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## Due November 16

### Problem 1

If

$A=\begin{pmatrix} 1/12 & 5/8\\ 5/8 & 1/12\end{pmatrix}$,

then the map $x\mapsto Ax$ is a contraction in $\R^2$.

### Problem 2

Let $F(x,t) = \dfrac{tx}{x^2+1}$. Then, for all $t\in\R$,

$|F(x,t) - F(y,t)| \le |t||x-y|$.

### Problem 3

The function $f(x) = \sqrt x$ on $[0,\infty)$ is uniformly continuous but not Lipschitz.

### Problem 4

Consider the operator $\Phi:C([-1,1])\to C([-1,1])$ given by

$\displaystyle \Phi(x)(t) = 1 + 2 \int_0^t s x(s) ds,$

for any $x(t)\in C([-1,1])$. Starting from the constant function $x_0(s)=1$, verify explicitly that the nth iteration of $x_{n+1} = \Phi(x_n)$ is the nth Taylor polynomial of $t\mapsto e^{t^2}$ around $t=0$.