Sample for the Third Examination

1. Solve each of the following for $x$. Express each answer exactly.
a) $4^{2x-1}=64\qquad \qquad \qquad \ \ \$
b) $\left( \frac{1}{5}\right) ^{4x-2}=125\qquad \qquad \qquad \ \ \$
c) $2^{x+1}=8^{x-3}$
d) $\ \log _{2}(4x+8)=5\qquad \qquad$
e) $\log _{3}(2x+1)=4\qquad \qquad \qquad \$
f) $\ln (e^{x-5})=e^{\ln (3x-9)}\qquad \qquad \ \ \ \ \ \ \$
g) $\ e^{\ln (x^{2}+2)}=3\ln (e^{x+4})\qquad \ \ \$
h) $\ e^{x^{2}-9}=1\qquad \qquad \qquad \ \ \ \ \ \ \ \ \ \ \ \$
i) $%% e^{x^{2}-5x+6}=1\qquad \qquad \qquad \ \ \ \ \ \ \ \ \$
j) $\ \log _{b}\left( x^{2}-6x+6\right) =0$
k) $\log _{2}\left( x^{2}-7x+18\right) =3\qquad \ \ \ \$
l) log $_{x}\left( 81\right) =4\qquad \qquad \qquad$
m) $\log _{x}\left( x^{3}\right) =5x-7\qquad \qquad$
n) $\ \ln (x^{2}-3x+2)-\ln \left( x^{2}-2x+1\right) +\ln (x-1)=1$
o) $\ln \left( x^{2}-7x+12\right) -\ln (x^{2}-16)+\ln (x+4)=2\qquad \qquad$
p) $\ln \left( x\right) =-1$

2. Solve each of the following equations. (Use a calculator to express each answer to two decimal places.)
a) $5^{\left( x-3\right) }=18\qquad \qquad$
b) $\log _{3}\left( 2x-1\right) =-\frac{1}{3}\qquad$
c) $e^{x+3}=81\qquad$
d) $\ln \left( 2x+1\right) =10$

3. Simplify each of the following:
a) $\frac{\left( 5^{2}\right) ^{3}\cdot 10^{-2}}{\left( 5\sqrt{2}\right) ^{4}}\qquad \qquad \qquad$
b) $\frac{\left( 3^{3}\right) ^{2}\cdot 6^{-4}}{%% \left( 2\sqrt{3}\right) ^{-4}}\qquad \qquad \qquad$
c) $(2^{\frac{3x}{2}%% })^{\frac{2}{x}}-\left( \frac{1}{2}\right) ^{x}\cdot \frac{4^{x}}{2^{x}}$
d) $\ \log _{a}\left( 3a\right) ^{4}-2\log _{a}\left( 9\right) +\log _{a}\left( \frac{1}{a^{2}}\right) \qquad \qquad \qquad$
e) $\log _{a}\left( 2a\right) ^{3}-2\log _{a}\left( 4\right) +\log _{a}\left( \frac{2}{a}\right) \qquad \qquad$
f) $\ln \left( x^{\frac{5}{2}}\right) -2\ln \left( x^{3}\right) +\frac{3}{2}\ln \left( x^{4}\right) \qquad \qquad \qquad \qquad \ \$
g) $\ \ln (e^{3x})-3e^{3\ln \left( \sqrt[3]{x}\right) }$
h) $\log (10^{3x})-2e^{2\ln (\sqrt{x)}}\qquad \qquad$
i) $\ \log _{83}(83)\qquad \qquad$
j) $\log _{3}\left( 243\right) \qquad \qquad$
k) $\log _{10}\left( .001\right) \qquad \qquad$
l) $\log _{a}(a^{5})\qquad \qquad$
m) $\log _{8}\left( \frac{1}{16}%% \right) \qquad \qquad$
n) $\log _{27}\left( \frac{1}{3}\right) \qquad \qquad \qquad$
o) $\ln \left( \sqrt{e}\right)$
p) $\ln (e^{3})\qquad \qquad \ \ \ \$
q) $\ln \left( \frac{1}{e}\right) \qquad \qquad \ \ \ \ \ \$
r) $\ 7^{2\log _{7}7}\qquad \qquad \qquad \qquad$
s) $\ \sqrt{e^{\frac{1}{2}\ln \left( 9\right) }}$
t) $\left( \frac{e^{x}+e^{-x}}{2}\right) ^{2}-\left( \frac{e^{x}-e^{-x}}{2}%% \right) ^{2}\qquad \qquad$

4. Given that $\ \ \ln \left( 2\right) \approx .\,\allowbreak 693\,$, $%% \ln \left( 3\right) \approx \allowbreak 1.\,\allowbreak 099$ and $\ln \left( 5\right) \approx 1.\,\allowbreak 609\,$ find each of the following to two decimal places (without using a calculator):
a) $\ln \left( 12\right) \qquad \qquad$
b) $\ln \left( 30\right) \qquad \qquad$
c) $\ln \left( \frac{3}{2}\right) \qquad \qquad$
d) $\ln \left( \sqrt[3]{2}\right)$
e) $\ln \left( e^{12}\right) \qquad \qquad$
f) $e^{\ln \left( 5\right) }\qquad \qquad$
g) $\ln \left( 5^{2}\right) \qquad \qquad$
h) $\left( \ln \left( 5\right) \right) ^{2}$
i) $\log _{3}\left( 5\right)$

5. Given that $\ \ \ln \left( a\right) \approx .\,\allowbreak 31$ and $\ln \left( b\right) \approx \allowbreak .69$ find each of the following to two decimal places :
a) $\ln \left( a^{2}b\right) \qquad$
b) $\ln \left( \frac{a}{b^{2}}%% \right) \qquad$
c) $\ln \left( \sqrt[3]{b}\right) \qquad$
d) $\ e^{\ln (\ln \left( ab\right) )}\qquad$
e) $\log _{a}\left( b\right)$

6. Given that $\ln \left( 2\right) =u$ and that $\ln \left( 7\right) =v$, each of the following in terms of $u$ and $v$ (Do not use a calculator):
a) $\ln \left( 28\right) \qquad \ \ \ \$
b) $\ln \left( \frac{7}{8}%% \right)$        
c) $\ln \left( \frac{\sqrt{8}}{49}\right) \qquad \ \ \ \$
d) $\ln \left( 7^{\ln \left( 2\right) }\right) \qquad \ \ \ \$
e) $e^{2u}\qquad \ \ \ \$
f) $\log _{2}\left( 7\right)$

7. Sketch the graph of each of the following. Make sure to indicate and label (i.e., give the coordinates or equation) all intercepts and asymptotes.
a) $\ g\left( x\right) =\left( \frac{1}{e}\right) ^{x}\qquad \qquad$
b) $%% p\left( x\right) =3^{x}$
c) $\ f\left( x\right) =2+e^{1-x}\qquad \qquad$
d) $h(x)=\left( \frac{3}{4}\right) ^{-x}$
e) $k\left( x\right) =\ln \left( 2x-3\right)$
f) $%% m\left( x\right) =\ln \left( 1-x\right) \qquad$
g) $r\left( x\right) =\ln \left( -x\right) \qquad \ \ \$
h) $\ s\left( x\right) =1+\ln \left( x-3\right) \ \ \ \ \ \ \$
i) $\ t\left( x\right) =\ln \left( \frac{1}{e}\right) ^{x}\qquad \ \ \ \$
j) $q\left( x\right) =e^{\ln \left( \left\vert x\right\vert \right) }\qquad \qquad$
k) $j\left( x\right) =3^{\log _{3}(x+1)}\qquad \ \ \$
l) $%% u(x)=e^{(x^{2})}$
m) $v(x)=\ln \left( x^{2}\right) \qquad \ \ \ \ \ \$
n) $%% n\left( x\right) =\left[ \ln \left( x\right) \right] ^{2}\qquad \ \ \ \$
o) $w\left( x\right) =\ln \left( e^{x^{2}-8x+15}\right)$

8. a) To the nearest tenth of a percent, at what rate of interest compounded continuously will $1000 grow to be worth $1200 after 4 years?
b) If $5000 is invested at an interest rate of 10% compounded annually and if the interest rate decreases to 8% compounded annually after two years, how much will the investment be worth at the end of 5 years?
c) If $5000 is invested at an interest rate of 6% compounded continuously and if the interest rate doubles after six years, how much will the investment be worth at the end of ten years?
d) If $2000 is invested at 8% interest, find the value of the investment after 2 years (to the nearest cent) if
      i) interest is compounded semiannually
      ii) interest is compounded continuously
e) The bacteria in a petri dish are dying at a constant rate due to an antibiotic which has been introduced. If half of the bacteria remaining in the dish dies every three hours, to the nearest tenth of an hour, how long will it take until only 10% of the original number of bacteria remain?
f) The number of bacteria in a petri dish doubles after two days. To the nearest tenth of a day, how long does is take until the number of bacteria is ten times to original amount.
g) A radioactive substance decays so that at the end of 5 years only 70% of the substance remains. To the nearest tenth of a year, how long will it take until only 50% of the substance remains?
h) Assume that the cost of a car is $30,000. With continuous compounding in effect, find the number of years it would take to double the cost of the car at an annual inflation rate of 2.4%. (Round the answer to the nearest hundredth.)

9. Convert each of the following angles to radian measure: (Where possible, given an exact answer; otherwise round to two decimal places.)
a) $108^{\circ }\qquad$
b) $720^{\circ }\qquad$
c) $315^{\circ }\qquad$
d) $-135^{\circ }\qquad$
e) $77^{\circ }30^{\prime }$
f) $%% 0^{\circ }\qquad$
g) $-335^{\circ }18^{\prime }35^{^{\prime \prime }}$

10. Convert the following radian measure angles to degrees: (Where possible, given an exact answer; otherwise round to two decimal places.)
a) $-\frac{\pi }{3}\qquad$
b) $\frac{\pi }{24}\qquad$
c) $\frac{11\pi }{6}\qquad$
d) $\frac{3\pi }{2}\qquad$
e) $3$ radians
f) $0$ radians

11. a) Find the radian measure of the central angle which subtends an arc of length 12 in a circle of radius 16.
b) Find the arc length subtended by a central angle of $\frac{\pi }{8}$ radians in a circle of radius 4.
c) Find the radius of the circle in which an angle of 3 radians cuts off an arc of length 6.
d) Given a circle of radius 2 inches, determine the radian measure of the central angle which cuts off an arc of $\frac{\pi }{9}$ inches.

12. If $\theta$ is an angle in the first quadrant such that $\sin (\theta )=0.8$, find the exact values of each of the other five trigonometric functions of $\theta .$
a) $\cos (\theta )=$ ___________
b) $\tan (\theta )=$ ____________
c) $\cot (\theta )=$ ____________
d) $\sec (\theta )=$ ____________
e) $\csc (\theta )=$ ____________

13. Given that the point $(-3,6)$ lies on the terminal side of an angle in standard position whose measure is $\theta$ radians, find the exact value of each of the following:
a) $\sin (\theta )$ = ____________
b) $\tan (\theta )$ = ____________

14. Without using a calculator, find the value of each of the trigonometric functions of the following angles. (Angles are deemed to be measured in radians unless otherwise noted.) Check your answers by using a calculator.
a) $\frac{3\pi }{4}\qquad$
b) $-\frac{\pi }{6}\qquad$
c) $240^{\circ }\qquad$
d) $-3\pi \qquad$
e) $840^{\circ }\qquad$
f) $-\frac{\pi }{2}%% \qquad$
g) $\frac{3\pi }{4}\qquad$
h) $-210^{\circ }$
i) The angle $\theta$ in standard position whose terminal side passes through the point $(-5,12)$.

15. Find the exact value of each of the following:
a) $\csc (\frac{\pi }{4})$
b) $\tan (\frac{\pi }{4})$
c) $\cos (240^{\circ })$
d) $\sec (180^{\circ })$
e) The angle in the third quadrant whose secant is $-\sqrt{2}$

16. Give the exact coordinates of the point on the unit circle determined by each of the following angles:
a) $-\frac{\pi }{2}\qquad \qquad$
b) $-\frac{2\pi }{3}\qquad \qquad$
c) $\frac{3\pi }{4}$

17. Using the table in Appendix B, find the values of the following:
a) $\cos (21^{\circ }10^{\prime })\qquad$
b) $\ \csc (83^{\circ })\qquad$
c) $\cot \left( 310^{\circ }\right) \qquad$
d) $\sin (-200^{\circ })\qquad$
e) $\tan \left( \frac{\pi }{8}\right)$
Check your answers by using a calculator.

18. Find an angle, $\theta$, in the indicated region, for which the following holds:
a) $\cos (\theta )=-\frac{\sqrt{3}}{2}\qquad$(second quadrant)
b) $\tan \left( \theta \right) =1\qquad$(third quadrant)
c) $\sin (\theta )=-\frac{\sqrt{2}}{2}$ (fourth quadrant)
d) $\cot \left( \theta \right) =0\qquad$( $\pi <\theta <2\pi$)
e) $\sec \left( \theta \right)$ is not defined   ( $\pi <\theta <2\pi$)

19. Find the values of each of the following trigonometric functions for the angle $\theta$ in standard position:
a) $\sec \left( \theta \right)$, where the terminal side of $\theta$ is in quadrant III and $\sin (\theta )=-\frac{3}{11}$:
b) $\csc \left( \theta \right)$, where teh terminal side of $\theta$ is in quadrant III and $\sin (\theta )=-\frac{3}{11}$:
c) $\tan (\theta )$, where the terminal side of $\theta$ is in quadrant IV and $\cos \left( \theta \right) =\frac{5}{9}$.
d) $\sec (\theta )$, where the terminal side of $\theta$ is in quadrant IV and $\sin \left( \theta \right) =r$
e) $\tan (\theta )$, where the terminal side of $\theta$ is in quadrant IV and $\cos \left( \theta \right) =\sqrt{1-r^{2}}$.

20. Use a calculator to find each of the following to the indicated accuracy:
a) $\cos (32^{\circ }20^{\prime })=$ __________ (four decimal places)
b) $\cot (241^{\circ })=$ __________ (four decimal places)
c) The angle in the second quadrant whose sine is .8774 = __________ (two decimal places)
d) The nonnegative angle less than $360^{\circ }$ in the third quadrant whose sine is -0.8290
e) The nonnegative angle less than $360^{\circ }$ in the fourth quadrant whose sine is -0.1569


In Problems 21 and 22, $ABC$ is a triangle with vertex angles $A$, $B$ and $C$, and with sides of lengths $a$, $b$ and $c$, with side $a$ opposite angle $A$, etc. In each case, we are given information about some of the sides and or angles and we are supposed to ''solve the triangle'' - i.e., determine the remaining sides (to the nearest tenth of a unit) and angles (to the nearest tenth of a degree).

21. In this problem, $C$ is a right angle and
a) $a=5$ and $b=12$;
b) $a=24$ and $c=25$;
c) $a=10$ and $\ A=\frac{\pi }{6}$;
d) $a=12$ and $B=63^{\circ }$;
e) $c=16$ and $A=\frac{\pi }{4}$;
f) $b=30$ and $A=30^{\circ }$.

22. a) $a=8$, $b=13$ and $c=15$;
b) $a=5$, $b=9$ and $C=51^{\circ }$;
c) $\ a=3$, $b=7$ and $C=105^{\circ }$.
d) $a=12,$ $A=30^{\circ }$ and $B=72^{\circ }$;
e) $a=15$, $B=45^{\circ }$ and $C=65^{\circ }$;
f) $a=10$, $b=16$ and $A=60^{\circ }$;
g) $a=16$, $b=11$ and $A=110^{\circ }$;
h) $a=12.20$, $b=4.70$, and $C=120^{\circ }45$.

23. a) A man looking through a telescope from the window of his 47th floor apartment sees a murder being committed on a lower floor in a building which is 1000 feet away. If the angle of depression of his telescope is set to $14^{\circ }20^{\prime }$, and if each story of both buildings is about 10 feet high, on what floor of the building did the police discover the body?
b) A man on the ground following the flight of a balloon finds that the balloon is momentarily obscured from view by a street sign. If the sign is 24 feet above the ground and if the man is standing 32 feet from the base of the sign's support, what is the angle of elevation of the balloon at the moment that it disappears from view?
c) A boy runs 120 feet in a straight line, then veers to the left $20^{\circ }$ and runs 60 feet in this new direction. To the nearest foot, how far is he now from his starting point?
d) Two towns Mudville and Crudville, on the same side of a straight river, are 20 miles apart. From Mudville, facing Crudville, Dudville, which is on the other bank of the river, forms an angle of $35^{\circ }$. From Crudville, facing Mudville, Dudville forms and angle of $47^{\circ }$. To the nearest tenth of a mile, how far, as the crow flies, is it from Mudville to Dudville?
e) From the roof of one building, an observer can see the top of another building with an angle of elevation $29^{\circ }$. If the second building is 500 yards from the first, how much taller is the second building than the first?