Calculus II, Spring 2016 (http://www.math.nthu.edu.tw/ wangwc/)

Solutions to selected problems in HW for Week 04

1. Let

f(x) = \{ \begin{matrix}0, & x = 0; \\ e^{−1/x^{2}}, & x ≠ 0. \end{matrix}

It is known that f^{(n)}(0) = 0 for all n. Verify this for f′(0) and f′′(0). Proof.

lim_{x→0} \frac{f(x) − f(0)}{x − 0} = lim_{x→0} \frac{e^{−1/x^{2}}}{x} = lim_{t→∞} \frac{e^{−t}}{1/\sqrt{t}} = 0 (L’Hôspital’s Rule)

⇒ f′(0) = 0.

Check that

f′(x) = \{ \begin{matrix}0, & x = 0; \\ 2x^{−3}e^{−1/x^{2}}, & x ≠ 0. \end{matrix}

lim_{x→0} \frac{f′(x) − f′(0)}{x − 0} = lim_{x→0} \frac{2x^{−3}e^{−1/x^{2}}}{x} = 2 lim_{t→∞} \frac{e^{−t}}{1/t^{2}} = 0 (L’Hôspital’s Rule)

⇒ f′′(0) = 0.