Instructions

This is a MIDTERM. You cannot ask for any external help and your work is strictly individual.

You may consult the the course textbooks (chapters relative to the contents covered), your personal notes, and HW solutions posted on the course website.

You may assume all statements we have covered in class and in the HW but cite their statement clearly when you use them.

All problems require proofs.

You may use integrals but they are not necessary.

You may use L'Hopital but be very careful with the hypotheses.

You hereby agree to follow the UVA Code of Conduct and abide by the Honor code.

You have 120 hours from the moment of download to do this Midterm, in any case you cannot turn in past <2020-11-26 Thu 11:00>. You can decide when download the midterm. Download and submission is through Gradescope. You MUST send a private Piazza post at the moment of download.

Please post any questions or concerns on Piazza in a private post.

1

Consider the function

\begin{equation*} f(x)=\left\{ \begin{aligned} &e^{-\frac{1}{x}} &&\text{if } x\gt 0\\ &0 &&\text{if } x\leq 0\\ \end{aligned}\right. \end{equation*}

Find all of its derivatives at \(0\). (Proof required)

2

Show that the sequence of functions

\begin{equation*} f_n(x)=2^{-n^2} x^n e^{-x^2} \end{equation*}

converge uniformly to \(0\).

3

Let \(F\colon \R \to \R\) be \(C^2\) (continuous, \(F'\) continuous, and \(F''\) continuous). Consider the map \(T\) defined on \(C^0\big([0,1];\R\big)\) with values in \(C^0\big([0,1];\R\big)\) given by

\begin{equation*} \Big(T(f)\Big):=\Big(x \to F(f(x))\Big) \end{equation*}

3.1

Show that \(F\) has bounded first and recond derivative on any set of the form \([-M;M]\).

3.2

Show that \(T\) is continuous as a map from \(\Big(C^0\big([0,1];\R\big),\, \|\cdot\|_{\sup}\Big)\) to \(\Big(C^0\big([0,1];\R\big),\, \|\cdot\|_{\sup}\Big)\).

3.3

Show that \(T\) is continuous as a map from \(\Big(C^1\big([0,1];\R\big),\, \|\cdot\|_{C^1}\Big)\) to \(\Big(C^1\big([0,1];\R\big),\, \|\cdot\|_{C^1}\Big)\).

3.4 Hints:

To solve the last two points you may assume initially that \(F'\) \(F''\) are bounded.

Then use the first point to remove that assumption.

4

Let \(f\colon(-1,1)\to \R\).

4.1

Suppose that \(f\) is differentiable at \(x_0\). Show that

\begin{equation*} \lim_{t\to 0} \frac{f(x_{0}+t)-f(x_{0}-t)}{2t} = f'(x_{0}) \end{equation*}

Careful, no information about the behavior of \(f\) outside of \(x_0\) is given, except for what follows from differentiability at \(x_0\).

4.2

Suppose that \(f\) is twice differentiable at \(x_0\). Show that

\begin{equation*} \lim_{t\to 0} \frac{f(x_{0}+t)+f(x_{0}-t)-2f(x_{0})}{t^{2}} = f''(x_{0}) \end{equation*}

Careful, no information about the behavior of \(f\) outside of \(x_0\) is given, except for what follows from differentiability at \(x_0\).

4.3

Let \(p_{1},p_{2},p_{3},p_{4}\) be four numbers in \(\R\). Show that a sufficient and necessary condition for

\begin{equation*} \lim_{t\to 0} \frac{\sum_{i=1}^{4}a_{i}f(x_{0}+p_{i}t)}{t^{3}}= f'''(x_{0}) \end{equation*}

for any \(f\colon(-1,1)\to \R\) three times differentiable at \(x_{0}\in(-1,1)\) is that

\begin{equation*} \begin{aligned} &\sum_{i=1}^{4}a_{i}p_{i}^{k}=0 \qquad \forall k\in\{0,1,2\} \\ & \sum_{i=1}^{4}a_{i}p_{i}^{3}=3! \end{aligned} \end{equation*}

Careful, no information about the behavior of \(f\) outside of \(x_0\) is given, except for what follows from differentiability at \(x_0\).

Hint: To check necessity test against \(f\) that are polynomials. To show that it is sufficient, Taylor expand.

5

Consider the function \(F\colon l^\infty_c(\N;\R)\to l^\infty(\N;\R)\) given by

\begin{equation*} F(\vec{x})_k= e^{-x_k^2}. \end{equation*}

Find a continuous extension of \(F\) to \(l^\infty_0(\N;\R)\) and show that it is unique.

Hint: Show that the derivative of \(e^{-x^2}\) is bounded.