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%                \hbox {$\d \lim_{x \to a} f(x) = b$}%
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\magnification= \magstep1
\rightline{4/24/2005}
\parindent=0pt
\font\largerbold = cmbx10 scaled \magstep2

\centerline{\bf Some Math 135 review problems for the first 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.

\medskip


%{\bf Precalc}



%\smallskip

{\bf Definition of derivative}
\smallskip

1.~Write the definition of derivative as a limit and {\it use this
definition}\/ to find the derivative of $f(x)=x^2-x$. 

%\smallskip

2.~Write the definition of derivative  as a limit and {\it use this
definition}\/ to find the derivative of $f(x)={1\over{3x+4}}$. 

%\smallskip

3.~Write the definition of derivative  as a limit and {\it use this
definition}\/ to find the derivative of $f(x)=4+3x^2$. 

%\smallskip

4.~~Write the definition of derivative  as a limit and {\it use this
definition}\/ to find the derivative of $f(x)=\sqrt{1-2x}$.

\smallskip
 
{\bf Computations of limits}
\smallskip
1.~Evaluate the limits exactly. Give brief evidence supporting your
answers which is not based on a calculator graph or calculator
computations. 

\line{a) ${\lim\limits_{x\to\infty}{{2x^2-5}\over{3x^2+1}}}$ \hfil
\ b) ${\lim\limits_{x\to 4}{{x-4}\over{\sqrt{x}-2}}}$
\hfil
\ c) ${\lim\limits_{x\to 5}{{x^2-3x-10}\over{x-5}}}$ \hfil
\ d) ${\lim\limits_{x\to +\infty}{{2x^3-x+3}\over{x^3+2}}}$ \hfil 
\ e) ${\lim\limits_{x\to 2^-}{{x^2-4}\over{|2-x|}}}$\hfil 
\hfil
\ f) ${\lim\limits_{x\to 2^+}{{x^2-4}\over{|2-x|}}}$\ 
}

%\smallskip

2.~Find the equations of all vertical and horizontal asymptotes of
$f(x)={{\sqrt{x^2+4}}\over{x-3}}$.

%\smallskip

3.~Find the equations of all vertical and horizontal asymptotes of
$f(x)={{3+5e^{-2x}}\over{7-e^{-2x}}}$. Computations with exp and log
should be simplified as much as possible; approximations are not
acceptable.

\smallskip
{\bf Computations of derivatives}
\smallskip

1.~Find ${{dy}\over{dx}}$.
%\smallskip
\overfullrule=0pt

\line{a) $y={{2x^2+5}\over{5x^3+1}}$ \hfil
\ b)  $y=e^{-\sqrt{x^2-1}}$ \hfil
\ c) $xy^3=\cos(7x+5y)$ \hfil
\ d) $xe^y=\sin(xy)$ \hfil
\ e) $y=x\ln(3x+5)$}

\line{f) $2x^3+5x^2y+y^3=2$ \hfil
\ g) $y=\ln(x^4+3x+1)$  \hfil
\ h) $y={{3x^3-1}\over{3x^4+1}}$ \hfil
\ i) $y=x^5\tan(3x)$ \hfil
\ j) $y=(4x+3)\sqrt{x^3+7}$}


2.~Suppose $f(x)$ is a differentiable function satisfying $f'(1)=3$ and
$f'\left({\pi\over 4}\right)=-2$, If $g(x)=f(\tan x)$, find
$g'\left({\pi\over 4}\right)$. 

3.~An equation of the tangent line to $y=f(x)$ when $x=1$ is known to be 
$2x+y+3=0$. Find $f(1)$ and $f'(1)$. 

4.~Suppose $U$ is a differentiable function with $U(8)=5$ and $U'(8)=3$,
and that $V(x)=U(x^3)$. What are $V(2)$ and $V'(2)$? Use this
information to write an equation of the tangent line to $y=V(x)$ when $x=2$.

5.~A function is defined implicitly by the equation
$x^2-y^2-5xy+x+y=10$. Find $y'$ in terms of $x$ and $y$. Find an
equation for the line tangent to the graph of this function at the
point $(1,2)$. 

\smallskip
{\bf Log/exp etc.}
\smallskip

1.~Find the range of $f(x)=e^{-2x}+e^{3x}$.

%\smallskip

2.~The rate of bacterial growth at any time is directly proportional to
the number of bacteria present at that time. If the population of a
certain kind of bacteria doubles in 4 hours and there are 500 bacteria
at 1 AM, how many bacteria will there be at 5:30 AM?

\smallskip
{\bf Continuity \& differentiability}

\smallskip

1.~Here $f(x)=\cases{x+3 &if $x\le-2$\cr
{1\over 2}x^2+A & if $-2< x$\cr}$ where $A$ is a constant to be
determined. Find $A$ so that $f(x)$ is continuous for all values
of $x$. Sketch a graph of $y=f(x)$ using that value of $A$ for $-4\le
x\le 2$. Is $f(x)$ differentiable at $x=-2$ using that value of $A$?

2.~Here $f(x)=\cases{Ax^2-1 & if $x<-1$\cr
x+B & if $-1\le x\le 1$\cr
2& if $1<x$\cr}$ where $A$ and $B$ are constants to be determined. Find
numbers $A$ and $B$ so that $f(x)$ is continuous for all values of
$x$. Sketch a graph of $y=f(x)$ for $-3\le x\le 3$. 


3.~In this problem $f(x)=\cases{1+x^2 & if $x<2$ \cr 
A+Bx & if $-2\le
x<1$\cr 
x^2 & if $x\ge 1$\cr}$. Find $A$ and $B$ so that $f(x)$ is
continuous at all points. Sketch a graph of $y=f(x)$ for  $-3\le x\le
3$.  For which values of $x$ is $f(x)$ {\it not}\/ differentiable?

\smallskip
{\hsize=2.3in\vtop{
4.~In the graph of $y=f(x)$ to the right, identify with 
{\bf m} any point which is a relative minimum; {\bf M} any point which is a relative maximum; {\bf C} any point which is a critical point; {\bf I} any point which is an inflection point; {\bf NC} any point at which $f(x)$ is {\it not}\/
continuous; and {\bf ND} any point at which $f(x)$ is {\it not}\/
differentiable.
Some points may have more than one label.}}
\vskip -1.6in
\rightline{\epsfxsize=3in \epsfbox{curvy.eps}}

\bigskip%\medskip
\rightline{\epsfxsize=1.2in \epsfbox{ann.eps}}

\vskip -1.34in
{\bf Related rates}

{\hsize =4.in\vtop{
1.~Two circles have the same center. The inner circle has radius $r$
which is increasing at the rate of 3 inches per second. The outer
circle has radius $R$ which is increasing at the rate of $2$ inches
per second. Suppose $A$ is the area of the region {\it between}\/
the circles. At a certain time, $r$ is 7 inches and $R$ is 10
inches. What is $A$ at that time? How fast is $A$ changing at that
time? Is $A$ increasing or decreasing at that time?}}
%\vskip -1.7in
\vskip -.1in
\smallskip
\rightline{\epsfxsize=1.in \epsfbox{triangle2.eps}}
\vskip -1.35in
{\hsize =4.2in\vtop{
2.~Suppose that the right triangle with height $h$ and base $b$ as shown
here {\it always}\/ has an area of 6 square inches. At a certain time,
the length of $b$ is 3 inches and it is increasing at the rate of
$.04$ inches per minute. What is the length of $h$ at that time? How
fast is the length of $h$ changing at that time? Is it increasing or
decreasing?  
What is the angle $\theta$ at that time? How
fast is the angle $\theta$ changing at that time? Is it increasing or
decreasing?  }}

\vskip -1.2in





\vfil\eject\end


{\largerbold SECOND HALF}

{\bf Linear approximation}

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 $F(5.02)$. 

Suppose $f(x)=\sin(x^2)$. 

{\bf Optimization}

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 flabs 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. 

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
{\hsize =4.2in\vtop{
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 -.7in
\rightline{\epsfxsize=1in \epsfbox{box.eps}}

{\hsize =3.5in\vtop{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.}}

\vskip -.9in
\rightline{\epsfxsize=1.8in \epsfbox{triangle.eps}}

{\bf Intermediate Value Theorem; MVT}

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 be
used to verify that $f(77)$ is bigger than $f(33)$. Do {\it not}\/
verify this by direct computation of function vlaues. 


{\bf Curve sketching}

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.

{\bf Antiderivatives \& initial value problems}

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. 

Suppose $f''(x)=\cos x+\sin(2x)$ and $f(00=0$ and $f'(\pi)=0$. Find a
fomrula for $f(x)$. Computations with sine and cosine should be simplified
as much as possible; approximations are not acceptable.

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. 


{\bf Riemann summs}

{\bf Definite integral}

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
$\int_9^{\pi/7}x\sin(7x)\,dx$. 

Suppose $f(x)=-x\cos x+\sin x$. Compute $f'(x)$, and use your answer
to evaluate $\int_0^{2\pi} x\sin x \, dx$ exactly. 

{\bf Area}

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

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

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

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

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



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

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

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

{\bf Substitution}

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

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

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