FINAL EXAM, Version 1 MATH 3
12/13/2007 Instructor: Frank Bauerle, Ph.D.
No books, graphing utilities or notes allowed. Show your work. Show your work.
Show your work.
Your Name:
Your TA: Max
Problem 1:
Problem 2:
Problem 3:
Problem 4:
Problem 5:
Problem 6:
Problem 7:
Problem 8:
Problem 9:
Problem 10:
Problem 11:
Problem 12:
Problem 13:
Problem 14:
Problem 15:
10 10 10 15 10 10 15 20 25 15 15 10 15 15 15 TOTAL:
210 Your score
Good Luck and have a great winter break! 1. (10 points) Fill in the table and give the exact values for sin 0:' l cos 0:' and tan 0:' for
0:'
= 0°
l
0:'
= 30°
l
0:'
= 45°, 0:' = 60°
and
0:'
= 90°.
Also give the radian measures of all
the angles.
oZ
V1)cl~t
~, V\
tX
COSo(
-t{).~ c<­
0°
30°
450
600
"01)
2. (10 points) Find the reference angle, quadrant of the angle and use the reference angle
principle to get the exact value of the given trig expression.
(a) Quadrant of 240° = Reference angle of 240 0
=
tan405° = -5rr
(b) Quadrant of -6- = -5rr Reference angle of -6­
-5rr csc-- =
6
3. (10 points) Find the exact value of the following by using the appropriate formula.
(Calculator approximations will give 0 credit, but can be used to verify your answer.).
You do not need to simplify the expressions.
(a) sin 15°
(b) sin 22.5°
1
4. (15 points) Consider the function f(x) =
~ sin(2x
1) - 1. Find the following:
(a) (1 points) The amplitude of f.
(b) (1 point) The vertical shift of f
(c) (2 points) The period of f.
(d) (2 points) The phase shift of f·
(e) (6 points) The graph of f.
(f) (3 points) Now graph g(x) =
above.
~ csc(2x -
2
1) - 1 into the same coordinate system
5. (10 points) Find the exact values of the following expressions. (Note: Calculator
approximations will give zero credit, but can be used to verify your answer.)
(a) sec (arcsin ~l
(b) cos[2 arccos ( -
2
'3) 1 6. (10 points) Verify the identity
1
1
1 - cos x
1 + cos x --- +
3
= 2csc2 x
7. (15 points) Consider the function f(x) = ex +1
(a) (2 points) Show that the function f(x) is one-to-one. (You can do this alge­
braically or graphically)
(b) (4 pOints) Find f-l(x).
(c) (8 points) Graph both f(x) and f-l(x) in the same coordinate system.
(d) (1 point) How are the graphs related?
4
8. (20 points) Assume Mathland has 1000 inhabitants in the year 2004 and 1600 inhabi­
tants on the same day one year later in 2005.
(a) For this part assume the population of Mathland will grow LINEARLY:
i. How many inhabitants will Mathland have on the same day in 20107
ii. In what year will the population have reached 10000 inhabitants?
(b) For this part assume the same number of inhabitants in 2005 and 2006 but assume
that the population grows EXPONENTIALLY:
i. How many inhabitants will Mathland have on the same day in 2010?
ii. In what year will the population have reached 10000 inhabitants?
5
9. (25 points) Give all real solutions to the following equations.
(a) tan x = .8 (b) 5 cos2 () + 6 cos () - 8 = 0
6
10. (15 points) Assume you are given the curve x = y2 and the point P(l,O).
(a) (4 points) Sketch the curve and the point.
(b) (9 points )Find the point( s) Q on the given curve that are closest to the point
P. Hint: Give a function of one variable that represents the distance of a point
on the curve to the point P as a function of one variable. Then minimize this
function.
(c) (2 points) What geometric property exists between the line segment connecting
P with such Q and the tangent line to the curve y .JX at the point Q?
7
11. (15 points) For the rational function f(x)
X
4-x
2
do the following:
(a) Find all x-intercepts (if any).
(b) Find the y-intercept.
(c) Find all vertical asymptotes (if any).
(d) Find the horizontal asymptote.
(e) Where does the graph intersect the horizontal asymptote ?
(f) Sketch the graph (label the axes, asymptotes and all intercepts):
8
12. (10 points) A principal P will be invested at an annual rate of 7%, interest compounded
quarterly. If no further deposits or withdrawals are made and the conditions of the
investment remain unchanged, how much money needs to be invested now so that the
balance in the account will be 15, 000 after exactly eight years? You can round up to
the nearest dollar.
13. (15 points) Two points P and Q are on opposite sides of a riv~r (see the sketch). From
P to another point R on the same side is 290 ft. Angle~and~re found to be 22° and
116°, respectively. Compute the distance from P to Q, across the river. (You can
round your answer to the nearest foot.)
p
-
.------­
......-" --...
-......­
,
~ .....
~
-.
,
•
9
14. (15 points) a) Find a possible equation of the given curves (s,~ 0..,.. (t)~i ~ )
Equation:
Equation:
b) Give two reasons why the following graph cannot represent a polynomial function
~
with degree three.
Reason 1:
Reason 2:
10 15. (15 points)
(a) (2 points) Find the common difference of the arithmetic sequence 10,6,2, -2, ....
(b) (3 points) Find the fifth term in the geometric sequence
2
5"'
4
8
25' 125' ...
(c) (4 points) Evaluate the sum
9
2:(1 + 2k)
k=5
(d) (6 points) Find the sum of the first 500 terms in the sequence 1,2,3,4, ....
11