Calculus I, Fall 2012 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 11

1. Section 6.6: Problems: 17, 21, 27, 33, 37-46, 51, 54, 55.

2. (s6.6-extra1) Cauchy’s Mean Value Theorem Prove the following variant of the Mean Value Theorem: Suppose f and g are continuous on [a, b] and differentiable on (a, b), then there exists c ∈ (a, b) such that

| \begin{matrix}f(b) − f(a) & f′(c) \\ g(b) − g(a) & g′(c) \end{matrix} | = 0.  (1)

Note that, this c need not satisfy \frac{f(b) − f(a)}{b − a} = f′(c), nor \frac{g(b) − g(a)}{b − a} = g′(c), but

only \frac{f(b) − f(a)}{g(b) − g(a)} = \frac{f′(c)}{g′(c)}, which is the same as (1) provided g(b) − g(a) ≠ 0.

Hint: Apply standard Mean Value Theorem to

F (x) = | \begin{matrix}f(b) − f(a) & f(x) − f(a) \\ g(b) − g(a) & g(x) − g(a) \end{matrix} | on [a, b].

3. (s6.6-extra2) Use Cauchy’s Mean Value Theorem to prove the strong form of l’Hôpital’s rule.

4. Section 6.7: Problems: 5, 7, 8, 11, 12, 17(a), 18.

5. Section 6.9: Problems: 9, 13, 21, 23, 31, 33, 39, 43, 57, 61, 67, 71, 79.

6. (s6.9-extra1) Show that

\frac{d csc^{−1} y}{dy} = \frac{−1}{|y|\sqrt{y^{2} − 1}}, |y| > 1

Explain why the negative sign is chosen. You will need to start from the ‘restricted’ domain of csc to see this.