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\fancyhead[L]{\Large \textbf{Math 158 Spring 2014}}
\fancyhead[R]{\large \textbf{Homework 2}}


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\section*{Some new problems}


\bp Read through section 4.20 in the text and do exercises 1, 8, 9 in
section 4.19 (page 191) and exercises 2, 3, 19, 21, 28 in section 4.21
(page 194-196)
\ep


\bp True or False:  Let $\ds f(x)=(x-1)^{\frac{2}{3}}$.  Since $f(0)=1$ and
  $f(2)=1$, there is some point $c\in (0,2)$ with $f'(c)=0.$
\ep

\bp Prove:
\ben
\item If $f$ satisfies $|f(s)-f(t)|\leq |s-t| \text{ for all }s,t,$
  then $f$ is continuous.  
\item If $f$ satisfies $|f(s)-f(t)|\leq |s-t|^2 \text{ for all
}s,t,$ then $f$ is constant.
\een
\ep

\bp It is hard, or maybe even impossible, to
determine $$\int_{-\frac{\sqrt{\pi}}{2}}^{-\frac{\sqrt{\pi}}{2}}
\cos(x^2)dx$$ exactly.  But $g(x)=\cos(x^2)$ can be approximated by a
polynomial.   Your problem: find a polynomial $p(x)=ax^4+bx^3+cx^2+dx+e$ that
satisfies $p(0)=g(0)$, $p'(0)=g'(0)$, $p''(0)=g''(0)$,
$p'''(0)=g'''(0)$, and $p''''(0)=g''''(0)$.
Use a
computer to graph $g$ and $p$ in the same picture.  Approximate $
\int_{-\frac{\sqrt{\pi}}{2}}^{-\frac{\sqrt{\pi}}{2}}\cos(x^2)dx$ by
computing $$\int_{-\frac{\sqrt{\pi}}{2}}^{-\frac{\sqrt{\pi}}{2}}
p(x)dx$$ instead.
\ep

\bp Suppose that $f(4)=0$, $g(4)=0$, $f'(4)=7$, and $g'(4)=-1.$  Prove
that $\ds \lim_{x \to 4}\frac{f(x)}{g(x)}$ exists and compute it.
\ep


\section*{Some review problems}

\bp Let $f$ be a function defined on an open neighborhood of $c$.
Define the statement ``$f$ is differentiable at $c$'' and the number
$f'(c).$
\ep

\bp Prove that if $f$ is differentiable at $x=c$ then $f$ is
continuous at $x=c.$
\ep

\bp Prove that if $f:\R \to \R$ is differentiable and $f(c)\geq f(x)$ for all
  $x\in \R$, then $f'(c)=0$.
\ep

\bp Prove that if $f'(x)>0$ for all $x\in (0,1)$, then $f$ is
increasing on $(0,1)$.
\ep


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