© | Dror Bar-Natan: Classes: 2004-05: Math 157 - Analysis I:

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Homework Assignment 20

Assigned Tuesday March 8; not to be submitted.

this document in PDF: HW.pdf

Required reading. All of Spivak's Chapter 20.

In class review problem(s) (to be solved in class on Tuesday March 15). Chapter 20 problem 16:

1. Prove that if $f''(a)$ exists, then
   $$f''(a)=\lim_{h\to 0}\frac{f(a+h)+f(a-h)-2f(a)}{h^2}.$$
   The limit on the right is called the Schwarz second derivative of $f$ at $a$. Hint: Use the Taylor polynomial $P_{2,a}(x)$ with $x=a+h$ and with $x=a-h$.
2. Let $f(x)=x^2$ for $x\geq 0$ and $-x^2$ for $x\leq 0$. Show that
   $$\lim_{h\to 0}\frac{f(0+h)+f(0-h)-2f(0)}{h^2}$$
   exists, even though $f''(0)$ does not.
3. Prove that if $f$ has a local maximum at $a$, and the Schwartz second derivative of $f$ at $a$ exists, then it is $\leq 0$.
4. Prove that if $f'''(a)$ exists, then
   $$\frac{f'''(a)}{3} = \lim_{h\to 0}\frac{f(a+h)-f(a-h)-2hf'(a)}{h^3}.$$

Recommended for extra practice. From Spivak's Chapter 20: Problems 3, 4, 5, 6, 9, 18 and 20.

Just for fun. According to your trustworthy professor, $$P_{2n+1,0,\sin}(x)=\sum_{k=0}^n(-1)^k\frac{x^{2k+1}}{(2k+1)!}$$ should approach $\sin x$ when $n$ goes to infinity. Here are the first few values of $P_{2n+1,0,\sin}(157)$:

$n$	$P_{2n+1,0,\sin}(157)$
0	157.0
1	-644825.1666
2	794263446.1416
3	-465722259874.7894
4	159244913619814.5429
5	-35629004757275297.7787
6	5619143855101017161.3172
7	-658116552443218272478.0047
8	59490490719826164706638.3418
9	-4275606060900548165855463.4918
10	250142953226934230105633222.4574
100	$\sim 5.653\cdot 10^{63}$

In widths of hydrogen atoms that last value is way more than the diameter of the observable universe. Yet surely you remember that $\vert\sin 157\vert\leq 1$; in fact, my computer tells me that $\sin 157$ is approximately -0.0795485. In the light of that and in the light of the above table, do you still trust your professor?

The Small Print. For $n=(200, 205, 210, 215, 220)$ we get $P_{2n+1,0,\sin}(157)=(1.8512\cdot 10^8, -13102.9, 0.648331, -0.0795805, -0.0795485)$.

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