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\begin{document}
\noindent
{\bf Math 105}  \hfil {\bf Review questions --- test 2}

\vskip.25in
\noindent
{\bf Note:  Unless instructions say otherwise , answers involving $e$ or
$\ln$ can be left that way.  You do not have to use a
calculator to get a decimal answer.}


1. Find each of the following. Your answer should be an integer or fraction.

\smallskip
(i) $e^{-2\ln3}$ \hfil (ii) $\ln{\sqrt{e}}$  \hfil (iii) $\ln(\ln e)$ \hfil
(iv) $\sqrt{e^{\ln 4}}$

\next
2. Which of the following is equal to $e^{-2\ln t}\>$?

\medskip
(a) ${1\over{2t}}$  \hfil (b) $\Big({1\over2}\Big)^t$ \hfil (c) $\sqrt t$
\hfil (d) ${1\over{t^2}}$
   \hfil   (e) $2^t$


\next
3. Which of the following is equal to $\ln(e^2) + \ln(e^3)$?

\smallskip
(a) $e$ \hfil (b)  6 \hfil (c) 8 \hfil (d) 9 \hfil (e) 5

\next
4. If $a$ is a positive number, which of the following is equal to
$\sqrt{a}\,$?

\smallskip
(a) $e^{a/2}$  \hfil (b) $e^{a\ln2}$ \hfil (c)  $e^{-a\ln2}$ \hfil (d)
$e^{(1/2)\ln a}$  \hfil (e) $e^{\ln(a/2)}$

\next
5.  Solve each of the following for $x$.

\smallskip
(i) $e^{2x} = 3$ \hfil (ii) $\ln(1 + \ln x) = 0$ \hfil (iii) $\ln(x^2) = 8$
\hfil

\medskip
(iv) $e^{\ln(2x+3)} = 7$ \hfil (v) $e^{x^2 - 2x -3} = 1$ \hfil (vi)
$\ln(x^3) = 6$

\next
6. Suppose \$10,000 is deposited in account paying an annual interest rate
of 4\%, compounded
continuously.  How long will it take for the account to grow to \$25,000?

\next
7. A certain amount of money is put into an account paying 5\% annual interest,
compounded continuously.  How long will it take the initial investment to
triple?

\next
8.  The radioactive substance einsteinium-253 decays exponentially --- in
other words, the
amount $y$ remaining after $t$ days is given by a formula of the form $y =
Ae^{-kt}$, where
$A$ is the initial amount and $k$ is the decay constant. Suppose an initial
amount of 30 mg of
the substance decays to 10mg   in 11.7 days.

\smallskip
(i) Find the decay constant of einsteinium-253.

\smallskip
(ii) Find its half life.

\smallskip
(iii)  If you start with 100mg of einsteinium-253, how much will there be
after 20 days?

\next
9. The radioactive substance Cesium-137 decays exponentially and has a half
life of 30 years.
If $t$ denotes time (in years) and $y$ denotes the amount of Cesium remaining
from an initial amount of 100 mg, which of the following equations correctly
expresses $y$ as a function of $t$?

\smallskip
(a) $y = 100e^{-\big((\ln2)/30\big)t}$ \hfil (b) $y =
100e^{\big(30/(\ln2)\big)t}$ \hfil
   (c)  $y = 100e^{-2(\ln30)t}$

\medskip
\hfil (d)  $y = 50e^{(\log_2{30})t}$  \hfil (e)  $y = {{10}\over3}e^{-t\ln2}$

\next
10. A certain radioactive substance satisfies the exponential decay law,
$\ y~=~Ae^{kt}$, where $t$ is the time (in days) and $y$ is the amount (in
grams) at time $t$.
Suppose an inital amount of 100 grams decays to 25 grams at the end of 10
days.  Which of the
following equals the amount remaining at the end of 15 days?

\bigskip
(a) $25e^{-5\ln4}$ \hfil (b) $100e^{-5\ln2}$ \hfil (c) $100e^{-(2/3)\ln2}$

\bigskip
\hfil (d) $25e^{1.5\ln2}$ \hfil (e) $100e^{-1.5\ln4}$


\next
11. An account paying 4\% annual interest compounded continuously starts
out with an initial deposit, after which no further deposits or withdrawals
are
made. How many years will it take for the initial amount to double?

\smallskip (a) ${-2\over{\ln(0.04)}}$  \hfil (b)  ${(0.04)\over{\ln2}}$
\hfil (c) ${{\ln2}\over{0.04}}$

\medskip
\hfil (d) ${2\over{\ln(0.04)}}$ \hfil (e) the answer depends on the initial
amount \hfil

\next
12. Suppose you invest a certain amount in an account that offers
continuously compounded
interest and your money triples in exactly 14 years.  (After the initial
investment you do not
add or take away from the account until that time.)  What is the annual
interest rate?


\next
13. Find $f'(x)$ for each of the following functions directly from the
basic defining formula for
the derivative without using any other rules or short-cut formulas.

\smallskip
(i) $f(x) = (2x+1)^2$ \hfil (ii) $f(x) = {1\over{x^2 + 1}}$ \hfil (iii)
$f(x) = \sqrt{x+1}$

\next
14.  Find $f'(x)$ for each of the following functions using any rule we
have had so far.

\smallskip \noindent
(i) $f(x) = {{x^2}\over 3} + {3\over{x^2}}$ \hfil (ii) $f(x) = \ln{(4x^3)}$
\hfil
    (iii) $f(x) = \ln{(1/\sqrt{x})}$

\bigskip \noindent
 (iv) $f(x) = x^5 - 3x^4 + 2x^3 + x^2 - 5x + 7$ \hfil (v) $f(x) = e^{3x}$

\bigskip \noindent
(vi) $f(x) = \sqrt{x} - {3\over{\sqrt x}} $\hfil (vii) $f(x) = 5^x$ \hfil
(viii) $f(x) =  \ln{(7/x)}$

\next
15. Find the second derivative $f''(x)$ for each of the following functions
using any rule we have
had so far.

\smallskip
(i) $f(x) = 2x^3 - 5x^2 + 7x -1$ \hfil (ii) $f(x) = \ln(x^2)$ \hfil (iii)
$f(x) = x^2 + e^x$

\medskip
(iv) $f(x) = \sqrt{x}$  \hfil (v) $f(x) = \ln x - x^2$ \hfil (vi) $f(x) =
e^{2x}$

\next
16. For the graph of $y = x^2$, which of the following equals the slope of
the secant line
between the point on the graph where $x = 1$ and the point on the graph
where $x = 1+h$?

\smallskip (a) ${{1-h}\over{h}}$  \hfil (b) $1 + h$ \hfil (c) $2 + 2h$
\hfil (d) $2 + h$  \hfil (e) ${{1- h^2}\over {h}}$

\next
17. Suppose $f(x) = \sqrt{x+1}$.  Which of the following limits represents
the slope of the graph
of $y = f(x)$ at the point $(3,2)$?

\medskip
(a)  $\lim_{h\to0}\,{{\sqrt{1+h} - 1} \over h}$
\hfil (b)  $\lim_{h\to0}\,{{\sqrt{3+h} - 3} \over h}$
\hfil (c) $\lim_{h\to0}\,{{\sqrt{4+h} - 2} \over h}$

\bigskip
\hfil (d)  $\lim_{h\to0}\,{{\sqrt h - 3} \over h}$
\hfil (e)  $\lim_{h\to0}\,{{\sqrt h - \sqrt3} \over h}$ \hfil

\next
18. If $f(x) = \sqrt{x}$, which of the following expressions is equal to
$f'(x)$?

\medskip (a) $\lim_{h\to0} {{\sqrt{x+h} -\sqrt{x}} \over h}$ \hfil (b)
$\lim_{h\to0} {{\sqrt{x} - \sqrt{h}} \over {x-h}}$
\hfil (c) $\lim_{h\to0} {{\sqrt{x + h} - \sqrt{h}} \over h}$

\bigskip
\hfil (d) $\lim_{h\to0} {{\sqrt{x+h} -\sqrt{x}}\over{\sqrt{x}}}$
  \hfil (e) $\lim_{h\to0} {{\sqrt{x} + \sqrt{h}}\over{\sqrt{x} - \sqrt{h}}}$

\next
19. If $f(x) = {1\over{x^3}}$, what is $f'(2)$?

\next
20. Find the slope of the graph of
$y = {x^2 \>+\> {4\  \over \  x^2}}$
at the point $(2,5)$.

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

\next
22. Find the equation of the line tangent to the graph of
$y = x^{3/2} - 1$
at the point $(4,7)$.

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

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

\next
25. For what $x$ does the graph of
$y = x^2 + 3x + 5$
have slope 9?

\next
26.  At what point is the tangent line to the graph of
$y = x^2 + 3x + 1$
parallel to the line $x + y = 1$?

\next
27. If a ball is dropped from a height of 100 feet, its height after $t$
seconds is given by the function
$f(t) = -16t^2 + 100.$

\smallskip
(i) Find the average velocity of the ball during the time period from 1 to 3/2
seconds.

\smallskip
(ii) What is  the instantaneous velocity of the ball after 1 second? After
3/2 seconds?

\smallskip
(iii) What is  the acceleration of the ball after 1 second? After 3/2
seconds? After 1.762235 seconds?


\next
28. If a ball is thrown upward from a height of 5 feet with an initial
speed of 30 feet per second, its height after $t$ seconds is given by the
function
$f(t) = -16t^2 + 30t + 5.$

\smallskip
(i)  Find the average velocity of the ball during the time period between
$t$ and
$t+h$ seconds.

\smallskip
(ii) Find the instantaneous velocity of the ball at the end of $t$ seconds.

\next
29. An object is moving along a straight line so that its distance $s$ from
a fixed reference
point at time $t$ is given by the formula
$$s = 20 - \sqrt{t} + 3t^{3/2}.$$

(i) Find its (instantaneous) velocity when $t=1$ and when $t=4$.

(ii) Find its average velocity between times $t=1$ and $t=4$

(iii) Find its acceleration when $t=1$.

\next
30.  Suppose the weight of a tumor in a laboratory animal is given by the
formula
$w = 0.02 + 0.35t + 0.002t^2,$
where $t$ is the time in days.  How fast is the weight increasing when $t =
3$?  When $t=10$?





\next
31.  Suppose you put \$100 into an account that pays 4\% simple interest
annually.

(a) If you leave
the money in the account without making any further deposits or
withdrawals, how much will
you have at the end of $t$ years. (Your answer should be a function of $t$.)

(b) Use a calculator to find the amount you will have at the end of 4 years
and at the end of 10
years.

\next
32.  Compute each of the following without a calculator.  Your answer
should be an integer or
simple fraction.

\smallskip
(a) $\displaystyle\left({1\over9}\right)^{1/2}$ \hfil
(b) $\displaystyle\left({1\over9}\right)^{-2}$ \hfil
\hfil
(c) $(0.1)^3$  \hfil (d) $(0.1)^{-3}$

\next
33.  Suppose you put \$5000 into an account that pays 6\% annual interest,
and make no further
deposits or withdrawals.

(a) If the interest is compounded 12 times a year, how much will be in the
account at the end of 1 year?

(b) If the interest is compounded 4 times a year, how much will be in the
account at the end of 3 years?

\next
34.  Suppose an account pays 5\% annual interest.  How much should you put
in the account now
in order to have \$1000 at the end of two years, if

\smallskip
(a) the interest is compounded once a year?

\smallskip
(b) the interest is compounded 6 times a year?












\vfil \eject \noindent
\centerline{\bf Answers}

\next
{\bf 1.}  (i) 1/9 \hfil (ii) 1/2 \hfil (iii) 0 \hfil (iv) 2  \hfil
{\bf 2.} d \hfil {\bf 3.}  e \hfil {\bf 4.} d

\next
{\bf 5.}  (i) ${{\ln3}\over 2}$ \hfil (ii) 1 \hfil (iii) $\pm e^4$ \hfil
(iv) 2 \hfil (v) $-1$ and 3
\hfil (v) $e^2$

\next
{\bf 6.} ${\ln 2.5 \over 0.04}$ \hfil {\bf 7.} ${{\ln3}\over{0.05}}$

\next
{\bf 8.} (i) ${\ln3 \over 11.7}$ \hfil (ii)
${{11.7\ln2}\over{\ln3}}$
     \hfil (iii) $100e^{-20\ln3/11.7} \approx 15.29$

\next
{\bf 9.} a \hfil {\bf 10.} e \hfil
{\bf 11.} c \hfil {\bf 12.} ${\ln3 \over 14} \approx 0.0785$ or 7.85\%

\next
{\bf 13.} (i) $f'(x) = 8x+4$ \hfil (ii) $f'(x) = {{-2x}\over{(x^2+1)^2}}$
\hfil
   (iii) $f'(x) = {1\over{2\sqrt{x+1}}}$

\next
{\bf 14.} (i) $f'(x) = {{2x}\over3} - {6\over{x^3}}$ \hfil (ii) $f'(x) =
{3\over x}$ \hfil
(iii) $f'(x) = -{1\over{2x}}$

\bigskip
(iv) $f'(x) = 5x^4 -12x^3 + 6x^2 + 2x - 5$ \hfil (v) $f'(x) = 3e^{3x}$

\bigskip
(vi)  $f'(x) = {1\over{2\sqrt{x}}} + {3\over{2x^{3/2}}}$ \hfil (vii) $f'(x)
= 5^x \ln5$
\hfil (viii) $f'(x) = -{1\over x}$

\next
{\bf 15.} (i) $f''(x) = 12x - 10$ \hfil (ii) $f''(x) = -{2\over{x^2}}$
\hfil (iii) $f''(x) = 2 + e^x$

\bigskip
(iv) $f''(x) = -(1/4)x^{-3/2}$ \hfil (v) $f''(x) = -\left({1\over{x^2}} +
2\right)$ \hfil
     \hfil (vi) $f''(x) = 4e^{2x}$

\next
{\bf 16.} d \hfil {\bf 17.} c \hfil
{\bf 18.} a \hfil  \hfil {\bf 19.} $-3/16$

\next
{\bf 20.} 3 \hfil {\bf 21.} $1/e$ \hfil {\bf 22.} $y = 3x - 5$ \hfil {\bf
23.} $1/3$

\next
{\bf 24.} $y = 2x +1$ \hfil {\bf 25.} 3 \hfil {\bf 26.} $(-2,-1)$

\next
{\bf 27.} (i) $-40$ \hfil (ii) $-32$, $-48$  \hfil (iii)   $-32$ at all times

\next
{\bf 28.} (i) $-32t -16h + 30$ \hfil (ii) $-32t + 30$

\next
{\bf 29.} (i) 35/4  when $t=4$ and 4 when $t=1$
\hfil (ii) 20/3  \hfil (iii) 5/2

\next
{\bf 30.} 0.362 when $t = 3$, \ 0.39 when $t=10$.

\next
{\bf 31.}
(a) $100(1.04)^t$ \ \ 
(b) 116.99, 148.02 \ \ 

\next 
{\bf 32.} (a) 1/3 \ \  (b) 81 \ \ (c) 0.001 \ \  (d)
1000

\next 
{\bf 33.} (a) \$5308.39, \ (b) \$5978.09

\next 
{\bf 34.} (a) \$900.16, \ (b) \$898.06

\end{document}
