Sweet Home for Calculus I (Fall, 2008)

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.

3. 增列MSN帳號：jnw120@hotmail.com

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.