Home for Calculus I (Fall, 2009)

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.

5. If a function f:X-->Y is not one-to-one or onto, one may be able to modify (shrink) the domain X or the range Y such that on smaller X or Y, the function f is one-to-one and onto. With such modification, we find define the inverse of f. Read what we did for sinx and tanx.

Week 2

1. Homework #1 (due 9/29): p.100:54,57;p.114:78;p.134:60;p.142-143:1,6,15,31,32;p.145-146:14,17,25,27.

2. Precise definitions of limits. Finding the relation between δ and ε is the key to prove the existence of the limit.

3. 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 one-sided limit.

4. The composite of two continuous functions is continuous.

5. Definition of removable discontinuities.

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 gave the definition of differentiability. 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, but not the other way around.

3. Derivatives of some elementary functions.

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

5. The chain rule (important): derivative of a composite function.

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 10/6 (Chapter 2).

Week 4

1. Homework #2 (due 10/13):p.189:47,48;p.204:102,103;p.212:59,60;p.219:19;p.233-234:51,61;p.237:68;p.242:21,22,23.

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

3. Derivatives of inverse trigonometric functions.

4. Linear approximations.

5. Related rate problems.

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, that is, c is a critical point.

Week 5

1. Second quiz on 10/20.

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

3. Rolle's theorem and mean value theorem.

4. Generalized mean value theorem (Cauchy mean value theorem) and Taylor's foirmula.

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

6. Limits of indeterminate forms.

7. Applications of the generalized mean value theorem to prove L'Hopital's rule.

Week 6

1. Homework #3(due 10/29): Taylor's formula, p.253:54;p.254:58;p.262:55,59,64;p.266:8;p.276-277:68,76,84;p.287:25;p.290-291:52,53,55;p.298:22,24,34;p.306:26,27.

2. Reminder: the first midterm is scheduled on 11/10!

3. Examples of indeterminate limits. Remember to check that the limit is an indeterminate form before applying the  L'Hopital rule.

4. Newton's method, quadratic convergence.

5. Definition of a definite integral.

6. Riemann sum, upper and lower Riemann sums.

7. Fundamental Theorem of Calculus.

Week 7

1. Fundamental theorem of Calculus -- evaluation of a definite integral.

2. Method of change of variable.

3. Applications of integration.

4. Evaluate the area of a bounded region.

5. Find the volume of a body: slicing method (it is called the disk method in the case of the solid of revolution) and shell method.

6. The formula of the length of a differentiable curve.

7. Homework #4 (due 11/10): p.367:67;p.375:51,52;p.386:78;p.390-393:67,73,5,7,8,22;p.406-407:4,7,45;p.415-416:13,24,35;p.423:15,27;p.434-435:22,26,40;p.445-446:20,26,28,29.

Week 8

1. Centroid of a bounded region.

2. Pappus' theorem for the solid of revolution.

3. Centroid of a differentiable curve.

4. Area of the surface of revolution.

5. Pappus' theorem for the surface of revolution.

Week 9

1. First midterm: 11/10.

2. Transcendental functions.

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

Week 10

1. Quiz#4 on 11/24.

2. Hyperbolic functions.

3. Integration by parts.

4. Method of partial fraction.

Week 11

1. Homework #5 (due 12/1): p.559-560:60,95,96;p.569-570:29,44,48,49;p.579-580:27,34,39;p.586:27,32,38;p.591-592:14,15,17,22,44,48.

2. Method of partial fraction.

3. Trigonometric substitution.

4. tan(θ/2) subsitution.

5. Numerical integration: trapezoidal rule, Simpson's rule.

Week 12

1. Introduction of improper integrals.

2. Improper integrals: unbounded domains, unbounded functions.

3. Tests of convergence: comparison test, limit comparison test.

4. Quiz#5 (12/8): techniques of integration (not include improper integrals).

Week 13

1. Further applications of integration: solving ordinary differential equations.

2. Separable differential equations.

3. Quiz#5 and #6 (12/10): Chapter 8.

Week 14

1. First order differential equations: finding an integrating factor.

2. Analysis of an autonomous differential equation: equilibrium solutions (or values), stable and unstable equilibrium solutions.

3. Logistic model.

4. Numerical method: Euler's method.

5. No class on 12/17.

Week 15

1. Homework #6 (due 12/22): §9.1:15,16;§9.2:3,10,11,18,22,25,26;§9.4:14,20;§9(Additional and Advanced Exercises):5,6.

2. Conic sections: parabola, ellipse, hyperbola.

3. Definition of eccentricity.

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

5. Polar coordinates.

6. Definition of a sequence.

Week 16

1. Convergence of a sequence.

2. Definition of a series.

3. Convergence of a series.

4. Convergence implies the necessary condition, a_n→0.

5. Integral test.

6. Comparison and limit comparison tests.

7. Ratio and root tests.

Week 17

1. Homework #7:

§11.1,86,88,91,116,119,120;§11.2,19,30,33,36,38,39,40,49,50,57,58,66,68,77;§11.3,17,18,21,39,41,42;§11.4,25,26,30,35,36;§11.5,38,44,46,47;§11.5,6,7,9,10,41,43,44,51,54,55,57,58.

2. Final exam on 1/12.