at the point$(a,b,2c)$. 13-c-10 c. Tangent planes, linear approximation, directional derivatives, and the gradient vector 0.640 0.451 5.810 12 0.771 0.131 Let$f(x,y)=\begin{cases} \frac{\sin(x^3)-\sin(y^3)}{x^2+y^2} &\text{if }(x,y)\ne(0,0),\\ 0 &\text{if }(x,y)=(0,0). \end{cases}$(a) Calculate$\nabla f(0,0)$. (b) Use the definition of directional derivative to calculate$D_{\mathbf{u}}f(0,0)$, where$\mathbf{u}=\frac{1}{\sqrt{2}}(\mathbf{i}-\mathbf{j})$. (c) Is$f(x,y)$differentiable at$(0,0)$? 13-c-11 c. Tangent planes, linear approximation, directional derivatives, and the gradient vector 0.482 0.659 7.990 12 0.900 0.418 A model of the monkey of fortune and prosperity (福祿猴) is given by the following implicit function:$\begin{align*} F(x,y,z)=\left(x^2+\frac34y^2+z^2-4\right)\Big(x^2+(y+1)^2+(z-3)^2-1\Big)\\ -\frac{5}{16}=0. \end{align*} $(a) Find the equation of the tangent plane to the surface at$P\left(\frac12,0,2\right)$. (b) The surface and the plane$y=0$intersect a curve. Find the parametric equations for the tangent line to the curve at$P\left(\frac12,0,2\right). 13-c-12 c. Tangent planes, linear approximation, directional derivatives, and the gradient vector 0.461 0.632 6.460 10 0.863 0.402 Consider the function \begin{align*} f(x,y)=\left\{ \begin{array}{cl} \frac{x^2y}{x^4+2y^2} & \text{if } (x,y)\neq(0,0)\2mm] 0 & \text{if } (x,y)=(0,0).\\ \end{array} \right. \end{align*} (a) Find the limit \lim\limits_{(x,y)\to(0,0)}f(x,y) or explain why the limit does not exist. (b) Compute the directional derivative D_{\mathbf{u}}f(0,0), where \mathbf{u}=(\cos\theta,\sin\theta) is any direction. 13-c-13 c. Tangent planes, linear approximation, directional derivatives, and the gradient vector 0.442 0.741 8.020 10 0.962 0.520 A differentiable function f(x,y) has the following properties: f(0,0)=1. D_{\mathbf{u}}f(0,0)=2, where \mathbf{u}=\left(\frac35,\frac45\right). D_{\mathbf{v}}f(0,0)=\frac{3}{\sqrt{2}}, where \mathbf{v}=\left(\frac1{\sqrt{2}},\frac1{\sqrt{2}}\right). (a) What is the maximal rate of increase of f(x,y) at (0,0)? (b) Use the linearization of f(x,y) at (0,0) to estimate f(0.07,-0.05). 13-c-14 c. Tangent planes, linear approximation, directional derivatives, and the gradient vector 0.476 0.643 13.320 20 0.881 0.405 Let f(x,y,z)=(x^2+z^2)\sin\frac{\pi x y}{2} + y z^2 and a point {\bf p} = (1,1,-1). Answer the following: (a) Find the gradient of f at {\bf p}. (b) Find the approximate value of f(0.98,1.02,-0.97). (c) Find the plane tangent to the level surface through {\bf p} defined by f(x,y,z)=f({\bf p})=3. (d) If a bird flies through {\bf p} directly to the point (2,-1,1) with speed 5, what is the rate of change of f as seen by the bird at {\bf p}? 13-c-15 c. Tangent planes, linear approximation, directional derivatives, and the gradient vector 0.453 0.440 4.460 10 0.667 0.214 Let surface S be given by S=\{(x,y,z)\in\mathbb{R}^3|\sin(xyz)=x+2y+3z\}. (a) On the surface, compute \frac{\partial z}{\partial x} and \frac{\partial y}{\partial x}. (b) Find an equation of the tangent plane to the surface S at (2,-1,0). (c) Suppose, when restricted to the surface S, a differentiable function f attains a local maximum value at the point (2,-1,0) with f(2, -1, 0)=10 and f_x(2, -1, 0) = 2. Let (x_0,y_0,z_0) be a point which is close to the point (2,-1,0) and lies on another surface \sin(xyz) = z+2y+3z+10^{-2}. Use the linear approximation to estimate f(x_0, y_0, z_0). 13-c-16 c. Tangent planes, linear approximation, directional derivatives, and the gradient vector 0.491 0.479 6.230 13 0.724 0.233 Let f(x,y)= \begin{cases} \frac{x^3+y^3}{x^2+y^2}~~~~&\text{if} ~~(x,y)\neq(0,0).\\ 0 &\text{if} ~~(x,y)=(0,0). \end{cases} (a) Is f(x,y) continuous at (0,0)? Justify your answer. (b) Find the gradient vector \nabla f(0,0). (c) Is f_x(x,y) continuous at (0,0)? Justify your answer. (d) Find the maximum and minimum directional derivatives of f at the point (0,0) among the directions of all the unit vectors \bf{u}. 13-d-1 d. Extreme values 0.416 0.536 8.488 15 0.744 0.328 Suppose f(x,y) = x^2 + c x y + 2 y^2 where c is a constant. (a) Find all values of c such that (0,0) is a stationary point of f. (b) Find all values of c such that (0,0) is a saddle point of f. (c) Find all values of c such that f has a local minimum at (0,0). (d) Find all values of c and all (x_0,y_0)\ne (0,0) such that f has a local minimum at (x_0,y_0). 13-d-2 d. Extreme values 0.291 0.820 8.516 10 0.966 0.675 Find stationary points of f=3xy-x^3-y^3+2. Determine which are local maximum, local minimum or a saddle point. 13-d-3 d. Extreme values 0.436 0.607 6.220 10 0.824 0.389 Let f(x,y)=\sin x\cos (x+y) and D=\left\{(x,y)|0\leq x\leq\tfrac{\pi}{2},0\leq y\leq\tfrac{\pi}{2}\right\}. Classify all the critical points of f on D. 13-d-4 d. Extreme values 0.484 0.561 5.930 10 0.803 0.319 Find the maximum and minimum values of xy+z^2 on the ball x^2+y^2+\left(z-\frac{1}{2}\right)^2\leq 1. 13-d-5 d. Extreme values 0.329 0.737 7.800 10 0.902 0.573 Find the local maximum, and local minimum values and saddle point(s) of f(x,y)=y^3+3x^2y-3x^2-3y^2+3. 13-d-6 d. Extreme values 0.309 0.705 8.740 12 0.860 0.551 Find and classify all critical points of f(x,y)=4x^3+2xy^2+\frac{2}{3}y^3+6x^2. Reminder: each critical point must be shown to be either a local maximal point, a local minimal point, or neither of the above. 13-d-7 d. Extreme values 0.432 0.552 5.600 10 0.769 0.336 Let g(x,y)=4x^3-13y^3+6x^2y+3xy^2-12x^2-12xy-30y^2. Find the critical points of g(x,y), and classify them. 13-d-8 d. Extreme values 0.293 0.848 10.730 12 0.994 0.701 Find all the critical points of f(x,y)=4+x^3+y^3-3xy. Then determine which gives a local maximum or a local minimum or a saddle point. 13-d-9 d. Extreme values 0.410 0.717 9.160 12 0.922 0.513 Find all critical points of the function f(x,y)=xye^{-x^2-y^2} and classify them. 13-d-10 d. Extreme values 0.219 0.864 14.290 16 0.974 0.755 Let f(x,y)=y^3+3x^2y-6x^2-6y^2+1. (a) Find all critical points for f(x,y). (b) Determine whether they are local maximum, local minimum, or saddle points. 13-d-11 d. Extreme values 0.278 0.783 8.260 10 0.923 0.644 Find the local maximum and minimum values and saddle point(s) of the function \[ f(x,y)=x^3-y^3+3x^2+3y^2-9x. 13-d-12 d. Extreme values 0.489 0.660 8.520 12 0.905 0.415 Find the local extreme values and saddle points off(x,y)=x^2y-xy^2+xy-y^2$. 13-d-13 d. Extreme values 0.386 0.637 7.980 12 0.830 0.444 Let$f(x, y)= 1+3x^2-2x^3+3y-y^3$. (a) Find the local maximum and minimum values and saddle point(s) of$f(x,y)$. (b) Find the extreme values of$f(x, y)$on the region$D$bounded by the triangle with vertices$(-2,2)$,$(2,2)$and$(2,-2)$. 13-e-1 e. Lagrange multipliers 0.656 0.665 11.444 15 0.993 0.337 A rectangular box has three of its faces on the coordinate planes and one vertex in the first octant on the paraboloid$z = 4 - 5 x^2 - 6 y^2$. Determine the maximum volume of the box. 13-e-2 e. Lagrange multipliers 0.415 0.567 5.799 10 0.775 0.360 Use Lagrange multiplier to find the maximum and the minimum of$f(x,y)=3x^2-2y^2$for$x$,$y$on the curve$2x^2-2xy+y^2=1$. (You don't have to give the locations of these extrema.) 13-e-3 e. Lagrange multipliers 0.484 0.561 5.930 10 0.803 0.319 Find the maximum and minimum values of$xy+z^2$on the ball$x^2+y^2+\left(z-\frac{1}{2}\right)^2\leq 1$. 13-e-4 e. Lagrange multipliers 0.392 0.509 8.230 15 0.705 0.313 Find the extreme values of$f(x,y)=x^2y-xy+xy^2$on$x^2+xy+y^2-x-y=1$. 13-e-5 e. Lagrange multipliers 0.494 0.411 6.250 15 0.658 0.164 Consider the part of an elliptic paraboloid defined by$z=\frac{x^2}{16}+\frac{y^2}{8}$,$z\leq 6$. Find the points on the surface segment which are respectively the farthest from and the closest to the point$(0,0,8)$. 13-e-6 e. Lagrange multipliers 0.589 0.585 5.920 10 0.879 0.290 Find the points on the intersection of the plane$x+y+2z=2$and the paraboloid$z=x^2+y^2$that are nearest to and farthest from the origin. 13-e-7 e. Lagrange multipliers 0.482 0.680 11.180 15 0.921 0.439 Find the points on the surface$xy^2z^3 =2$that are closest to the origin and also the shortest distance between the surface and the origin. 13-e-8 e. Lagrange multipliers 0.350 0.357 4.330 12 0.532 0.182 Among all planes that are tangent to the surface$x^2yz=1$, are there the ones that are nearest or farthest from the origin? Find such tangent planes if they exist. 13-e-9 e. Lagrange multipliers 0.539 0.627 9.430 14 0.897 0.358 Find the points on$-x^2+2y^2+2z^2=1$that are closest to the point$(0,2,2)$. 13-e-10 e. Lagrange multipliers 0.550 0.529 7.660 14 0.804 0.254 {\it Viviani's curve}, sometimes also called {\it Viviani's window}, is the intersection of the cylinder$(x-1)^2+y^2=1$and the sphere$x^2+y^2+z^2=4$, as Figure. (a) Find the tangent line equation of the Viviani's curve at$P(1,1,\sqrt{2})$. (b) Find the points on the Viviani's curve that are nearest to and farthest from$Q(2,0,2)$. 13-e-11 e. Lagrange multipliers 0.370 0.742 9.340 12 0.927 0.557 Find the maximum and the minimum of the function$f(x,y)=3x^2-2y^2$on the curve$2x^2-2xy+y^2=1$. 13-e-12 e. Lagrange multipliers 0.286 0.485 5.130 10 0.628 0.342 By the Extreme Value Theorem, a continuous function on a sphere attains both absolute maximum and minimum values. Find the extreme values of$f(x,y,z)=\ln(x+2)+\ln(y+2)+\ln(z+2)$on the sphere$x^2+y^2+z^2=3$. 13-e-13 e. Lagrange multipliers 0.453 0.440 4.460 10 0.667 0.214 Let surface$S$be given by$S=\{(x,y,z)\in\mathbb{R}^3|\sin(xyz)=x+2y+3z\}$. (a) On the surface, compute$\frac{\partial z}{\partial x}$and$ \frac{\partial y}{\partial x}$. (b) Find an equation of the tangent plane to the surface$S$at$(2,-1,0)$. (c) Suppose, when restricted to the surface$S$, a differentiable function$f$attains a local maximum value at the point$(2,-1,0)$with$f(2, -1, 0)=10$and$f_x(2, -1, 0) = 2$. Let$(x_0,y_0,z_0)$be a point which is close to the point$(2,-1,0)$and lies on another surface$\sin(xyz) = z+2y+3z+10^{-2}$. Use the linear approximation to estimate$f(x_0, y_0, z_0)$. 編號 副主題 鑑別度 難易度 平均 總分 高分群答對率 低分群答對率 題目 14-a-1 a. Double integrals (as iterated integrals) 0.293 0.832 8.701 10 0.978 0.685 Evaluate${\displaystyle \int_0^2 \int_{x^3}^8 \frac{x^5}{\sqrt{x^6+y^2}} dy dx. }$14-a-2 a. Double integrals (as iterated integrals) 0.543 0.343 3.350 10 0.615 0.072 Evaluate$\int_0^{\infty}\frac{\tan^{-1}\pi x-\tan^{-1}x}{x}\,dx$. [Hint. Express it as an iterated integral.] 14-a-3 a. Double integrals (as iterated integrals) 0.273 0.832 8.560 10 0.968 0.695 Sketch the region of integration and evaluate the integral$\int_0^3\int_{\sqrt{x+1}}^2 e^{\frac{x}{y+1}}\,dydx$. 14-a-4 a. Double integrals (as iterated integrals) 0.577 0.615 7.840 12 0.904 0.327 (a) Evaluate the iterated integral$ \int_{0}^{8} \int_{\displaystyle x^{\frac{1}{3}}}^{2} \frac{1}{1 + y^{4}} \; dy \; dx. $(b) Evaluate the double integral$ \iint_{{\cal R}} (x^{2}+y^{2}) \;dA, $where${\cal R}$is the region bounded by the ellipse$9x^{2}+y^{2} = 36$. 14-a-5 a. Double integrals (as iterated integrals) 0.466 0.562 8.710 15 0.795 0.329 Evaluate the integrals. (a)$\displaystyle\int_0^1\int_{\tan^{-1} y}^{\frac{\pi}{4}}\cos x \cdot\tan(\cos x)dxdy$. (b)$\displaystyle\int_1^{\sqrt{2}}\int_0^{\sqrt{2-y^2}}\frac{x+y}{x^2+y^2}dxdy +\int_0^1\int_{1-y}^1\frac{x+y}{x^2+y^2}dxdy+\int_1^{\sqrt{2}}\int_0^{\sqrt{2-x^2}}\frac{x+y}{x^2+y^2}dydx$. 14-a-6 a. Double integrals (as iterated integrals) 0.428 0.695 8.330 12 0.909 0.481 Evaluate$\displaystyle\iint\limits_Dy\sin\left(\frac{\pi x}{y}\right)dA$, where$D$is the region bounded by$y=x$and$y=\sqrt{x}$. 14-a-7 a. Double integrals (as iterated integrals) 0.365 0.772 8.270 10 0.954 0.589 Evaluate the double integral$\int_1^2\int_{\sqrt{x}}^x\frac{\sin y}{y}dydx +\int_2^4\int_{\sqrt{x}}^2\frac{\sin y}{y}dydx$14-a-8 a. Double integrals (as iterated integrals) 0.282 0.839 8.820 10 0.980 0.698 Evaluate the iterated integral $$\int_{0}^{1}\int_{x}^{\sqrt{x}}\frac{\sin y}{y}\,\mathrm{d}y\,\mathrm{d}x.$$ 14-a-9 a. Double integrals (as iterated integrals) 0.245 0.864 10.580 12 0.986 0.741 Evaluate the following integrals. (a)$\displaystyle\int_0^1\!\!\int_{y}^1\tan(x^2)\,\mathrm{d}x\,\mathrm{d}y$. (b)$\displaystyle\int_{\frac{1}{\sqrt{2}}}^1\int_{\sqrt{1-x^2}}^x1\,\mathrm{d}y\,\mathrm{d}x+\int_1^{\sqrt{2}}\!\!\int_0^x1\,\mathrm{d}y\,\mathrm{d}x +\int_{\sqrt{2}}^2\int_0^{\sqrt{4-x^2}}1\,\mathrm{d}y\,\mathrm{d}x$. 14-a-10 a. Double integrals (as iterated integrals) 0.319 0.811 9.090 11 0.971 0.651 Sketch the region of integration and evaluate the integral$\displaystyle\int_0^8\int_{\sqrt{1+x}}^3\cos\left(\frac{x}{y+1}\right)dydx$. 14-a-11 a. Double integrals (as iterated integrals) 0.400 0.734 15.030 20 0.933 0.534 Evaluate the integrals. (a)$\int_0^1\int_{\sqrt{y}}^{2\sqrt{y}}e^{x^3} \,dx \,dy +\int_1^4\int_{\sqrt{y}}^2e^{x^3} \,dx \,dy$. (b)$\iint_{R}\cos \left(\frac{y-2x}{y+x}\right) \,dA$, where$R$is the trapezoidal region with vertices (1,0),(2,0),(0,2) and (0,1). 14-b-1 b. Triple integrals (as iterated integrals) 0.296 0.808 8.120 10 0.956 0.660 Evaluate the triple integral$\iiint_{E} xyz\,dV$with $$E=\left\{0 \le x \le \sqrt{4-y^2},\,0 \le y \le 2,\,\sqrt{x^2 + y^2} \le z \le \sqrt{8 - x^2 -y^2}\right\}.$$ 14-b-2 b. Triple integrals (as iterated integrals) 0.459 0.629 9.310 14 0.859 0.400 (a) Find the volume of the region${\cal R}$, where${\cal R}$is bounded by the surfaces$z=0$,$x=0$,$x=y^2$, and$z=1-y^2$. (b) Let${\cal R}$be the part of the solid ball$x^2+y^2+z^2\leq 1$which is above the cone$z=\sqrt{\frac{x^2+y^2}{3}}$. Evaluate$\iiint_{\cal R} (x^2+y^2)dV.$14-b-3 b. Triple integrals (as iterated integrals) 0.312 0.639 6.460 10 0.795 0.483 Write the integral$\displaystyle\int_0^1\int_{\sqrt{x}}^1 \int_0^{1-y}f(x,y,z)dzdydx$in 5 other orders. (a)$\displaystyle\int\int\int f(x,y,z)dzdxdy$(b)$\displaystyle\int\int\int f(x,y,z)dxdydz$(c)$\displaystyle\int\int\int f(x,y,z)dxdzdy$(d)$\displaystyle\int\int\int f(x,y,z)dydzdx$(e)$\displaystyle\int\int\int f(x,y,z)dydxdz$14-b-4 b. Triple integrals (as iterated integrals) 0.450 0.685 8.200 12 0.910 0.460 Evaluate the following integrals. (a)$\displaystyle\iiint_Ey\cos((y-z)^2)\,\mathrm{d}V$, where$E$is the solid tetrahedron bounded by four planes$x=1, y=1,z=0$, and$x+y-z=1$, as Figure. (b)$\displaystyle\int_0^{\frac1{\sqrt{2}}}\!\int_x^{\sqrt{1-x^2}}\!\int_{2\sqrt{x^2+y^2}}^{1+x^2+y^2}1\,\mathrm{d}z\,\mathrm{d}y\,\mathrm{d}x$. 14-b-5 b. Triple integrals (as iterated integrals) 0.384 0.736 12.510 16 0.928 0.544 Evaluate the integrals. (a)$\int_0^2 \int_0^1 \int_y^1 e^{-z^2} \,dz \,dy \,dx$. (b)$\iiint_E x^2 dV$, where$E$is the solid that lies in the first octant within the cylinder$x^2+y^2=1$and below the cone$z^2=4x^2+4y^2$. 14-b-6 b. Triple integrals (as iterated integrals) 0.328 0.704 7.190 10 0.868 0.541 Let$E$be the tetrahedron bounded by the planes$x+y+z=3$,$x=2z$,$y=0$, and$z=0$which is completely occupied by a solid with the density function$\rho(x,y,z)=y$. Find the total mass of the solid. 14-c-1 c. Change of variables in multiple integrals - (i) Double integrals in polar coordinates 0.616 0.554 6.855 12 0.862 0.246 Evaluate$\iint\limits_{\Omega}e^{-4x^2-9y^2}dxdy$, where$\Omega$is the region satisfying$2x\leq 3y$and$x\geq 0$. 14-c-2 c. Change of variables in multiple integrals - (i) Double integrals in polar coordinates 0.543 0.605 6.120 10 0.876 0.333 Evaluate$\int_{\tan^{-1}2}^{\frac{\pi}{2}}\int_0^{\frac{3}{\cos\theta+\sin\theta}}r^3\cos\theta\sin\theta\,drd\theta$. 14-c-3 c. Change of variables in multiple integrals - (i) Double integrals in polar coordinates 0.466 0.562 8.710 15 0.795 0.329 Evaluate the integrals. (a)$\displaystyle\int_0^1\int_{\tan^{-1} y}^{\frac{\pi}{4}}\cos x \cdot\tan(\cos x)dxdy$. (b)$\displaystyle\int_1^{\sqrt{2}}\int_0^{\sqrt{2-y^2}}\frac{x+y}{x^2+y^2}dxdy +\int_0^1\int_{1-y}^1\frac{x+y}{x^2+y^2}dxdy+\int_1^{\sqrt{2}}\int_0^{\sqrt{2-x^2}}\frac{x+y}{x^2+y^2}dydx$. 14-c-4 c. Change of variables in multiple integrals - (i) Double integrals in polar coordinates 0.476 0.694 7.270 10 0.932 0.456 Evaluate the double integral$\displaystyle\iint_R\mathrm{e}^{3x^2+y^2}\,\mathrm{d}A$, where$R$is the region inside the ellipse$3x^2+y^2=1$and above the lines$y=x$and$y=-\sqrt{3}\,x$, as Figure. 14-c-5 c. Change of variables in multiple integrals - (ii) Triple integrals in cylindrical coordinates 0.484 0.676 10.189 15 0.918 0.434 Find the volume of the solid common to the balls$\rho\leq 2\sqrt{2}\cos\phi$and$\rho\leq2$. 14-c-6 c. Change of variables in multiple integrals - (ii) Triple integrals in cylindrical coordinates 0.431 0.648 6.675 10 0.863 0.433 Find the volume of the solid bounded below by the cone$z^2 = 4(x^2 + y^2)$and above by the ellipsoid$4(x^2 + y^2)+z^2 = 8$. 14-c-7 c. Change of variables in multiple integrals - (ii) Triple integrals in cylindrical coordinates 0.296 0.808 8.120 10 0.956 0.660 Evaluate the triple integral$\iiint_{E} xyz\,dV$with $$E=\left\{0 \le x \le \sqrt{4-y^2},\,0 \le y \le 2,\,\sqrt{x^2 + y^2} \le z \le \sqrt{8 - x^2 -y^2}\right\}.$$ 14-c-8 c. Change of variables in multiple integrals - (ii) Triple integrals in cylindrical coordinates 0.384 0.736 12.510 16 0.928 0.544 Evaluate the integrals. (a)$\int_0^2 \int_0^1 \int_y^1 e^{-z^2} \,dz \,dy \,dx$. (b)$\iiint_E x^2 dV$, where$E$is the solid that lies in the first octant within the cylinder$x^2+y^2=1$and below the cone$z^2=4x^2+4y^2$. 14-c-9 c. Change of variables in multiple integrals - (iii) Triple integrals in spherical coordinates 0.366 0.791 8.380 10 0.974 0.608 Evaluate$\iiint\limits_{B}(x^2+y^2+z^2)^2\,dV$, where$B$is the ball with center the origin and radius$1$. 14-c-10 c. Change of variables in multiple integrals - (iii) Triple integrals in spherical coordinates 0.459 0.629 9.310 14 0.859 0.400 (a) Find the volume of the region${\cal R}$, where${\cal R}$is bounded by the surfaces$z=0$,$x=0$,$x=y^2$, and$z=1-y^2$. (b) Let${\cal R}$be the part of the solid ball$x^2+y^2+z^2\leq 1$which is above the cone$z=\sqrt{\frac{x^2+y^2}{3}}$. Evaluate$\iiint_{\cal R} (x^2+y^2)dV.$14-c-11 c. Change of variables in multiple integrals - (iii) Triple integrals in spherical coordinates 0.404 0.325 3.300 10 0.527 0.123 Let$S$be the surface of the solid bounded by$x^2+y^2+z^2=1$and$z\geq \frac{1}{2}$. Find the total flux of$\textbf{F}(x,y,z)=x^2\textbf{i}+y^2\textbf{j}+z^2\textbf{k}$across$S$. 14-c-12 c. Change of variables in multiple integrals - (iii) Triple integrals in spherical coordinates 0.554 0.702 7.440 10 0.979 0.425 Find the volume of the cherry, which is enclosed by the spherical coordinate surface$\rho=1-\cos\phi$. 14-c-13 c. Change of variables in multiple integrals - (iv) General 0.462 0.731 11.324 15 0.962 0.500 Evaluate the integral$\iint\limits_{\Omega}\sin(3x^2-2xy+3y^2)dxdy$, where$\Omega$is the ellipse$3x^2-2xy+3y^2\leq 2$. You may try the change of variables$x=u+kv$,$y=u-kv$for some constant$k$. 14-c-14 c. Change of variables in multiple integrals - (iv) General 0.327 0.295 2.860 10 0.459 0.131 Let$E$be the tetrahedron bounded by planes$-x+y+z=0$,$x-y+z=0$,$x+y-z=0$, and$-x+5y+7z=6$. (a) Find the volume of$E$. (b) Evaluate$\iiint\limits_{E}z\,dV$. 14-c-15 c. Change of variables in multiple integrals - (iv) General 0.211 0.882 9.010 10 0.988 0.777 Evaluate$\iint\limits_{A}e^{xy}\,dxdy$, where$A$is the region enclosed by$xy=1$,$xy=4$,$y=1$and$y=3$. 14-c-16 c. Change of variables in multiple integrals - (iv) General 0.577 0.615 7.840 12 0.904 0.327 (a) Evaluate the iterated integral$ \int_{0}^{8} \int_{\displaystyle x^{\frac{1}{3}}}^{2} \frac{1}{1 + y^{4}} \; dy \; dx. $(b) Evaluate the double integral$ \iint_{{\cal R}} (x^{2}+y^{2}) \;dA, $where${\cal R}$is the region bounded by the ellipse$9x^{2}+y^{2} = 36$. 14-c-17 c. Change of variables in multiple integrals - (iv) General 0.663 0.562 7.220 12 0.893 0.230 A lamina with constant density$\rho$occupies the region${\cal R} := \left\{(x,y)|1 \leq x^{\frac 53}y \leq 9 \: , \: 1 \leq xy \leq 4, \: x>0\right\} \; . $Find the coordinates$(\bar x , \bar y)$of its center of mass. 14-c-18 c. Change of variables in multiple integrals - (iv) General 0.456 0.705 8.860 12 0.933 0.477 Evaluate the double integral$\iint\limits_R(x+y)^2\sin^2(x-y)\,dA$, where$R$is the square region with vertices$(\frac\pi2, 0)$,$ (\pi,\frac\pi2)$,$(\frac\pi2,\pi)$, and$(0,\frac\pi2)$. 14-c-19 c. Change of variables in multiple integrals - (iv) General 0.703 0.524 5.370 10 0.875 0.172 Find the area of the region in the first quadrant enclosed by the curves$xy=a$,$xy=b$,$xy^{1.4}=c$and$xy^{1.4}=d$where$0<a < b$and$0< c < d$. 14-c-20 c. Change of variables in multiple integrals - (iv) General 0.497 0.463 4.480 10 0.712 0.215 Find the mass of the solid$S$bounded by the paraboloid$z=x^2+2y^2$and the plane$z=2+4y$if$S$has density function$\rho(x,y,z)=|x|$. 14-c-21 c. Change of variables in multiple integrals - (iv) General 0.515 0.689 9.090 13 0.946 0.431 Evaluate the double integral$\iint_R\sqrt{\frac{y}x}\,\mathrm{e}^{\sqrt{xy}}\,\mathrm{d}A$, where$R$is the region bounded by$xy=1$,$xy=4$,$y=x$, and$y=2x$in the first quadrant. 14-c-22 c. Change of variables in multiple integrals - (iv) General 0.323 0.783 9.630 12 0.945 0.622 Evaluate$\displaystyle\iint_Re^{\frac{x+2y}{2x-y}}dA$, where$R$is the region in the$xy$-plane bounded by the four lines$2x-y=1$,$2x-y=2$,$x-3y=0$and$3x-4y=0$. 14-c-23 c. Change of variables in multiple integrals - (iv) General 0.400 0.734 15.030 20 0.933 0.534 Evaluate the integrals. (a)$\int_0^1\int_{\sqrt{y}}^{2\sqrt{y}}e^{x^3} \,dx \,dy +\int_1^4\int_{\sqrt{y}}^2e^{x^3} \,dx \,dy$. (b)$\iint_{R}\cos \left(\frac{y-2x}{y+x}\right) \,dA$, where$R$is the trapezoidal region with vertices (1,0),(2,0),(0,2) and (0,1). 14-d-1 d. Applications of multiple integrals 0.663 0.562 7.220 12 0.893 0.230 A lamina with constant density$\rho$occupies the region${\cal R} := \left\{(x,y)|1 \leq x^{\frac 53}y \leq 9 \: , \: 1 \leq xy \leq 4, \: x>0\right\} \; . $Find the coordinates$(\bar x , \bar y)$of its center of mass. 編號 副主題 鑑別度 難易度 平均 總分 高分群答對率 低分群答對率 題目 15-a-1 a. Line integrals - (i) By parameterizing curves 0.475 0.699 10.968 15 0.937 0.462 For$y>0, let \begin{align*} \textbf{F}(x,y,z)&=(e^{-x}\ln y-z)\textbf{i}+\left(2yz-e^{-x}/y\right)\textbf{j}+(y^2-x)\textbf{k}\mbox{ and}\\ \textbf{G}(x,y,z)&=e^{-x}\ln y\,\textbf{i}+\left(2yz-e^{-x}/y\right)\textbf{j}-x~\textbf{k}. \end{align*} 1. Show that the vector function\textbf{F}$is a gradient on$\{(x,y,z)|~y>0\}$by finding an$f$such that$\nabla f=\textbf{F}$. 2. Evaluate the line integral$\displaystyle\int_C\textbf{G}(\textbf{r})\cdot \textbf{dr}$, where$C$is the curve given by$\textbf{r}(u)=(1+u^2)\textbf{i}+e^u\textbf{j}+(1+u)\textbf{k}$,$u\in[0,1]$. 15-a-2 a. Line integrals - (i) By parameterizing curves 0.548 0.355 4.185 12 0.629 0.081 Find the line integral $\int_{{\cal C}}\,(2 x \sin(\pi y) - e^{z})\,dx + (\pi x^2 \cos(\pi y) - 3 e^{z})\,dy - x e^z\,dz$$\mbox{along the curve}\;\;{\cal C} = \left\{(x, y, z) | z = \ln \sqrt{1 + x^2},\; y = x,\; 0 \le x \le 1 \right\}. $15-a-3 a. Line integrals - (i) By parameterizing curves 0.398 0.417 4.170 10 0.616 0.218 Let$C$be the upper half of the curve$(x^2+y^2)^2=x^2-y^2$. Evaluate$\displaystyle\int_{C}y\,ds$. 15-a-4 a. Line integrals - (i) By parameterizing curves 0.432 0.746 11.570 15 0.962 0.531 1. Find a scalar function$f(x,y,z)$such that$\nabla f=\sin y\,{\bf i}+x\cos y\,{\bf j}-\sin z\,{\bf k}$. 2. Find the line integral$\displaystyle\int_C\sin y\,dx+x\cos y\,dy+(y-\sin z)dz$, where$C:\ {\bf r}(t)=\left\langle t,\frac{\pi}{2}\cos t,\frac{\pi}{2}\sin t\right\rangle$,$0\leq t\leq \pi$. 15-a-5 a. Line integrals - (i) By parameterizing curves 0.605 0.439 6.280 16 0.741 0.136 Let$S$be a toroidal shell (not a solid torus!) obtained by rotating about the$z$-axis the circle$C$in the plane$y=0$of radius$a$centred at$(b,0,0)$, where$0 < a < b\:$. It can be parametrized by$ \textbf r(\theta, \alpha) = \left\langle\:(b +a\cos\alpha) \cos\theta , (b +a\cos\alpha) \sin\theta , a\sin\alpha \right\rangle$,$0 \leq \theta, \alpha \leq 2\pi$. Suppose that the temperature at each point of the surface is proportional to its distance from the plane$z=0\:$, i.e., the temperature$T(x,y,z)$for every point$(x,y,z) \in S$is given by$T(x,y,z) = \lambda |z|$for some constant$\lambda >0$. 1. Find the average temperature along the circle$C$which is$\frac{\displaystyle\int_C T \ ds}{\displaystyle\int_C 1\ ds}$. 2. Find the area of$S$,$A(S)$. Then evaluate the average temperature of the toroidal shell$S$which is$\displaystyle\frac{\iint_S T\ dS}{A(S)}$. 15-a-6 a. Line integrals - (i) By parameterizing curves 0.502 0.361 3.720 10 0.612 0.109 Let$C$be the curve of intersection of$x^2+y^2+z^2=4$,$x^2+y^2=2x$,$z\geq 0$, oriented$C$to be counterclockwise when viewed from above. Evaluate$\displaystyle\int_Cy^2dx+z^2dy+x^2dz$. 15-a-7 a. Line integrals - (i) By parameterizing curves 0.642 0.573 7.540 13 0.894 0.252 Compute the line integral $\int_Cz^2\,\mathrm{d}x-x^2\,\mathrm{d}y+2yz\,\mathrm{d}z,$ where$C$is the curve of intersection of the upper half sphere$z=\sqrt{4-x^2-y^2}$and the circular cylinder$x^2+y^2=2y$, orientated counterclockwise viewed from the above, as Figure. 15-a-8 a. Line integrals - (i) By parameterizing curves 0.654 0.442 4.720 12 0.769 0.115 Let$C$be the polar curve defined by$r^2=\cos 2\theta$in the first quadrant. Evaluate$\displaystyle{\int_Cy\ ds}$. 15-a-9 a. Line integrals - (i) By parameterizing curves 0.543 0.451 4.250 9 0.722 0.179 Let$C$be the closed curve formed by$y=x^2$, where$0\leq x\leq 1$, and$x=y^2$, where$0\leq y\leq 1$. Given$C$the counterclockwise orientation, evaluate$\displaystyle\int_C xy\,ds$and$\displaystyle\oint_C xy \,dx$. 15-a-10 a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields) 0.475 0.699 10.968 15 0.937 0.462 For$y>0, let \begin{align*} \textbf{F}(x,y,z)&=(e^{-x}\ln y-z)\textbf{i}+\left(2yz-e^{-x}/y\right)\textbf{j}+(y^2-x)\textbf{k}\mbox{ and}\\ \textbf{G}(x,y,z)&=e^{-x}\ln y\,\textbf{i}+\left(2yz-e^{-x}/y\right)\textbf{j}-x~\textbf{k}. \end{align*} 1. Show that the vector function\textbf{F}$is a gradient on$\{(x,y,z)|~y>0\}$by finding an$f$such that$\nabla f=\textbf{F}$. 2. Evaluate the line integral$\int_C\textbf{G}(\textbf{r})\cdot \textbf{dr}$, where$C$is the curve given by$\textbf{r}(u)=(1+u^2)\textbf{i}+e^u\textbf{j}+(1+u)\textbf{k}$,$u\in[0,1]$. 15-a-11 a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields) 0.548 0.355 4.185 12 0.629 0.081 Find the line integral $\int_{{\cal C}}\,(2 x \sin(\pi y) - e^{z})\,dx + (\pi x^2 \cos(\pi y) - 3 e^{z})\,dy - x e^z\,dz$$\mbox{along the curve}\;\;{\cal C} = \left\{(x, y, z) | z = \ln \sqrt{1 + x^2},\; y = x,\; 0 \le x \le 1 \right\}. $15-a-12 a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields) 0.454 0.750 8.110 10 0.976 0.523 Evaluate$\displaystyle\int_C(yze^{xyz}+x)\,dx+xze^{xyz}\,dy+xye^{xyz}\,dz$, where$C$is the curve${\bf r}(t)=\left\langle t,\,\cos(\pi t),\,\tan^{-1}t\right\rangle$,$0\leq t\leq 1$. 15-a-13 a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields) 0.490 0.584 8.450 14 0.829 0.338 Consider the vector field$\textbf F (x,y)=P(x,y)\,\textbf i +Q(x,y)\,\textbf j=\frac{x+3y}{x^2+y^2}\,\textbf i +\frac{-3x+y}{x^2+y^2}\,\textbf j $. 1. Show that$\textbf F$is conservative on the half plane$D=\{(x,y)|x<0\}$. 2. Compute$\displaystyle\int_{C_0}\textbf F \cdot d\textbf r$, where$C_0$is the unit circle$x^2+y^2=1$oriented counterclockwise. Is$\textbf F$conservative on$\mathbb{R}^2\backslash \{(0,0)\}$? 3. Compute$\displaystyle\int_C \textbf F\cdot d\textbf r$, where$C$is a piecewise smooth path consisting of the ellipse$\frac{x^2}{4}+y^2=1$and the triangle formed by the lines$x=-3$,$x+y=-2$, and$y-x=2$. The orientation of$C$is shown in the figure. 15-a-14 a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields) 0.295 0.238 4.740 20 0.385 0.091 Let$\textbf{F}=\frac{(x-y)^2y}{(x^2+y^2)^2}\textbf{i} +\frac{-(x-y)^2x}{(x^2+y^2)^2}\textbf{j}$. 1. Verify that$\textbf F$is conservative on the right half plane$x>0$. Find a potential function of$\textbf F$on the right half plane. 2. Evaluate$\displaystyle\oint_{C_1}\textbf{F}\cdot d\textbf{r}$where$C_1$is the ellipse$\frac{x^2}{4}+(y-2)^2=1$. 3. Evaluate$\displaystyle\int_{C_2}\textbf{F}\cdot d\textbf{r}$where$C_2$is the curve with polar equation$r=e^{|\theta|}$,$-\frac{9\pi}{4}\leq\theta\leq\frac{9\pi}{4}$. 15-a-15 a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields) 0.454 0.752 9.800 12 0.979 0.524 1. Find the potential function of the vector field $\mathbf{F}(x,y,z)=\frac{2x(1-\mathrm{e}^y)}{(1+x^2)^2}\,\mathbf{i}+\left(\frac{\mathrm{e}^y}{1+x^2}+(y+1)\mathrm{e}^y\right)\,\mathbf{j}+\mathbf{k}.$ 2. Evaluate$\displaystyle\int_C\mathbf{F}\cdot d\mathbf{r}$, where$C$is any curve from$(0,0,0)$to$(1,1,1)$. 15-a-16 a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields) 0.577 0.560 10.180 18 0.848 0.271 Consider the vector field defined by$\textbf{G}(x,y)=(3x^2+y)\textbf{i}+(2x^2y-x)\textbf{j}$,$(x,y)\in\mathbb{R}^2$. 1. Is$\textbf{G}(x,y)$conservative? 2. Find a function$\mu(x)$with$\mu(1)=1$such that$\mu(x)\textbf{G}(x,y)$is conservative. 3. Set$\textbf{F}(x,y)$to be the conservative vector field in (b). Find the potential function$f(x,y)$of$\textbf{F}$with$f(1,0)=3$. 4. Let$C$be the curve with defining equation in polar coordinate given by $r=\sec\theta+\frac{\sqrt{2}}{\pi}\theta,\ \theta\in\left[0,\frac{\pi}{4}\right].$ Evaluate the integral$\displaystyle\int_C\textbf{F}\cdot d\textbf{r}$. 15-a-17 a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields) 0.456 0.548 6.870 12 0.776 0.320 Let$\textbf{F}(x,y)=\frac{y^3}{(x^2+y^2)^2}\textbf{i}-\frac{xy^2}{(x^2+y^2)^2}\textbf{j}$. 1. Show that$\textbf{F}$is conservative on the domain$D=R^2-\{(0,y)| y\leq 0\}$. 2. Compute$\displaystyle\int_C\textbf{F}\cdot d\textbf{r}$, where$C$is the part of the polar curve$r=1+\sin\theta$,$0\leq \theta\leq \pi$. 15-a-18 a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields) 0.324 0.828 11.390 13 0.990 0.666 1. Find the value$\lambda$such that the vector field$\textbf{F}=(x^2+4xy^\lambda)\textbf{i}+(6x^{\lambda-1}y^2-2y)\textbf{j}$is conservative. 2. For this$\lambda$, find a potential function of$\textbf{F}$. 3. For$\lambda$in (a), evaluate$\displaystyle\int_C\textbf{F}\cdot \mathrm{d}\textbf{r}$, where$C$is the path described by$\frac{x^2}{9}+(y-1)^2=1$counterclockwise from$(0,0)$to$(3,1)$. 15-a-19 a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields) 0.468 0.673 8.220 12 0.907 0.438 Let$\displaystyle\mathbf{F}(x,y)=\frac{x+y}{x^2+y^2}\,\mathbf{i}+\frac{-x+y}{x^2+y^2}\,\mathbf{j}$. 1. Is$\mathbf{F}(x,y)$conservative on the half plane$D=\{(x,y)|x>0\}$? 2. Evaluate the line integral$\displaystyle\int_{C_1}\mathbf{F}\cdot\mathrm{d}\mathbf{r}$, where$C_1$is the part of the parabola$y=(x-2)^2$from$(2,0)$to$(4,4)$. 3. Evaluate the line integral$\displaystyle\int_{C_2}\mathbf{F}\cdot\mathrm{d}\mathbf{r}$, where$C_2$is the unit circle$x^2+y^2=1$oriented counterclockwise. Is$\mathbf{F}(x,y)$conservative on$\mathbb{R}^2-\{(0,0)\}$? 15-a-20 a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields) 0.163 0.889 10.120 11 0.970 0.807 Let$\textbf{F}=z\cos(xz)\textbf{i}+ze^{yz}\textbf{j} +(x\cos(xz)+ye^{yz})\textbf{k}$. 1. Find a scalar function$\varphi(x,y,z)$such that$\nabla \varphi=\textbf{F}$. 2. Evaluate$\displaystyle\int_C\textbf{F}\cdot d\textbf{r}$, where$C$is the curve${\bf r}(t)=(\cos (\pi t^2), \ln (t+1) ,\tan^{-1}(t)), 0\leq t\leq 1.$15-a-21 a. Line integrals - (ii) By the fundamental theorem for line integrals (conservative vector fields) 0.573 0.539 8.800 16 0.825 0.252 Let$\textbf F(x,y)=P(x,y)\ \textbf i+Q(x,y)\ \textbf j$, where$P(x,y)=\frac{xy}{\sqrt{x^2+y^2}}$,$Q(x,y)=\frac{x^2+2y^2}{\sqrt{x^2+y^2}}$. 1. Compute$\frac{\partial P}{\partial y}$and$\frac{\partial Q}{\partial x}$. Is$\textbf F$conservative on the right half plane$D=\{(x,y)|x>0\}$? Justify your answer. 2. Compute$\displaystyle\int_C\textbf F\cdot d\textbf r $, where$C$is any curve in the right half plane$D$from$(1,1)$to$(2,2)$. 3. Compute$\displaystyle\oint_C\textbf F\cdot d\textbf r$, where$C$is a positively oriented circle centered at$(0,0)$with radius$r>0$. 4. Compute$\displaystyle\oint_C \textbf F{\bf\cdot} d\textbf r $, where$C$is any positively oriented simple closed curve,$C\subset \mathbb{R}^2\backslash\{(0,0)\}$. (Hint: You need to discuss whether$C$encloses$(0,0)$or not.) 5. Is$\textbf F$conservative on$\mathbb{R}^2\backslash\{(0,0)\}$? Justify your answer. 15-a-22 a. Line integrals - (iii) By Green's theorem 0.521 0.718 11.836 15 0.978 0.458 Let$C$be a piecewise-smooth Jordan curve that does not pass through the origin. Evaluate$\displaystyle\oint_C \frac{-y^5}{(x^2+y^2)^3} dx + \frac{xy^4}{(x^2+y^2)^3} dy $for the following two cases, where$C$is traversed in the counterclockwise direction. 1.$C$does not enclose the origin. 2.$C$does enclose the origin. 15-a--23 a. Line integrals - (iii) By Green's theorem 0.449 0.676 7.173 10 0.901 0.452 Evaluate$\displaystyle\oint_{r=1-\cos\theta}(x^2y+y)dx-(xy^2-x)dy$with the curve oriented counterclockwise. 15-a-24 a. Line integrals - (iii) By Green's theorem 0.672 0.502 5.510 10 0.838 0.167 Let the vector field${\bf F}(x,y)=\frac{x^2y}{(x^2+y^2)^2}\,{\bf i}-\frac{x^3}{(x^2+y^2)^2}\,{\bf j}$,$C_1$be the curve$|x|+|y|=1$and$C_2$be the curve$x^2+(y-2)^2=1$. Find$\displaystyle\oint_{C_1}{\bf F}\cdot d{\bf r}$and$\displaystyle\oint_{C_2}{\bf F}\cdot d{\bf r}$. 15-a-25 a. Line integrals - (iii) By Green's theorem 0.272 0.860 13.650 15 0.996 0.724 Let$D$be the bounded region in the first quadrant enclosed by$y=0$,$x=1$, and$y=\sqrt{x}$with positively oriented boundary$C(i.e. counter clockwise.). Evaluate \begin{align*}\oint_{C}\left[9x^2y(x^3+1)^{\frac{1}{2}}-xy^2(x^3+1)^{\frac{3}{2}}\right]dx \\ +\left[2(x^3+1)^{\frac{3}{2}}+2(y^3+1)^{\frac{3}{2}}\right]dy.\end{align*} 15-a-26 a. Line integrals - (iii) By Green's theorem 0.586 0.602 8.790 14 0.894 0.309 1. Evaluate the line integral\displaystyle I_1=\int_{C_1}(-\sin x+\mathrm{e}^y)\,\mathrm{d}x+(2x+x\mathrm{e}^y)\,\mathrm{d}y$, where$C_1$is the line segment from$(0,0)$to$(2,2)$. 2. Find the area of the region between$C_1$and$C_2$, where$C_2$is a curve from$(0,0)$to$(2,2)$parameterized by $\mathbf{r}(t)=\frac2\pi(t-\sin t)\,\mathbf{i}+(1-\cos t)\,\mathbf{j},\quad 0\leq t\leq \pi.$ (c) Evaluate the line integral$\displaystyle I_2=\int_{C_2}(-\sin x+\mathrm{e}^y)\,\mathrm{d}x+(2x+x\mathrm{e}^y)\,\mathrm{d}y$. 15-a-27 a. Line integrals - (iii) By Green's theorem 0.430 0.544 6.420 12 0.759 0.329 Determine whether the statement is true of false. Fill$\textbf T$(true) or$\textbf F$(false) in the blanks. If the statement is false, write down a reason, or give a correct statement, or find a counterexample. 1. Let$\mathbf{F}(x,y)=P(x,y)\,\mathbf{i}+Q(x,y)\,\mathbf{j}$. If$\displaystyle\frac{\partial Q}{\partial x}=\frac{\partial P}{\partial y}$throughout the domain of$\mathbf{F}(x,y)$, then the line integrals of$\mathbf{F}(x,y)$is independent of path on the domain. 2. Let$f(x,y)$be a smooth function. Suppose that a smooth curve$C$gives an orientation from initial point$p$to terminal point$q$. If$-C$denotes the curve consisting of the same points as$C$but with the opposite orientation (from initial point$q$to terminal point$p$), then$\displaystyle\int_{-C}f(x,y)\,\mathrm{d}s=-\int_Cf(x,y)\,\mathrm{d}s$. 3. For a unit circle$C: x^2+y^2=1$, we have$\displaystyle\oint_Cx\,\mathrm{d}y=0$by symmetry. 4. Any smooth function$f(x,y,z)$satisfies$\mathrm{div}(\nabla f)=0$. 15-a-28 a. Line integrals - (iii) By Green's theorem 0.623 0.620 7.700 12 0.931 0.308 Find the value$k\in\mathbb{R}$such that the line integral $I(k)=\int_{C_k}(1+y^2+y\,\mathrm{e}^{xy})\,\mathrm{d}x+(2x+y+x\,\mathrm{e}^{xy})\,\mathrm{d}y$ achieves the minimum value, where$C_k$is the curve$y=k\sin x$from$(0,0)$to$(\pi,0)$. 15-a-29 a. Line integrals - (iii) By Green's theorem 0.392 0.738 9.350 12 0.934 0.542 Evaluate$\displaystyle\oint_C(x^2y^2+y)dx-(2xy^3-3x)dy$, where$C$is described by the polar equation$r=1-\cos\theta$oriented counterclockwise. 15-a-30 a. Line integrals - (iii) By Green's theorem 0.506 0.626 7.820 12 0.879 0.373 Evaluate the line integral$\displaystyle\int_C\sin \pi x\ dx+(e^{y^2}+x^2)dy$along the following choices of the curve$C$. 1.$C=C_0$is the line segment from$(-1,0)$to$(0,0)$. 2.$C=C_1\cup C_2$, where$C_1$is the polar curve$r=2\sin\theta$,$0\leq\theta\leq\frac{\pi}{2}$and$C_2$is the cardioid$r=1+\sin\theta$,$\frac{\pi}{2}\leq\theta\leq \pi$. 15-a-31 a. Line integrals - (iii) By Green's theorem 0.548 0.567 8.640 15 0.841 0.293 Evaluate the line integral$\displaystyle\int_C\left(-x-y+\frac{y^2}{2}\right)dx+(x+2xy+3)dy$, where$C$consists of the arc$C_1$of the quarter circle$x^2+y^2=1, x\geq 0, y\leq 0$, from$(0,-1)$to$(1,0)$followed by the arc$C_2$of the quarter ellipse$4x^2+y^2=4$,$x\geq 0$,$y\geq 0$, from$(1,0)$to$(0,2)$. (Hint: You may use Green's Theorem, but note that$C$is not closed.) 15-b-1 b. Surface integrals 0.660 0.544 8.670 15 0.874 0.214 1. Find the area of the part of the sphere$x^2+y^2+z^2=2$that lies above the plane$z=1$. 2. Let the canopy be the part of the upper hemisphere$x^2+y^2+z^2=2$that lies above the square$-1\leq x\leq 1$,$-1\leq y\leq 1$, and let$C$be the boundary of canopy oriented counterclockwise when viewed from above. Evaluate$\displaystyle\oint_C{\bf F}\cdot d{\bf r}$, where${\bf F}(x,y,z)=(xz+\tan x^2)\,{\bf i}+(\sin x\cos y+e^{y^2})\,{\bf j}+\left(-\frac{y^2}{2}+\sin\sqrt{z}\right)\,{\bf k}$. 15-b-2 b. Surface integrals - (i) By parameterizing surfaces 0.616 0.538 8.250 15 0.846 0.230 Let$S$be the triangular region with vertices$(0,0,0)$,$(a,0,0)$, and$(a,a,a)$,$a>0$, with upward unit normal$\textbf{n}$, and$C$be the positively oriented boundary of$S$. Let $\textbf{F}=\left(y-z\cos(x^2)\right)\textbf{i} +\left(2x-\sin(z^2)\right)\textbf{j} +\left(3z-\tan(y^2)\right)\textbf{k}.$ 1. Find a parametrization of$S$and find the upward unit normal$\textbf{n}$. (Hint. Consider the projection of$S$to$xy$-plane.) 2. Evaluate$\nabla\times\textbf{F}$. 3. Evaluate$\displaystyle\oint_{C}\textbf{F}\cdot \textbf{dr}$. 15-b-3 b. Surface integrals - (i) By parameterizing surfaces 0.726 0.512 7.181 15 0.875 0.149 Let$S_1$be the surface$\{(x,y,z)|~z=x^2+y^2,~z\leq y\}$,$S_2$be the surface$\{(x,y,z)|~z=y,~x^2+y^2\leq z\}$, and$\textbf{V}(x,y,z)=-y\textbf{i}+x\textbf{j}+z\textbf{k}$. 1. Compute directly the downward flux of$\textbf{V}$across$S_1$. 2. Use the divergence theorem to compute the upward flux of$\textbf{V}$across$S_2$. 15-b-4 b. Surface integrals - (i) By parameterizing surfaces 0.592 0.663 8.576 12 0.959 0.367 Evaluate the surface integral$\displaystyle\iint\limits_{S}(x^2+y^2)zd\sigma$, where$S$is the part of the plane$z=4+x+y$that lies inside the cylinder$x^2+y^2=4$. 15-b-5 b. Surface integrals - (i) By parameterizing surfaces 0.492 0.315 4.070 15 0.561 0.069 Let${\bf F}(x,y,z)=\frac{1}{\left(x^2+y^2+z^2\right)^{3/2}}(x\,{\bf i}+y\,{\bf j}+z\,{\bf k})$. 1. Let$S_1$be the part of the sphere$x^2+y^2+z^2=1$in the first octant bounded by the planes$y=0$,$y=\sqrt{3}x$and$z=0$, oriented upward. Find the flux of${\bf F}$across$S_1$. 2. Let$S_2$be the part of the surface$\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}=1$in the first octant bounded by the planes$y=0$,$y=\sqrt{3}x$and$z=0$, oriented upward. Find the flux of${\bf F}$across$S_2$. 15-b-6 b. Surface integrals - (i) By parameterizing surfaces 0.493 0.720 7.620 10 0.966 0.473 Find the area of the surface$\{ x^2 + y^2 + z^2 = 4,\,1 \le x^2+y^2 \le 3,\,z\geq 0 \}$. 15-b-7 b. Surface integrals - (i) By parameterizing surfaces 0.605 0.439 6.280 16 0.741 0.136 Let$S$be a toroidal shell (not a solid torus!) obtained by rotating about the$z$-axis the circle$C$in the plane$y=0$of radius$a$centred at$(b,0,0)$, where$0 < a < b\:$. It can be parametrized by$ \textbf r(\theta, \alpha) = \left\langle\:(b +a\cos\alpha) \cos\theta , (b +a\cos\alpha) \sin\theta , a\sin\alpha \right\rangle$,$0 \leq \theta, \alpha \leq 2\pi$. Suppose that the temperature at each point of the surface is proportional to its distance from the plane$z=0\:$, i.e., the temperature$T(x,y,z)$for every point$(x,y,z) \in S$is given by$T(x,y,z) = \lambda |z|$for some constant$\lambda >0$. 1. Find the average temperature along the circle$C$which is$\frac{\displaystyle\int_C T \ ds}{\displaystyle\int_C 1\ ds}$. 2. Find the area of$S$,$A(S)$. Then evaluate the average temperature of the toroidal shell$S$which is$\displaystyle\frac{\iint_S T\ dS}{A(S)}$. 15-b-8 b. Surface integrals - (i) By parameterizing surfaces 0.726 0.403 3.670 10 0.766 0.040 Let$S$be the surface$x^2+y^2+z^2=a^2$,$x\geq 0$,$y\geq 0$,$z\geq 0$($a>0$), and let$C$be the boundary of$S$. Find the centroid of$C$. 15-b-9 b. Surface integrals - (i) By parameterizing surfaces 0.625 0.451 4.190 10 0.763 0.138 Evaluate$\displaystyle\iint_SxdS$where$S$is the part of the cone$z=\sqrt{2(x^2+y^2)}$that lies below the plane$z=1+x$. 15-b-10 b. Surface integrals - (i) By parameterizing surfaces 0.621 0.637 6.980 10 0.947 0.326 Evaluate the surface integral$\displaystyle\iint\limits_S(x^2+y^2)dS$, where$S$is the surface$z=\sqrt{x^2+y^2}$with$0\leq z\leq 1$. 15-b-11 b. Surface integrals - (i) By parameterizing surfaces 0.677 0.393 3.470 10 0.732 0.054 Let$S$be a cone has radius$a$and height$h$without base. Evaluate the integral of the distance of the points to its axis over$S$. 15-b-12 b. Surface integrals - (i) By parameterizing surfaces 0.583 0.499 4.890 10 0.790 0.207 Compute the surface integral $\iint_S xz\,\mathrm{d}S,$ where$S$is the part of the cone$z=\sqrt{x^2+y^2}$inside the circular cylinder$x^2+y^2=2x$. 15-b-13 b. Surface integrals - (i) By parameterizing surfaces 0.623 0.646 8.190 12 0.958 0.335 Evaluate the surface integral$\displaystyle\iint_S\sqrt{x^2+y^2}\,\mathrm{d}S$, where$S$is the part of the surface$\displaystyle z=\tan^{-1}\left(\frac{y}x\right)$inside the circular cylinder$x^2+y^2=1$and in the first octant. 15-b-14 b. Surface integrals - (i) By parameterizing surfaces 0.573 0.455 5.600 12 0.741 0.169 Find the area of the sphere$x^2+y^2+z^2=4$lying inside the cylinder$(x-1)^2+y^2=1$. 15-b-15 b. Surface integrals - (i) By parameterizing surfaces 0.480 0.563 6.060 10 0.803 0.323 Find the area of the part of the surface$x^2+y^2+z^2=1$that lies within the cylinder$x^2+y^2+x=0$and above$z=0$. 15-b-16 b. Surface integrals - (ii) By Stokes' theorem 0.616 0.538 8.250 15 0.846 0.230 Let$S$be the triangular region with vertices$(0,0,0)$,$(a,0,0)$, and$(a,a,a)$,$a>0$, with upward unit normal$\textbf{n}$, and$C$be the positively oriented boundary of$S$. Let $\textbf{F}=\left(y-z\cos(x^2)\right)\textbf{i} +\left(2x-\sin(z^2)\right)\textbf{j} +\left(3z-\tan(y^2)\right)\textbf{k}.$ 1. Find a parametrization of$S$and find the upward unit normal$\textbf{n}$. (Hint. Consider the projection of$S$to$xy$-plane.) 2. Evaluate$\nabla\times\textbf{F}$. 3. Evaluate$\displaystyle\oint_{C}\textbf{F}\cdot \textbf{dr}$. 15-b-17 b. Surface integrals - (ii) By Stokes' theorem 0.654 0.606 7.711 12 0.933 0.279 Let$\textbf{V}=(2x-y)\textbf{i}+(2y+z)\textbf{j}+x^2y^2z^2\textbf{k}$and let$S$be the upper half of the ellipsoid$\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{25}=1$. Find the flux of curl$\textbf{V}$in the direction of the upper unit normal$\textbf{n}$(pointing away from the origin.). 15-b-18 b. Surface integrals - (ii) By Stokes' theorem 0.727 0.603 9.790 15 0.967 0.239 Let$S$be the part of the sphere$x^2+y^2+(z-2)^2=8$that lies above the$xy$-plane and that has outward normal (i.e. with${\bf k}$-component$\geq 0$). Let${\bf F}(x,y,z)=\left\langle -y^3\cos xz,\,x^3e^{yz},\,-e^{xyz}\right\rangle$. Find$\displaystyle\iint_S\mbox{curl}\,{\bf F}\cdot d{\bf S}$. 15-b-19 b. Surface integrals - (ii) By Stokes' theorem 0.613 0.554 6.810 12 0.860 0.247 Let$f(x,y,z)=x + x y + y z + z x$, and$g(x,y,z)=x+2y+3z$. 1. Show by direct calculation that$\mbox{curl}\,(f\,\nabla g)\,=\,\nabla f\,\times\,\nabla g$. 2. Find$\displaystyle\iint_{S}\,(\nabla f\,\times\,\nabla g)\cdot\,\textbf n\,dS$where$S$is the surface$z=\sqrt{4 - x^2 - y^2}$, and$\textbf n$is the unit normal on$S$pointing upwards. 15-b-20 b. Surface integrals - (ii) By Stokes' theorem 0.074 0.424 2.050 5 0.461 0.387 Determine the statement is true ($\bigcirc$) or false ($\times$). 1. If$f(x,y)$is continuous on the rectangle$R=\{(x,y)|\ a\leq x\leq b,\ c\leq y\leq d\}$except for finitely many points, then$f(x,y)$is integrable on$R$and $\iint_Rf(x,y)dA=\int_c^d\int_a^bf(x,y)dxdy =\int_a^b\int_c^df(x,y)dydx.$ 2. If$\textbf{F}(x,y)=P(x,y)\textbf{i}+Q(x,y)\textbf{j}$and$\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}$on an open connected region$D$, then$\textbf{F}$is conservative on$D$. 3. If curl$\textbf{F}=$curl$\textbf{G}$on$\mathbb{R}^3$, then$\displaystyle\int_C\textbf{F}\cdot d\textbf{r}=\int_C\textbf{G}\cdot d\textbf{r}$for all closed path$C$. 4. If$\textbf{F}$and$\textbf{G}$are vector fields and curl$\textbf{F}=$curl$\textbf{G}$, div$\textbf{F}=$div$\textbf{G}$, then$\textbf{F}-\textbf{G}$is a constant vector field. 5. Let$B$be a rigid body rotating about the$z$-axis with constant angular speed$\omega$. If$\textbf{v}(x,y,z)$is the velocity at point$(x,y,z)\in B$, then curl$\textbf{v}$is parallel to$\textbf{k}$. 15-b-21 b. Surface integrals - (ii) By Stokes' theorem 0.585 0.372 3.750 10 0.664 0.079 Evaluate$\displaystyle\int_C(y+\sin^3x)dx+(z^2+\cos^4y)dy +(x^3+\tan^5z)dz$where$C$is the curve$\textbf{r}(t)=\sin t\ \textbf{i}+\cos t\ \textbf{j}+\sin 2t\ \textbf{k}$,$0\leq t\leq 2\pi$. (Hint:$C$lies on the surface$z=2xy$.) 15-b-22 b. Surface integrals - (ii) By Stokes' theorem 0.443 0.411 4.250 10 0.633 0.190 Let$C$be the curve formed by the intersection of the plane$z=x$and the surface$z=x^2+y^2$.$C$is oriented counterclockwise when viewed from above. Evaluate$\displaystyle\oint_C(xyz+\tan^{-1}x)dx+(x^2+\sinh y)dy +(xz+\ln z)dz$. 15-b-23 b. Surface integrals - (ii) By Stokes' theorem 0.469 0.722 11.680 15 0.957 0.488 1. Find curl$\textbf{F}$, where$\textbf{F}(x,y,z)=(y^2-z^2)\textbf{i} +(z^2-x^2)\textbf{j}+(x^2-y^2)\textbf{k}$. 2. Compute the line integral $\oint_C(y^2-z^2)\,\mathrm{d}x+(z^2-x^2)\,\mathrm{d}y+(x^2-y^2)\,\mathrm{d}z,$ where$C$is the hexagon which is the boundary of the intersection of the plane$x+y+z=\frac32$and the unit cube$B=\{(x,y,z)|0\leq x\leq 1,0\leq y\leq 1,0\leq z\leq 1\}$, oriented as pictured. 15-b-24 b. Surface integrals - (ii) By Stokes' theorem 0.642 0.573 7.540 13 0.894 0.252 Compute the line integral $\int_Cz^2\,\mathrm{d}x-x^2\,\mathrm{d}y+2yz\,\mathrm{d}z,$ where$C$is the curve of intersection of the upper half sphere$z=\sqrt{4-x^2-y^2}$and the circular cylinder$x^2+y^2=2y$, orientated counterclockwise viewed from the above, as Figure. 15-b-25 b. Surface integrals - (ii) By Stokes' theorem 0.435 0.530 7.290 14 0.747 0.312 Compute$\displaystyle\iint_S\text{curl}\textbf{F}\cdot\text{d}\textbf{S}$, where${\bf F}=e^{xz} {\bf i}+(x^2+z^2){\bf j}+(y+\cos z){\bf k}$and where$S=\left\{(x,y,z)\Big|\frac{x^2}{4}+\frac{y^2}{9}+z^2=1\ \text{and }x+2z\geq 0\right\}$oriented so that the boundary is counterclockwise when viewed from above. 15-b-26 b. Surface integrals - (ii) By Stokes' theorem 0.468 0.444 6.260 14 0.678 0.210 Let$\textbf F = (x-y) \textbf i + (y-z) \textbf j + (z-x) \textbf k$be a vector field on$\mathbb{R}^3$. 1. Compute$\text{curl} \textbf{F}$and div$\textbf{F}$on$\mathbb{R}^3$. 2. Let$S_1$be the part of paraboloid$z=x^2+(y-1)^2$that is below the plane$z=5-2y$with downward orientation. Find the flux of$\textbf{F}$across$S_1$,$\displaystyle\iint_{S_1} \textbf{F} \cdot d\textbf S$. 3. Let$S_2$be a surface defined by the equation$\frac{x^2}{9}+\frac{y^2}{36}-z^2=1$for$z \in [0,1]$and endowed with the orientation given by the downward normal vector. Find the flux of$\text{curl}\textbf{F}$across$S_2$,$\displaystyle\iint_{S_2} \text{curl}\textbf{F} \cdot d\textbf S$. 圖 1 圖 2 15-b-27 b. Surface integrals - (ii) By Stokes' theorem 0.528 0.543 7.650 14 0.808 0.279 Let$\textbf F = (x-y) \textbf i + (y-z) \textbf j + (z-x) \textbf k$be a vector field on$\mathbb{R}^3$. 1. Compute$\text{curl} \textbf{F}$on$\mathbb{R}^3$. 2. Let$S_1$be a parametric surface given by$\textbf r(r,\theta) = r \cos\theta \textbf i + 2r \sin\theta \textbf j + (9-r^2) \textbf k$for$r\in [0,3]$and$\theta \in [0,2\pi]$, which comes with the standard orientation given by the normal vector$\textbf{r}_r \times \textbf{r}_\theta$. Find the flux of$\text{curl} \textbf{F}$across$S_1$. 3. Let$S_2$be a surface defined by the equation$\frac{x^2}{9}+\frac{y^2}{36}-z^2=1$for$z \in [0,1]$and endowed with the orientation given by the downward normal vector. Find the flux of$\text{curl}\textbf{F}$across$S_2$. 圖 1 圖 2 15-b-28 b. Surface integrals - (ii) By Stokes' theorem 0.630 0.513 7.490 15 0.828 0.198 Evaluate$\displaystyle\iint\limits_S \nabla\times\textbf{F}\cdot d\textbf{S}$, where$\textbf{F}(x,y,z)=(y+\sin x)\,\textbf{i}+(z^2+\cos y)\,\textbf{j}+x^3\,\textbf{k}$and where$S$is the surface$z=2xy$inside the cylinder$x^2+y^2=1$and with the normal pointing in the positive$z$-direction. 15-b-29 b. Surface integrals - (iii) By the divergence theorem 0.726 0.512 7.181 15 0.875 0.149 Let$S_1$be the surface$\{(x,y,z)|~z=x^2+y^2,~z\leq y\}$,$S_2$be the surface$\{(x,y,z)|~z=y,~x^2+y^2\leq z\}$, and$\textbf{V}(x,y,z)=-y\textbf{i}+x\textbf{j}+z\textbf{k}$. 1. Compute directly the downward flux of$\textbf{V}$across$S_1$. 2. Use the divergence theorem to compute the upward flux of$\textbf{V}$across$S_2$. 15-b-30 b. Surface integrals - (iii) By the divergence theorem 0.649 0.436 4.711 12 0.760 0.111 Evaluate the flux of $\textbf{V}(x,y,z)=(z^2x+y^2z)\textbf{i} +\left(\frac{1}{3}y^3+z\tan x\right)\textbf{j}+(x^2z+2y^2+1)\textbf{k}$ across$S$: the upper half sphere$x^2+y^2+z^2=1$,$z\geq 0$with normal pointing away from the origin. 15-b-31 b. Surface integrals - (iii) By the divergence theorem 0.492 0.315 4.070 15 0.561 0.069 Let${\bf F}(x,y,z)=\frac{1}{\left(x^2+y^2+z^2\right)^{3/2}}(x\,{\bf i}+y\,{\bf j}+z\,{\bf k})$. 1. Let$S_1$be the part of the sphere$x^2+y^2+z^2=1$in the first octant bounded by the planes$y=0$,$y=\sqrt{3}x$and$z=0$, oriented upward. Find the flux of${\bf F}$across$S_1$. 2. Let$S_2$be the part of the surface$\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}=1$in the first octant bounded by the planes$y=0$,$y=\sqrt{3}x$and$z=0$, oriented upward. Find the flux of${\bf F}$across$S_2$. 15-b-32 b. Surface integrals - (iii) By the divergence theorem 0.642 0.642 10.720 15 0.963 0.321 Let${\bf F}=\left\langle 3xy^2,\,y^3,\,e^{x^2+y^2}\right\rangle$. Let$S$be the part of the surface$z=1-x^2-y^2$that lies above$xy$-plane oriented upwards (that is, with normal having${\bf k}$-component$\geq 0$). Calculate the flux$\displaystyle\int_{S}{\bf F}\cdot d{\bf S}$of${\bf F}$across$S$. Note that$S$is not closed. 15-b-33 b. Surface integrals - (iii) By the divergence theorem 0.427 0.420 6.980 16 0.633 0.206 Consider the unit disk $S_1 = \left\{(x,y,z) \in \mathbb{R}^3| x^2 + y^2 \leq 1 , z = 0\right\}$ and the half cone $S_2 = \left\{(x,y,z) \in \mathbb{R}^3| 2z = 1-\sqrt{x^2+y^2} , z \geq 0\right\} \; ,$ and let$S = S_1 \cup S_2$be the closed surface of a cone with the positive (outward) orientation. Both$S_1$and$S_2$are endowed with the induced orientation from$S$. 1. Let$\textbf F = \left\langle 0 , y^2 , z-2yz\right\rangle\:$. Find$\displaystyle\iint_{S} \textbf F \cdot d\textbf S \:$. 2. Let$\textbf G =\textbf F + \textbf E$, where $\textbf E = \left\langle\frac x{(x^2 +y^2 +z^2)^{\frac 32}} , \frac y{(x^2 +y^2 +z^2)^{\frac 32}} , \frac z{(x^2 +y^2 +z^2)^{\frac 32}} \right\rangle\:$ defined on$\mathbb{R}^3\backslash \{(0 , 0 , 0)\}$. Find$\displaystyle\iint_{S_2} \textbf G \cdot d\textbf S \:$. (Note that the integral is only over$S_2$.) 15-b-34 b. Surface integrals - (iii) By the divergence theorem 0.621 0.602 9.430 15 0.913 0.292 Evaluate the flux integral$\displaystyle\iint\limits_S\mathbf{F}\cdot\mathbf{n}\,dS$, where $\mathbf{F}(x,y,z)=(x+z)\,\mathbf{i}-(z+y)\,\mathbf{j}+(y+z^3)\,\mathbf{k},$ and$S$is the sphere$(x-2)^2+y^2+z^2=4$with outward normal. 15-b-35 b. Surface integrals - (iii) By the divergence theorem 0.430 0.580 6.250 10 0.795 0.365 Let$S$be the surface$x^2+y^2+z^2=1$,$x,y,z\geq 0$, an eighth of a sphere, and$\textbf{F}=x^2\textbf{i}+y^2\textbf{j}+z^2\textbf{k}$. Find the outward flux of$\textbf{F}$across$S$. 15-b-36 b. Surface integrals - (iii) By the divergence theorem 0.451 0.725 11.290 15 0.950 0.499 1. Find div$\textbf{F}$, where$\textbf{F}(x,y,z)=(y^2x+\sin z)\textbf{i}+(x^2y-\cos x)\textbf{j}+\left(\frac{1}{3}z^3+y^2\right)\textbf{k}$. 2. Evaluate$\displaystyle\iint_S\textbf{F}\cdot d\textbf{S}$, where$S$is the top half of the sphere$\begin{cases} x^2+y^2+z^2=1\\ z\geq 0 \end{cases}$oriented upward. 15-b-37 b. Surface integrals - (iii) By the divergence theorem 0.741 0.449 6.400 15 0.819 0.078 Consider the vector field$\mathbf{F}(\mathbf{x})=\frac{\mathbf{x}}{|\mathbf{x}|^3}$, that is,$ \mathbf{F}(x,y,z)=\frac{x}{(x^2+y^2+z^2)^{\frac32}}\,\mathbf{i}+\frac{y}{(x^2+y^2+z^2)^{\frac32}}\,\mathbf{j}+\frac{z}{(x^2+y^2+z^2)^{\frac32}}\,\mathbf{k}.$1. Evaluate$\displaystyle\iint_{S_1}\mathbf{F}\cdot\mathrm{d}\mathbf{S}$, where$S_1$is the part of the sphere$x^2+y^2+z^2=1$inside the cone$z=\sqrt{\frac{x^2+y^2}{3}}$with upward orientation. 2. Evaluate$\displaystyle\iint_{S_2}\mathbf{F}\cdot\mathrm{d}\mathbf{S}$, where$S_2$is the part of the cone$z=\sqrt{\frac{x^2+y^2}{3}}$between planes$z=\frac12$and$z=\frac2{\sqrt{3}}$with outward orientation. 3. Use the Divergence Theorem to evaluate$\displaystyle\iint_{S_3}\mathbf{F}\cdot\mathrm{d}\mathbf{S}$, where$S_3$is the part of the paraboloid$z=\frac{6-x^2-y^2}{\sqrt{3}}$inside the cone$z=\sqrt{\frac{x^2+y^2}{3}}$with upward orientation. 15-b-38 b. Surface integrals - (iii) By the divergence theorem 0.491 0.544 9.280 16 0.789 0.299 Let$Ebe the space region bounded by the surfaces \begin{align*} S_1 &:= \{(x, y, z)|\ z = -1+\sqrt{x^2+y^2},\ z\leq0\},\\ S_2 &:= \{(x, y, z)|\ z = 1-x^2-y^2,\ z\geq0\},\\ \end{align*} and{\bf V}(x, y, z) = -y{\bf i} + x{\bf j} + z{\bf k}.$1. Evaluate$\displaystyle\iiint_{\Omega}\text{div}\textbf{V}\, dV$, where$\Omega$is the solid region enclosed by$S_1\cup S_2$. 2. State the Divergence Theorem and evaluate the total outward flux$\displaystyle\iint_{S_1\cup S_2}\bf{V}\cdot d\bf{S}$. 3. Compute the upward flux of$\bf{V}$across$S_2$. 15-b-39 b. Surface integrals - (iii) By the divergence theorem 0.211 0.155 1.400 10 0.260 0.049 Suppose that$f(x,y,z)$is a scalar function with continuous second partial derivatives. Fix a point$P_0=(x_0,y_0,z_0)$. Consider spheres$S_\rho$centered at$P_0$with radius$\rho>0$. 1. Parametrize$S_\rho$with spherical coordinates$\textbf r(\varphi,\theta)=(x_0+\rho\sin\varphi\cos\theta, y_0+\rho\sin\varphi\sin\theta, z_0+\rho\cos\varphi)$,$0\leq\varphi\leq\pi$, and$0\leq \theta\leq 2\pi$. Write down the double integral in$\varphi$and$\theta$that represents the average value of$f$on$S_\rho$. 2. Let function$A(\rho)$be the average value of$f$on$S_\rho$, for$\rho>0$. Evaluate$A'(\rho)$in terms of$\displaystyle\iint_{S_\rho}\nabla f\cdot d \textbf S$. 3. If$\nabla^2f=f_{xx}+f_{yy}+f_{zz}$is always positive, show that$A(\rho)$is increasing. If$\nabla^2f(x,y,z)=0$for all$(x,y,z)$, compute$A(\rho)$. 15-b-40 b. Surface integrals - (iii) By the divergence theorem 0.501 0.434 5.020 12 0.685 0.184 Let$S$be the boundary surface of the union of the balls$x^2 + y^2 + z^2 \le 1$and$x^2 + y^2 +(z-1)^2 \le 1$. 圖 1 1. Use spherical coordinates to parametrize$S$. 2. Find$\displaystyle\iint_S \textbf F \cdot d\textbf S$where$\textbf F = \textbf i + \textbf j + z^2\,\textbf k$and$S\$ is given the outward orientation.