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\magnification= \magstep1
\rightline{5/1/2005}
\parindent=0pt
\font\largerbold = cmbx10 scaled \magstep2
%\smallskip
\centerline{\bf Some Math 135 review problems for the second half of the course}

\medskip

These problems are mostly from previous Math 135 finals. Almost any
Math 135 final will have problems covering this material. Students who
are prepared to do problems of this type successfully should succeed
on the final. But I'm {\it not}\/ writing the final, and personal
``style'' does alter problem selection and presentation.

\smallskip
{\bf Linear approximation}

1.~Suppose you know that $f(5)=7$, $f'(5)=2$, $g(7)=3$, and
$g'(7)=8$. If $F(x)=g(f(x))$, compute $F(5)$ and $F'(5)$. Use linear
approximation or differentials to get an approximate value of
$F(5.02)$.

2.~Suppose $f(x)=\tan(x^2)$. Use linear approximation or differentials
to find an approximate value of $f\!\left(\sqrt{{\pi\over
4}}-.03\right)$.

\smallskip

{\bf Optimization}

1.~A box with an open top is to be made from a rectangular sheet of
cardboard 5 inches by 8 inches by cutting equal squares out of the
four corners and bending up the resulting four flaps to make the sides
of the box. Use calculus to find the largest volume of the box. Be
sure to explain briefly why your answer gives a maximum.

{\hsize =4.2in\vtop{
2.~Find the largest and smallest values of $f(x)=30x^3-90x^2+90x+100$
when $2\le x\le 3$. Justify your answer with calculus.


% domain range

%\bigskip

3.~A rectangle is inscribed as shown in the parabola
$y=3-x^2$. Of all such rectangles, find the dimensions of the one
whose area is a maximum. Explain briefly why your answer gives a
maximum.
}}

\vskip -1in
\rightline{\epsfxsize=1in \epsfbox{box.eps}}
\vskip -.05in
{\hsize =3.85in\vtop{4.~A triangle has two equal sides each 4 inches
long. What is the length of the third side if the triangle is to have
maximum area? Be sure to explain why your answer is a maximum. The
diagram may be useful. Use it to express an algebraic relationship
between $H$ and ${1\over 2}S$ and then compute the area of the triangle.}}

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\smallskip
{\bf Intermediate Value Theorem; MVT}

1.~Suppose $f(x)={1\over{101}}x^{101}+{5\over {37}}x^{37}+{8\over
5}x^5+46x+8$. What is $f'(x)$? For which $x$ is $f'(x)$ positive? For
which $x$ is $f'(x)$ negative? Describe briefly how calculus can
verify that $f(77)$ is bigger than $f(33)$. Do {\it not}\/
verify this by direct computation of function values. 

2.~Suppose $M(x)=x^5-7x+4$. Compute $M(2)$ and $M(-2)$ and explain
briefly why the results allow you to conclude that $M(x)=0$ has
at least one root. 
\smallskip
{\bf Curve sketching}

1.~Suppose $f(x)=5x^3-3x^5$. Compute $f'(x)$ and $f''(x)$. Where are each
of these functions equal to $0$? Find all relative max \& min values
of $f(x)$, briefly explaining your answers. Find all inflection points
of $f(x)$, briefly explaining your answers.

2.~In this problem the function $G$ is defined by the formula
$G(x)=(2-e^x)(1-e^x)$.  Computations with exp and log should be
simplified as much as possible; approximations are not acceptable.


a) What is $\lim\limits_{x\to-\infty}G(x)$?  What is
$\lim\limits_{x\to+\infty}G(x)$? 

b) For which $x$ is $G(x)=0$? 

\rightline{\bf This problem continues on the next page.}
\vfil\eject
c) Compute $G'(x)$. For which $x$ is $G'(x)=0$? 


d) Compute $G''(x)$. For which $x$ is $G''(x)=0$? 

e) Sketch a graph of $y=G(x)$, using the information you have obtained
in the previous parts of the problem. Indicate precisely any relative
maxima or minima or points of inflection with exact coordinates, and
any vertical or horizontal asymptotes with exact
equations. 
(Axes with $-5\le x\le 1$ and $0\le y\le 6$ were supplied.)

\vfil

%\smallskip
{\bf Antiderivatives \& initial value problems}

1.~Suppose $f''(x)=x+{1\over{x^2}}$ and $f(1)=1$ and $f'(1)=-2$. Find a
formula for $f(x)$. Computations with exp and log should be simplified
as much as possible; approximations are not acceptable. 

2.~Suppose $f''(x)=\cos x+\sin(2x)$ and $f(0)=0$ and $f'(\pi)=0$. Find a
formula for $f(x)$. Computations with sine and cosine should be simplified
as much as possible; approximations are not acceptable.

3.~Suppose $f''(x)=e^x-1$ and $f(0)=2$ and $f'(0)=-3$. Find a formula for
$f(x)$. Computations with exp and log should be simplified
as much as possible; approximations are not acceptable. 

\smallskip

{\bf Riemann sums}

1.~Suppose $f(x)=2x^2-1$. Compute the Riemann sum for $f(x)$ on the
interval $[-1,5]$ with partition $\{-1,2,4,5\}$ using the left-hand
endpoints as sample points. 

2.~Suppose $f(x)=\cos x$. Compute the Riemann sum for $f(x)$ on the
interval $[0,{3\over 2}\pi]$ obtained by partitioning the interval
into 6 equal subintervals and using the right-hand endpoints as sample
points.

%\smallskip
\vfil
{\bf Definite integral}

1.~Suppose $P$ and $Q$ are constants, and
$f(x)=P\sin(7x)+Qx\cos(7x)$. Find specific values of $P$ and $Q$ so
that $f'(x)=x\sin(7x)$. Use your answer to evaluate
$\int_9^{\pi/7}x\sin(7x)\,dx$. 

2.~Suppose $f(x)=xe^x-e^x$. Compute $f'(x)$, and use your answer
to evaluate $\int_0^1 xe^x \, dx$ exactly. 

%\smallskip
\vfil
{\bf Area}

1.~Sketch the region in the plane bounded by $y=4-x^2$ and the
$x$-axis. Find the area of this region.

2.~Sketch the region in the plane bounded above by $y=4-x^4$ and below by
$y=3$. Find the area of this region. 

3.~Sketch the region in the plane bounded by the $x$-axis, the line
$x=2$, and the curve $y={1\over 9}x^5$. Find the area of this region.

%\smallskip
\vfil
{\bf FTC}
%Evaluate. Computations with exp and log and square root should be
%simplified as much as possible. 

1.~$\int_1^4\left(2x-5\sqrt{x}\right)dx$ 

2.~$\int_1^4{{1-\sqrt{x}}\over x}dx$

3.~If $f(x)=\int_{-42}^x {{\sin(t^2)}\over{1+t^4}}dt$, compute $f(-42)$,
$f'(0)$, and $f'(\sqrt{\pi})$ exactly. 

4.~$\int_1^2\left(3\sqrt{x}-{1\over{x^4}}\right)dx$

%\smallskip
\vfil
{\bf Substitution}

1.~$\int (3x-6)e^{3x^3-12x} dx$

2.~$\int_0^{\sqrt{\ln 7}}xe^{(x^2)}dx$.
%\vfil
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