Sweet Home for Calculus I (Fall, 2008)

同學們，開工了，課程簡介 及教學計畫。

*上課內容摘要*

__Week 1 __

1. The definition of a function. f:X-->Y is called a function if f(x) is uniquely determined. X is called the domain of the definition and Y is called the range.

2. Introduce some basic types of functions.

3. Define the inverse of a function. Let f: X--> range(f) be one-to-one. Then there exists a unique function, f^{-1}: range(f)-->X, such that f^{-1}(f(x))=x for all x in X

and f(f^{-1}(y))=y for all y in range(f). f^{-1} is called the inverse of f.

4. Define the limit of a function.

__
Week 2__

1.
Homework #1 (due 10/2): p.85,* *53, 54, 56;
p.93, 34; p.94,
57* *; p.95, 60; p.107,
6, 8; p.109, 73, 82; p.130,
53, 59, 60; p.142, 17. (後面四題較難，需對連續函數充分瞭解）

2. Precise definitions of limits. Finding the relation between δ and ε is the key to prove the existence of the limit. You should work out some examples by yourselves.

4. Define a continuous function. A function f:(a,b)--> R and c in (a,b) is said to be continuous at c iff lim_{x→c}f(x)=f(c). The continuity at the endpoint is defined similarly with the limit

being replaced by the one-sided limit.

5. The composite of two continuous functions is continuous.

6. A
continuous function possesses the *intermediate value property, i.e., if
c is between f(a) and f(b), then there exists a x in (a,b) such that f(x)=c.*

7. Using the intermediate value property to find roots of a continuous function.

__
Week 3__

1. We began to discuss differentiability. We define f'(x_0)=lim_{x→x_0}[f(x)-f(x_0)]/[x-x_0] provided the limit exists. We can also define one-sided derivatives (similar to one-sided continuity).

2. Differentiability implies continuity.

3. Derivatives of some elementary functions.

4. Differentiation rules, e.g., product and quotient rules.

5. The chain rule (important) and parametric equations.

6. Implicit differentiation (good trick). For example, we can find the equation of the tangent line at a point of a given curve without knowing the explicit form of the function.

7. First quiz on 10/7 (Chapter 2).

__
Week 4__

1. Homework #2 (due 10/14): p.155, 51,57,58; p.168-169, 50,56; p.179, 18; p.180, 21; p.200, 65,66; p.202, 116,117; p.209, 43,44,46; p.210, 60.

2. Derivative of the inverse function. (e^x)'=e^x; (ln |x|)'=1/x.

3. Inverse functions of trigonometric functions. It is important to specify the domain and range of a trigonometric function and its inverse.

4. Derivatives of inverse trigonometric functions.

5. Linear approximation and differentials.

6. Optimization problem. Finding the maximum and minimum of a continuous function on a closed interval [a,b].

7. Let c be in (a,b) and f(c) is a local maximum or minimum, then either f is not differentiable at c or f'(c)=0.

__
Week 5__

1. First derivative test for local extrema. Looking at the sign of first derivative near the critical point to determine the type of critical point.

2. Rolle's theorem and mean value theorem.

3. Second derivative test for local extrema. Point of inflection.

4. Generalized mean value theorem.

5. Applications of the generalized mean value theorem to prove the second derivative test and L'Hopital's rule.

6. Second quiz on 10/21 (Chapter 3).

__Week 6__

1. Homework #3 (due 10/30): p.274, 68; p.275, 80; p.284, 55, 60; p.290, 53, 54, 58; p.299-300, 37, 59, 69; p.311, 25; p.314, 52; p.323-324, 27, 51, 53, 55, 61, 62, 63, 64.

2. Examples of indeterminate limits.

3. Newton's method.

4. Definition of a definite integral, Upper and Lower Riemann sum.

5. Fundamental Theorem of Calculus.

6. Method of find an antiderivative of a given function -- change of variable.

7. I will be absent on 10/30. TA will lead a problem session.

__Week 7__

1. Applications of integration.

2. Evaluate the area of a bounded region.

3. Find the volume of a body: slicing method (general method), disk method, and shell method (for the solid of the revolution).

4. First midterm on 11/11 in class.

__Week 8__

1. 有人問我 sec x怎麼積，這是答案。

2. The formula of arc length.

3. Computing the center of mass or centroid. Areas fo surfaces of revolution.

4. Theorems of Pappus. Relation between the centroid and the volume of revolution.

__Week 9__

1. In class midterm on 11/11.

2. Transcendental functions.

3. Simple ordinary differential equation, initial value problem, exponential functions.

__Week 10__

1. Homework #4 (due 12/2): p.515-517, 1, 8, 13, 19, 26; p.531-534, 49, 60, 77, 84, 85, 87; p.552-554, 10, 18, 25, 26, 29, 45, 48.

2. Third quiz on 12/2 as well.

3. Hyperbolic functions.

4. More integration techniques: integration by parts, partial fraction, trigonometric substitution.

__Week 11 __

1. Trigonometric substitution, rationalizing substitution.

2. Numerical integration: midpoint rule, trapezoidal rule, Simpson's rule.

3. Introduction of improper integrals.

4. Homework #5 (due 12/9): p.564-565, 38, 40, 44, 52; p.575, 15, 17, 21, 33, 37; p.620, 120, 122, p.622, 1, 2, 6.

__Week 12__

1. Convergence tests for improper integrals: comparison test and limiting comparison test.

2. Introduction of ordinary differential equations.

3. Separable differential equations.

4. Initial value problems.

5. Linear first order ordinary differential equations.

__Week 13__

1. Integrating factor.

2. Applications of ordinary differential equations.

3. Numerical methods: Euler's method and improved Euler's method.

4. Analysis of autonomous ordinary differential equations.

5. Method of the phase line.

6. Equilibrium solutions, stable and unstable.

7. Fourth quiz on 12/16.

__Week 14__

1. Homework #6 (due 12/23): p.632, 15, 16, 18; p.642-643, 18, 22, 26, 35, 36; p. 655-656, 6, 14, 20.

2. Conic sections: parabola, ellipse, hyperbola.

3. Definition of eccentricity.

4. Characterization of parabola, ellipse, and hyperbola in terms of eccentricity.

5. Polar coordinates.

__Week 15__

1. Integrals in terms of polar coordinates.

2. Conic sections in polar coordinates.

3. Definition of a sequence, convergence.

4. Test of convergence for a sequence.

5. Definition of a series, convergence.

6. Necessary condition for a convergent series.

7. Homework #7 (due 1/8): p.742-743: 86, 91, 115, 116, 119; p.754-755: 17, 18, 34, 36, 64, 66; p.760: 34, 35, 39, 40; p.765: 35, 36.

學期成績 (對成績不滿意嗎？ Life goes on. 聽個歌吧。)