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\title{ Math 3210-4, HW V \\ due Wednesday Nov 13}
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1) Let $f: \mathbb R \rightarrow \mathbb R$ such that $|f(x)-f(y)| \leq (x-y)^2$. Prove that $f$ is constant.
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2) Let $f$ be a differentiable function defined in a neighborhood of $x$. Assume that $f''(x)$ exists. Use L'Hospital's rule to prove that
\[
\lim_{h\rightarrow 0} \frac{f(x+h) + f(x-h) -2 f(x)}{h^2} = f''(x).
\]
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3) (Fixed Point Theorem.) Let $f: \mathbb R \rightarrow \mathbb R$ such that $|f'(x)| \leq C$ for some $0\leq C< 1$ and all $x$. A number $x$ is a fixed point for $f$ if
$f(x)=x$. Prove that $f$ cannot have two fixed points.
Let $x_1$ be any real number, and define
a sequence by $x_{n+1}=f(x_n)$. Prove that the sequence $\{x_n\}$ is Cauchy. (Hint: $|x_{n+1}-x_n| \leq C|x_n -x_{n-1}|$.) Prove that the limit is a fixed
point of $f$.
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4) (Concavity.) Let $f: (\alpha,\beta) \rightarrow \mathbb R$ be twice differentiable function such that $f''\geq 0$ on the interval. Let $c\in (\alpha,\beta)$ and let $g(x)$ be the linear
function whose graph is the tangent line of the graph of $f$ at $c$ i.e. $g(x)=f(c) + f'(c)(x-c)$. Prove that $f(x)\geq g(x)$ for $x\in (\alpha,\beta)$.
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5) (Newton Method.)
Let $f: \mathbb R \rightarrow \mathbb R$ be twice differentiable function. Let $[a,b]$ be a closed interval such that $f(a) <0$ and $f(b)>0$, $f'(x) \geq \delta>0$,
and $f''(x)\geq 0$ for $x\in [a,b]$. Prove that there is unique $c\in (a,b)$ such that $f(c)=0$. Define a sequence by $x_1=b$ and
\[
x_{n+1} = x_n -\frac{f(x_n)}{f'(x_{n})}
\]
Prove that the sequence is decreasing and bounded from below by $c$, it has a limit. (Hint: interpret the sequence using the tangent line of the graph of $f$ at $x_n$, and use
the previous exercise.)
Prove that the limit is $c$. Check that the conditions are
satisfied for $f(x)=x^2-2$ and the interval
$[1,2]$. What is the limit of the sequence $\{x_n\}$? Compute $x_n$ for $n=1,2,3,4$.
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6) Consider the power series $x-\frac{x^3}{3!} + \frac{x^5}{5!} - \ldots$, i.e. the sequence whose $n$-th term is $(-1)^{n-1}\frac{x^{2n-1}}{(2n-1)!}$. Compute the
radius of convergence of this series. Use the theorem of Taylor to prove that $\sin(x)=x-\frac{x^3}{3!} + \frac{x^5}{5!} - \ldots$ for every $x$.
Use this series to find a rational number that approximates $\sin(1/2)$ with an error less than $1/10^3$.
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