Home for Calculus I (Fall, 2010)

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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. 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 cosx.

5. Observation of limits.

6. Homework #1 (due 9/28): p.76:78;p.83-85:35,47,49,53,54,58;p.91-92:6,34,42;p.101-103:5-10,55,63,65,67,68.

Week 2

1. Precise definition of limit. Finding the relation between δ and ε is the key to prove the existence of the limit.

2. One-sided limits.

3. 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. Removable discontinuities.

5. The composite of two continuous functions is continuous.

6. Quiz #1 on 10/5.

Week 3

1. The intermediate value property for a continuous function. Locating roots of a function.

2. The sign preserving property for a continuous function.

3. Differentiability, derivatives.

4. Differentiability implies continuity.

5. Product rules, quotient rule.

6. Derivatives of trigonometric functions.

7. The chain rule.

8. Homework #2 (due 10/12):p.132-134,32,41,42,57,58;p.144-145,70,72;p.160-161,32,52,57,59;p.168-169,68,70,104;p.175,50,52;p.185,65,82;p.191,40,42.

Week 4

1. Implicit differentiation, derivatives of the inverse function.

2. Derivatives of inverse trigonometric functions.

3. Linearization or linear approximation.

4. Applications of derivatives--finding the maximum and minimum.

5. Definition of critical points.

6. Quiz#2 (10/19)

Week 5

1. Finding the extreme values of a function in [a,b].

2. Mean value theorem.

3. Monotonic functions and the first derivative test.

4. Concavity, point of inflection.

5. Generalized mean value theorem.

6. The second derivative test.

7. Graph of a function.

8. Homework #3 (due 10/26):p.199-200,29,43;p.211,44,63;p.229,77,82,84;p.236-237,16,61,62,67,68,70;p.242-243,71,73,80,81;p.252-253,32,53,111;p.261-262,69,70,76,85,88.

Week 6

1. Indeterminate forms and the L'Hopital rule. Remember to check that the limit is an indeterminate form before applying the  L'Hopital rule.

2. Newton's method.

3. Riemann sum. Upper and lower Riemann sum.

4. Integrability.

5. Quiz #3 (11/2).

6. Midterm is on 11/9.

Week 7

1. Let f(x) be continuous on [a,b], then f is integrable. Give an example that is not integrable.

2. Mean value theorem for an integral.

3. Definite integrals. Fundamental Theorem of Calculus. Indefinite integrals and antiderivative.

4. Change of variable or substitution method.

5. Volume of a solid: method of slicing.

6. Volume of a solid of revolution: the disk method and the shell method.

7. Arclength.

8. Homework #4 (due 11/11):p.271-274,39,48,59,60(a),63;p.277-278,9,10,11;p.324,87;p.335,80,83,84;p.343-344,67,69,70,79;p.371-373,2,10,18,44,60;p.380-381,30,45;p.386-387,19,21;p.411-413,8,17,28,34.

Week 8

1. Moments. Center of mass of a rod.

2. Centroid of a plane region.

3. Areas of surfaces of revolution.

4. Pappus's theorems.

5. Review for the midterm.

Week 9

1. Transcendental functions.

2. Definition of logarithmic function by integral.

3. Exponential growth and decay.

4. Simple differential equations, initial value problems.

Week 10

1. Examples of exponential growth and decay.

2. Hyperbolic functions, inverse hyperbolic functions, derivatives.

3. Techniques of integration -- integration by parts.

4. Homework #5 (due 11/25):p.434-435,29,31,34,43,46;p.442-443,73,77,83,84,86;p.460-461,46,49,69,71,73.

Week 11

1. Method of partial fraction.

2. Trigonometric integrals.

3. Trigonometric subsitutions.

4. Numerical integration.

5. Quiz #4 (11/30).

Week 12

1. Improper integrals, Type I and Type II.

2. Test for convergence : comparison test and limit comparison test.

3. Homework #6 (due 12/9):p.466-467,28,30,43,44,56,60,65,67,68;p.470-471,26,33,36,45,46,47,48,57,58;p.479-480,32,37,46,50;p.505,42,61,62,63,64.

4. Solving ordinary differential equations, initial value problems.

5. Geometrical interpretation of differential equations; Slope fields.

6. Linear first order differential equations (homogeneous and nonhomogeneous), integrating factor.

7. Separable ordinary differential equations.

Week 13

1. Euler's method (one-step method).

2. Applications: mixture problems, orthogonal trajectories.

3. Phase plane analysis, autonomous systems.

4. Stable and unstable equilibrium solutions.

5. Logistic population model.

6. Quiz #5, #6 (12/14).

Week 14

1. Homework #7 (due 12/23):p.527-528,12,19,22,26,30,32;p.534,6,9,13,16;p.540-541,14,18,20;p.546,9,10,11,12,13.

2. System of ordinary differential equations, phase-plane analysis.

3. Definition of a sequence.

Week 15

1. Convergence of a sequence.

2. Monotonic sequences, increasing or decreasing sequences.

3. Any increasing sequence which is bounded above is convergent.

4. Definition of a series, geometric series, harmonic series, telescoping series, p-series, convergence of a series.

5. Necessary condition of a convergent series: a_n→0.

6. Tests for convergence of a positive series: Integral test

7. Homework #8 (due 12/30):p.560-561,100,101,102,103;p.569-570,33,34,43,44,54,61,68,83,84,85,86,87,88,94;p.575-576,26,31,40,47,48,53,57.

8. Comparison test and limit comparison test.

9. Ratio test and root test.

Week 16

1. Note.

2. Alternating series test.

3. Absolute convergence and conditional convergence.

4. Power series.

5. Quiz #7 (1/4)

6. Final exam (1/11)

Week 17

1. Radius of convergence, interval of convergence.

2. Differentiation and integration term-by term.

3. Taylor's series and Maclaurin's series.

4. Applications of Taylor's series.