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\begin{document}
\noindent
{\bf Math 105}  \hfil {\bf Review questions --- test 3}

\vskip.25in
\noindent
1. Suppose a  beaker of hot water is put in a room whose temperature is kept
fixed at 65 degrees Fahrenheit.
According to Newton's law of cooling, the temperature of the water is given
by the formula
$$H = 65 + Ae^{kt}$$
where $H$ is the temperature, $t$ is the time in minutes, and $A$ and $k$
are constants.
At the end of 30 minutes the temperature of the water is 165 degrees, and
it is decreasing at the rate of 1 degree per minute.
What are the constants $k$ and $A$?

\next
2.  A hard-boiled egg at 98 degrees Celsius is put into a sink of water
that is kept at a steady temperature of 18 degrees
Celsius.  After 5 minutes, the egg's temperature is found to be 38 degrees
Celsius.  How much longer will it take the egg
to reach 20 degrees Celsius?

\next
3.  Charcoal from a tree killed in the volcanic eruption that formed Crater
Lake in Oregon contained 44.5\% of the radioactive
carbon found in a living sample of the same wood.  How long ago did the
volcanic eruption take place? (Use 5730 years as the
half life of radioactive carbon.)

\next
4.  Find the equation of the line tangent to each of the following graphs
at the given point.

\medskip \noindent
(a) $y = (2 - x^2)^{30}$ at  $(1,1)$.   \hfil (b) $y = {x\over{x+1}}$ at
$(2,2/3)$.

\medskip \noindent
\hfil (c) $y = \sqrt{x^2 + x + 2}$ at $(1,2)$. \hfil

\next
5.  Find $f'(x)$ in each of the following cases.

\medskip
(a) $f(x) = x^3e^x$ \hfil (b) $f(x) = {3\over{(x^2 + x + 1)^2}}$ \hfil (c)
$f(x) = x\sqrt{x^2 - 4}$

\medskip
(d) $f(x) = 3^{2x+5}$  \hfil (e) $f(x) = 2^{1/x}$ \hfil (f) $f(x) =
{{e^{2x}}\over{1 + e^{3x}}}$

\medskip
(g) $f(x) = {{2x+1}\over{x+5}}$ \hfil (h) $f(x) = \sqrt{e^{3x} + 1}$ \hfil
                                      (i) $f(x) = {1\over{x\ln x}}$

\medskip
(j) $f(x) = \ln{(4x^3)}$  \hfil   (k) $f(x) = \ln{(1/\sqrt{x})}$ \hfil

\next
6. If $f(0) = 4$ and $f'(0) = 2$, find the equation of the line tangent to
the graph of
$\ y = \sqrt{x + f(x)}\ $
at  $(0,2)$.

\next
7. The volume of a sphere is given by the formula $\ V = {4\over3}\,\pi
r^3$, \ where $r$ is the
radius.  If a spherical balloon is expanding at a rate of 2 cu. ft. per
minute, how fast is its
radius increasing when the radius is 3 feet?  (Assume the balloon always
maintains an exact
spherical shape.)

\next
8. Given the following table of values:

\medskip \hfil
\begin{tabular}{|c||c|c|c|c|} \hline
$x$ & $f(x)$ & $g(x)$ & $f'(x)$ & $g'(x)$ \\ \hline \hline
1 & 2 & 3 & $-1/2$ & $-2$ \\ \hline
2 & 5 & $-1/3$ & 1 & 3/5 \\ \hline
3 & $-1$ & 2 & 1/4 & 3/2 \\ \hline
\end{tabular}  \hfil

\medskip \noindent
Find the derivatives of the following functions at the points indicated:

\medskip \noindent
(a) $f(x)\cdot g(x)$ at $x=3$ \hfil (b) $f(g(x))$ at $x = 1$\hfil
(c) $g(f(x))$ at $x = 1$

\medskip \noindent
 (d) ${{f(x)}\over{g(x)}}$ at $x = 2$ \hfil (e) $\ln(f(x) + g(x))$ at $x = 1$
     \hfil (f) $ {1\over{x + f(x)}}$ at $x=2$

\next
9.  Suppose that the demand for a certain product is given by
$$q = 5000e^{-0.08p}$$
where $p$ is the price and $q$ is the number sold at that price.

\smallskip
(a) What quantity is sold when the price is \$10?

\smallskip
(b) At what rate is the demand changing with respect to the price when the
price is \$10?

\smallskip
(c) What is the revenue when the price is \$10?

\smallskip
(d) At what rate is the revenue changing with respect to the price when the
price is \$10?

\next
10.  Water is pouring into a hemispherical bowl of radius 10 inches.  The
volume of
water satifies the equation $\ V = {\pi\over3}\,(30x^2 - x^3),\ $where $x$
is the depth.
Suppose that when the depth is 1 inch it is increasing at the rate of 0.5
inches per minute, and
when the depth is 3 inches it is increasing at the rate of 0.2 inches per
minute.  How fast is
the volume increasing at each of those times?

\next
11.  The demand for ``I Love Kramer'' tee shirts  satisfies a function of
the form
$q = f(p)$, where $p$ is the price and $q$ is the number sold.  The revenue
is given by $R = pq$.
Given that $f(12) = 6000$ and $f'(12) = -400$, find $dR/dp$ when $p = 12$.

\next
12.  Determine where each of the functions is increasing and where it is
decreasing, and find
any relative maxima or minima.

\medskip
(a) $f(x) = 3x^2 - 6x - 9$ \hfil  (b) $f(x) = x + (4/x), \ x \ne 0$

\medskip
 (c)  $f(x) = x^2e^{-x}$  \hfil  (d) $f(x) = x\ln x, \ x > 0$

\medskip
(e) $f(x) = {{x^2}\over{x^2 + 4}}$  \hfil  (f) $f(x) = {{x}\over{x^2 + 4}}$

\next
13.  Find the absolute maximum or minimum value, as indicated, of each of the
following functions.  Justify your answer by a few words of explanation.

\medskip \noindent
(a) $f(x) = xe^{-2x}$, abs. max. \hfil (b) $f(x) = 2x - \ln x, \ x > 0$,
abs. min.

\medskip
\hfil (c) $f(x) = 3x^2 - 4x $, abs. min.  \hfil

\next 
14. Find the slope of the graph of $y = 1 + \ln x$ at the point $(e, 2)$.

\next
15.  Find where each of the following functions is increasing, where it is
decreasing, and where
its relative maxima and/or minima (if any) are.

\medskip
(a) $f(x) = \ln(1 + x^2)$ \hfil (b) $f(x) = x^3 - 3x + 7$

\next
16.  Show that $e^x \ge 1 + x$ for all $x$ by the following steps:

\smallskip
(a) Find where the function $f(x) = e^x - x$ has a relative minimum and
explain why the function must have an absolute minimum there.

\smallskip
(c) Find the absolute minimum value of $f(x)$ and
use that information to show that $e^x \ge 1 + x$ for all $x$.

\next 
17. For what $x$ does the graph of $y = x + \ln x$ have slope 4?


\next 
18. Find the equation of the line tangent to the graph of $y = x +
e^x$ at the point $(0,1)$.

\next
19. Let $C(x)$ be the cost of producing $x$ units of a certain
commodity.  Which of the following procedures will give you the
\textbf{marginal} cost of producing 100 units?

\smallskip
 (a)  compute $C(100)$ \hfil (b) compute $C'(100)$  \hfil (c) compute
$C(100) - C(0)$

\medskip
 (d) set $C'(x) = 100$ and solve for $x$  \hfil (e) set $C(x) = 100$ and
solve for $x$

\next
20. A manufacturing company determines that the revenue obtained from
the sale of $x$ items is given by $R(x) = 220x - 4x^2$ and the cost of
producing these items by $C(x) = 900 + 40x$.  For what output level
$x$ does the marginal revenue equal the marginal cost? Is this the
same as the break-even point?

\next
21. Suppose the cost of producing $x$ pounds of fertilizer is given by
the cost function
\[
C(x) = 0.5x^2 + 1.5x + 8.
\]

\smallskip
(i) Find the cost of producing 10 pounds.

\smallskip
(ii) Find the marginal cost for $x=10$.

\smallskip
(iii) Approximate the extra cost in going from 10 pounds to 11 pounds.

\next
22. Use linear approximation with the function $f(x) = \sqrt{x}$ to estimate:

\smallskip
(i) $\sqrt{50}$ \hfil (ii) $\sqrt{83}$  \hfil (iii) $\sqrt{98}$

\next
23. Use linear approximation with an appropriate choice of function (and
without a calculator)
to estimate:

\smallskip
(i) ${1\over{0.98}}$ \hfil (ii) ${1\over{2.108}}$ \hfil (iii) $(8.1)^{1/3}$

\next
24.  Suppose  the weight $W$ of a growing puppy is a function of time $t$
in days and you know
that after 30 days, $W = 1500$ grams and $dW/dt = 4$ grams per day.
Approximate the puppy's
weight in grams at the end of 32 days.

\next
25. Suppose the profit function for a certain manufactured item is given by
$$P(x) = 50x - 0.04x^2 - 10,000.$$

(i) Find the profit at production level $x=600$.

(ii) Find the marginal profit at production level $x=600$.

(iii) Use the previous answers to approximate the profit at production
level $x=601$ without
computing $P(601)$.






\pagebreak \noindent
\underbar{\bf Answers}

\medskip \noindent
1. $k = -0.01, \ A = 100e^{0.3}$ 

\next 
2. 13.3 minutes 

\next 
3. 6693 years

\next
4. (a) $y = -60x + 61$ \hfil (b) $y = {1\over9}\,x + {4\over 9}$ \hfil $y =
{3\over4}x + {5\over4}$

\next
5. (a) $x^2e^x(x+3)$ \hfil (b) $\displaystyle{{-6(2x+1)}\over{(x^2 + x +
1)^3}}$
\hfil
(c) $\displaystyle{{2x^2 - 4}\over{\sqrt{x^2 - 4}}}$

\smallskip
(d) $(2\ln 3)3^{2x+5}$ \hfil (e) $\displaystyle -{{2^{1/x}\ln 2}\over{x^2}}$
    \hfil (f) $\displaystyle{{e^{2x}(2 - e^{3x})}\over{(1 + e^{3x})^2}}$

\smallskip
(g) $\displaystyle{9\over{(x+5)^2}}$ \hfil
       (h) $\displaystyle{{3e^{3x}}\over{2\sqrt{e^{3x} + 1}}}$  \hfil
       (i) $\displaystyle-{{1 + \ln x}\over{(x\ln x)^2}}$

\smallskip
(j) $f'(x) = {3\over x}$ \hfil (k) $f'(x) = -{1\over{2x}}$

\next
6. $y = {3\over4}\,x + 2$

\next 
7. ${1\over{18\pi}}$

\next
8. (a) $-1$ \hfil (b) $-1/2$  \hfil (c) $-3/10$

\smallskip
(d) $-30$  \hfil (e) $-1/2$ \hfil (f) $-2/49$ \hfil

\next
9. (a) $5000e^{-0.8} \approx 2247$
       \hfil (b) $-400e^{-0.8} \approx - 180$

\medskip
 (c) $50,000e^{-0.8} \approx 22,466$  \hfil (d) $1000e^{-0.8} \approx 449.33$

\next
10.  At $x=1$, $dV/dt = 9.5\pi$.  At $x = 3$, $dV/dt = 10.2\pi$  

\next 
11. 1,200

\next
12.  (a) inc if $x > 1$, dec if $x < 1$, rel min at $x = 1$.

\smallskip
(b) inc if $x < -2$ or $x > 2$, dec if $-2 < x < 0$ or $0 < x < 2$,

  \quad  rel min at $x=2$, rel max at $x=-2$.

\smallskip
(c) inc if $0 < x < 2$, dec if $x < 0$ or $x > 2$,

\quad rel min at $x=0$, rel max at $x = 2$.

\smallskip
(d) inc if $x > 1/e$, dec if $0 < x  < 1/e$, rel min at $x = 1/e$.

\smallskip
(e) inc if $x > 0$, dec if $x < 0$, rel min at $x=0$.

\smallskip
(f) inc if $-2 < x < 2$, dec if $x < -2$ or $x > 2$,

 \quad rel min at $x=-2$, rel max at $x=2$.

\pagebreak
\next
13. (a) ${1\over{2e}} \approx 0.184$ \hfil (b) $1 + \ln2 \approx 1.69$
\hfil (c) $-4/3$

\next
14. $1/e$ 

\next 
15. (a) inc if $x > 0$, dec if $x < 0$, rel min at $x=0$.

\smallskip
 (b) inc if $x < -1$ or $x > 1$, dec if $-1 < x < 1$,

\quad rel min at $x=1$, rel max at $x = -1$.

\next
16. Absolute min at $x=0$, minimum value $= 1$.

\next 
17. $1/3$

\next
18. $y = 2x +1$

\next 
19. b  

\next 
20. 45/2, not the break-even point

\next
21. (i) 73 \hfil (ii) 11.5  \hfil (iii) 11.5

\next
22. (i) ${99\over14}\approx 7.07143$ \hfil (ii) ${82\over9}\approx
9.1111$ \hfil
   (iii) $9.9$

\next
23. (i) 1.02 \hfil (ii) $0.5 - 0.027 = 0.473$ \hfil (iii) $2 +
{{0.1}\over12} \approx2.0083$

\next
24. 1508 

\next 
25. (i) $P(600) = 5,600$ \hfil (ii) $P'(600) = 2$  \hfil
(iii) 5,602


\end{document}
