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\title{MTH436 - Homework 3}

\begin{document}
 \maketitle
 
 Solutions to the questions in Section B should be submitted by the start of class on 2/21/18.
 
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 \section{Warm-up Questions}
 
 \begin{que}
  Compute the derivatives (if they exist) of the following functions.
  \begin{enumerate}
   \item[(i)] $f(x) = x^2+1$ if $x$ is rational and $f(x) = 2x$ if $x$ is irrational.
   \item[(ii)] $f(x) = x|x|$
   \item[(iii)] $f(x) = \frac{\sin(x^2)}{x}$ for $x \neq 0$ and $f(0) = 0$.
   \item[(iv)] $f(x) = x^2 \sin \left( \frac{1}{x^2} \right)$ for $x \neq 0$ and $f(0) = 0$. Prove also that $f'$ is unbounded on the closed interval $[-1,1]$ and so is not continuous.
  \end{enumerate}

 \end{que}

 \begin{que}
  Prove that if $f,g \colon I \to \R$ are differentiable at $c$ then $(f+g)$ is differentiable and $(f+g)'(c) = f'(c) + g'(c)$.
 \end{que}
 
 \begin{que}
  Suppose $f \colon \R \to \R$ and $|f(x) - f(y)| \leq (x - y)^2$ for all $x,y \in \R$. Prove that $f$ is constant.
 \end{que}

 
 \begin{que}
  Prove that the derivative of an even function is an odd function, and vice versa.
 \end{que}

 \begin{que}
  Let $f \colon [0,\infty) \to \R$ be differentiable and suppose that $\lim_{x \to \infty} f'(x) = M$. Find $\lim_{x \to \infty} (f(x+1) - f(x))$.
 \end{que} 

 \begin{que}\mbox{}
 \begin{enumerate}
  \item[(i)] Let $f(x) = x + 2x^2 \sin \left( \frac{1}{x} \right)$ for $x \neq 0$ and $f(0) = 0$. Show that $f'(0)= 1$ but $f$ is not increasing in any neighbourhood of $0$.
  \item[(ii)] Let $f(x) = 2x^4 + x^4 \cos \left( \frac{1}{x} \right)$ for $x \neq 0$ and $f(0) = 0$. Show that $0$ is a global minimum for $f$ but for every neighbourhood $V$ of $0$ there exists $x,y \in V$ such that $f'(x)>0$ and $f'(y) < 0$.
  \end{enumerate}
 \end{que}

 
 
 
 
 
 
 
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 \section{Submitted Questions}
 
 \begin{que}
  Suppose $f \colon [a,b] \to \R$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Let $c$ be an interior point of $I$ and suppose $\lim_{x \to c} f'(x) = L$. Prove that $f'(c) = L$.
 \end{que}

 
 \begin{que}
  Let $f \colon [0,\infty) \to \R$ be differentiable. Prove or disprove the following.
  \begin{enumerate}
   \item[(i)] If $\lim_{x \to \infty}f(x) = L$ then $\lim_{x \to \infty} f'(x) = 0$.
   \item[(ii)] If $\lim_{x \to \infty} f(x) = L$ and $\lim_{x \to \infty} f'(x) = M$ then $M=0$.
   \item[(iii)] If $\lim_{x \to \infty} f'(x) = M$ then $\lim_{x \to \infty} \frac{f(x)}{x} = M$. 
  \end{enumerate}

 \end{que}

 
 
 
 
 
 
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 \section{Challenge Questions}
 
 \begin{que}
  Let $f \colon [0,2] \to \R$ be continuous on [0,2] and differentiable on $(0,2)$. Suppose further that $f(0) = 0$, $f(1) = 1$ and $f(2) = 1$. Show there exists $c \in (0,2)$ such that $f'(c) = \frac{1}{\pi}$.
 \end{que}

 \begin{que}
  Let $C^1[0,1]$ be the set of functions on $[0,1]$ for which $f'$ is continuous. Show that 
  \[
d(f,g) = \sup \{ |f(x) - g(x)| \mid x \in [0,1] \} + \sup \{ |f'(x) - g'(x)| \mid x \in [0,1] \}
\]
defines a metric on $C^1[0,1]$. 
 \end{que}

 
 
 
 
 
 
 \end{document}