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\title[Problem Set 5]{Problem Set 5.  Due Tuesday, 12 September}

\begin{document}

\maketitle



\noindent {\bf Reading.} {\em Quick Calculus}, pp. 138--142; 148--167.

\noindent {\bf Supplementary reading.} Simmons, Sections 5.1--5.3.

\begin{enumerate}

\item (2pts) Approximate the following numbers, using the tangent
line approximation.

\begin{enumerate}
\item $^3{\hskip-0.1in \sqrt{28}}$
\item $\sqrt{102}$
\end{enumerate}

\medskip

\item (2pts) Find the Taylor series (at $x=0$) for $f(x)=\frac{1}{1-x}$.

\medskip

\item (4pts) A sphere of radius $r$ has volume
$$
V(r)=\frac{4}{3}\pi r^3,
$$
and surface area
$$
A(r)=4\pi r^2.
$$
Approximate the volume and surface area of a sphere of radius
$7.02$cm.  You can check your answer by using a calculator to compute
the volume and surface area exactly.

\medskip


\item (4pts) Compute the following integrals.

\begin{enumerate}
\item $\int x^3\ dx$
\item $\int \sin(x)\ dx$
\item $\int e^x\ dx$
\item $\int \sqrt{x}\ dx$
\end{enumerate}

\medskip


\item (4pts) Compute the following integrals by substitution, using
the substitution given.

\begin{enumerate}
\item $\int \sqrt{5+7x}\ dx,\ u=5+7x$
\item $\int \frac{2x}{\sqrt{3+x^2}}\ dx,\ u=\sqrt{3+x^2}$
\item $\int 2xe^{x^2}\ dx,\ u=x^2$
\item $\int \frac{dx}{(x-4)^5},\ u=x-4$
\end{enumerate}

\medskip


\item (4pts) Integrals satisfy
$$
\int(f(x)+g(x))\ dx=(\int f(x)\ dx)+(\int g(x)\ dx),
$$
just like derivatives do.  Again like derivatives, they do {\bf not}
satisfy a simple product rule:
$$
\int( f(x)\cdot g(x))\ dx\neq(\int f(x)\ dx)\cdot(\int g(x)\ dx),
$$
Check that this is indeed not true by using $f(x)=x$ and $g(x)=x$, and 
computing both sides of the above equation.  
%That is, compute
%$$
%\int( x\cdot x)\ dx
%$$
%and show that it is not equal to
%$$
%(\int x\ dx)\cdot(\int x\ dx).
%$$
\end{enumerate}





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