18.100B Problem Set 6

Due in class Monday, April 8. You may discuss the problems with other students, but you should write solutions entirely on your own.

1. Give a proof or a counterexample for the following assertion: suppose that f is a real-valued differentiable function on the interval (-1,1), and that f′(0) = 1. Then there is an ϵ > 0 so that f is strictly increasing on the interval (−ϵ, ϵ).

2. Text, page 114, number 4.

3. Text, page 115, number 9.

4. For this problem, you may assume that for every non-negative integer N,

lim_{x→0} |x|^{−N} exp(−1/|x|) = 0.

(This is the same statement as lim_{t→∞} t^{N} exp(−t) = 0, which may look a bit more familiar.)

The problem concerns the function

f(x) = \{ \begin{matrix}exp(−1/x) & if x > 0 \\ 0 & if x ≤ 0. \end{matrix}

a) Sketch the graph of f.

b) Show that for each k ≥ 0 there is a polynomial p_{k}(x) so that the kth derivative of f is

f^{(k)}(x) = \{ \begin{matrix}(p_{k}(x)/x^{2k}) exp(−1/x) & if x > 0 \\ 0 & if x < 0. \end{matrix}

c) Show that f^{(k)}(0) exists for all k ≥ 0.

d) For which values of x is f(x) equal to the sum of the Taylor series

∑_{k=0}^{∞} \frac{f^{(k)}(0)}{k!}x^{k}?