\input amssym.def
\def\d{\displaystyle}
\input epsf
\def\d{\displaystyle}
\nopagenumbers
\magnification= \magstep1

\parindent=0pt
\font\largerbold = cmbx10 scaled \magstep2

\centerline {\largerbold Problems, Week 3}
\bigskip
%integration by parts. trig ints. partial fractions. lectures 4  and 5
%on the syllabus.
 


1.~a) There is a rational number\footnote*{\sevenrm A rational number is a
quotient of two integers}, $A$, so that

$$\int_2^3 {1\over {x(x+1)}} dx = \ln A\; .$$

\hskip 3in What is $A$?

\medskip

b) There is an algebraic number\footnote{**}{\sevenrm An algebraic number is a
number which can be obtained from integers using addition,
subtraction, multiplication, division \underbar{and} root
extractions.}, $B$, so that

$$\int_3^4 {1 \over {x(x+1)(x+2)}}\, dx = \ln B$$

\hskip 3in What is $B$?

\bigskip
\settabs 2 \columns

2.~A program called {\sl Macsyma} can do 
integration. It is a bit old, but it can do 4 of the following 5
integrals in a total of less than 5 seconds of CPU time. Please do
3 of these integrals. (You choose which ones!)
\medskip
\+a) $\displaystyle{\int x \sin (x^2) \, dx}$& %16msec
b)$\displaystyle{\int x \left( \sin x \right)^2 \,dx}$\cr %750msec
\+c) $\displaystyle{\int x^2 \sin x \, dx}$& %816msec
d) $\displaystyle{\int x^2 \left( \sin x \right)^2 \, dx}$\cr %1116msec
\+e) $\displaystyle{\int x^2 \sin (x^2) \, dx}$& \cr %1416msec answer
                                                    %in terms of erf


\bigskip

\line{3.\hfil $\underline{{\rm A\ lesson\ in\ approximating\ }\sin x}$ \hfil}
\medskip
a) For $x$ near $0$, $\sin x$ is well-approximated by its tangent line
at $x=0$. What is this tangent line?

\medskip

b) For many purposes, approximation over an interval is preferred
over approximation near a point. There are many criteria for assessing
the accuracy of such approximations. One is {\it mean-square error}.
The mean-square error between a straight line $y = Ax$ going through the origin
and the function $\sin x$ over the interval $[0, 1]$ is given by the
definite integral 

$$ \int_0^1 \left( \sin x - Ax \right)^2 \, dx$$

Find the $A$ which $\underline{\rm minimizes}$ this integral. (Hint:
expand the integrand and compute everything. You'll get a
simple function of $A$. Minimize this function.)

\medskip

c) Construct a table, the first column of which is  $x$, as $x$ ranges from
$0$ to $1$ in increments of $1 \over 10$. The other three columns of the
table should be the corresponding values of $\sin x$, of the tangent line
approximation to $\sin x$ obtained in a), and of the mean-square
approximation, $Ax$, to $\sin x$ obtained in b). {\it Use a
calculator!} 

\medskip

d) Make a rough sketch over the unit interval $[0,1]$ of the three
curves whose data points you have tabulated in c).

\bigskip


4.~a) When $D$ is very large, show that the integral
$\displaystyle{\int_1^2 e^{-Dx} \, dx}$ is very small. (You may wish
to draw a picture, but other verification is also necessary.)

\medskip

b) When $D$ is very large, show that the integral
$\displaystyle{\int_1^2 x e^{-Dx} \, dx}$ is very small. (You may wish
to draw a picture, but other verification is also necessary.)

\medskip

c) When $D$ is very large, show that the integral
$\displaystyle{\int_1^2 {1 \over {1+5 x^{48}}} e^{-Dx} \, dx}$ is very
small\footnote{$\dag$}{\sevenrm It is {\sevenbf not} always necessary
or even possible to compute every integral exactly. But if the
integral can be estimated easily $\ldots$}. (You may wish to draw a
picture, but other verification is also necessary.)

\bigskip

5.~Consider the function ${G}(x)= e^x \sin Nx$ \underbar{on the
interval} $[0,1]$. 
\medskip
a) With a sketch or otherwise, describe this function when $N=3$ and
when $N=10$ and 
when $N=10^{10}$.
\medskip
b) Compute $\d{\int_0^1 {G} (x)\, dx}$ if $N=10^{10}$. 
\smallskip
c) Explain b)'s result in relation to a).

\bigskip

6.~a) Compute $\d{\int_1^2 {{dx} \over {x^2}}}$.
\medskip
b) Compute $\d{\int_1^2 {{dx} \over {x(x-m)}}}$ if $m$
is a small positive number.
\medskip
c) What happens to your answer to b) when $m \to 0^+ \,$?
\medskip
d) Compute $\d{\int_1^2 {1 \over {x^2 + n}} \, dx}$ if $n$ is a small
positive number. 
\medskip
e) What happens to your answer to d) when $n \to 0^+ \,$?
\medskip
f) Pictures? Comments??

\bigskip

7.~Many integrals can be done with ``rationalizing substitutions'' which
change the integrals into integrals involving rational functions. In
turn, these integrals can be computed (theoretically) using partial
fractions. Here are two examples of such integrals:
\medskip
a) Integrate:
$$ \int {1 \over { (\sqrt {x} + 1) ( \sqrt {x} +2 )}} \, dx$$
\rightline {Try $x = t^2$.}
\medskip
b) Integrate:
$$ \int {{e^{x} + 1} \over { e^{2x} + 1}} \, dx$$
\rightline {Try $t = e^x$.}

\bigskip

8.~The following information is known about a function $T$ and its
derivatives: 
\medskip

\hskip 1.7in \vbox{
\def\tablerule{\noalign{\hrule depth0pt height.5pt width1.7in}}
\halign to \hsize{ \hfil $ # $\strut\quad \vrule\thinspace
& \hfil $ #
$\quad \vrule\thinspace &
\hfil $ # $\quad \vrule\thinspace
& \hfil $ # $\cr 
\ x \negthinspace\hfil &\  T(x)
\negthinspace\negthinspace\negthinspace\hfil  &\
T'(x)\negthinspace\negthinspace\negthinspace\negthinspace  \hfil &\
T''(x)\negthinspace\negthinspace\negthinspace\negthinspace  \hfil\cr
\tablerule
\tablerule
1 & 2 & -2 & 2 \cr
\tablerule
2 & 3 & 6 & 5 \cr
\tablerule
3 & 7 & 4 & -4\cr
\tablerule
4 & 2 & 5 & 7 \cr}}

\medskip
a) Compute $\d{\int_2^3 T'(x)\, dx}$.
\medskip
b) Compute $\d{\int_2^3 T''(x)\, dx}$.
\medskip
c) Compute $\d{\int_2^3 x dx}$.
\medskip
d) Compute $\d{\int_2^3 x T''(x)\, dx}$.

\rightline{\sevenrm Better not look at b) and c)! Integrate by parts.}

\medskip
e) Compute $\d{\int_2^3 x^2T'''(x)\, dx}$.

\rightline{\sevenrm And again and again.}

\medskip


\vfil\eject\end


% Local Variables: 
% mode: tex
% TeX-master: t
% End: 
