Download PreCalculus Final Exam Review Sheet

PreCalculus Final Exam Review Sheet
1. Find the domain of the following functions:
1
a.
b.
16  x 2
x5
c.
2. Graph the following functions. Be sure to label all key points.
a.
b.
f ( x)  2( x  3) 2  6
f ( x)  3 x  4  2
x5
x  5x  6
d.
r  3  4 cos
r  3sin 3
f.
f ( x)  3 cos2x  2 
g.
f ( x)  2 tan( x  4 )
h.
f ( x) 
i.
f ( x)  3  2 x  1
j.
f ( x)  5 csc   1
c.
f ( x) 
e.
2
x 2  3x  2
x2  7x  6
x 2  3x  9
x 2  x  20
3. Give the following polynomial: f ( x)  x 4  7 x 2  6 x
a. List all the possible rational zeros.
b. Find all the roots.
c. Sketch the curve. Label all key points clearly.
4. Explain the use of the horizontal line test.
5. Explain the use of the vertical line test.
6. How do you determine whether a function is odd or even?
7. Find the inverse of the following functions. How can you tell if it has an inverse?
3
f ( x) 
f ( x)  3 x  5
a.
b.
c.
f ( x)  ( x  2) 2 ; x  2
x
8. Given f ( x)  3x  5 and g ( x)  x 2  7 , evaluate the following:
f  g (x)
g  f (x)
a.
b.
c.
g  g ( 4)
9. Write the rule for the following sequences:
a. arithmetic: a5  12; a8  24
b. geometric: a3  1; a5  16
10. Use summation notation to express the sums. Then find the sum.
1
 2  8  ...  2048
a.
7 + 11 + 15 + … + 63
b.
2
c.
100 + 10 + 1 + …
11. Given the rectangular coordinates of a point, plot the point and find two sets of polar coordinates for the
point, 0    2
a.
(-1, 1)
b.
(3, -1)
c.
(5, 12)
12. Convert the following polar coordinates to rectangular coordinates:
3 

 3 
a.
b.
 1,  
 4,

4 

 2 
c.
3 

 0,  
6 

13. Convert the rectangular equation to polar form:
y4
a.
b.
x2  y 2  9
14. Convert the polar equation to rectangular form:


a.
b.
r  4sin 
6
15. Simplify using the properties of logarithms:
ln e 2 x 1
log 5 252 x
a.
b.
16. Solve the following logarithmic equations:
a.
ln x  ln( x  2)  ln e3
c.
ln 4x  1
c.
x  10
c.
r  2csc
c.
5  ln e3 x
2
5 x  6
b.
log5 x  log5 ( x  2)  log5 ( x  6)
d.
5x  2  32 x 1
17. A deposit of $160,000 is compounded monthly at an annual percentage rate of 6.75%. How long will it take
for the money to triple?
18. A deposit of $5,500 is compounded continuously at an annual percentage rate of 7%. How long will it take
until you have $160,000?
19. An investment of $10,000 is compounded continuously. What annual percentage rate will produce a balance
of $25,000 in 10 years?
20. Evaluate the derivative of f ( x)  3x 2  5x  6 using the difference quotient.
21. Evaluate the derivative of f ( x) 
1
using the difference quotient.
2x  5
22. Evaluate the following using your calculator if necessary. Remember the quadrants for which the inverse
functions are defined.
9
tan
a.
b.
c.
sin5.36
sec75
2
d.
sin 1 (1.5686)
e.

3
cos 1  

 2 
23. Find the following without using your calculator:
3
 8 
cos
sin   
a.
b.
4
 3 
d.
x 

csc  arccos

2

e.
sin(arctan x)
24. Solve for the remaining parts of the following triangles
a.
b.
f.
arctan(9.85)
c.
tan
f.
x

tan  arcsin 
2

11
6
c.
d.
e.
a
10
115º
15
25. The length of the shadow of a tree is 125 feet when the angle of elevation of the sun is 33 . Approximate the
height of the tree.
26. From a point 100 feet in front of a public library, the angles of elevation to the base of the flagpole and to
the top of the pole are 28º and 39º 45’, respectively. The flagpole is mounted on the front of the library’s roof.
Find the height of the pole.
27. Fire tower A is 8 km southwest of fire tower B. A fire is spotted tower A in the direction of N 32º E. From
tower B, the fire is in the direction of N 63º W. Find the distance from the fire to the nearer tower.
28. Verify the following identities.
sec2   1
a.
 sin 2 
sec2 
c.
cot 2 
1  sin 

1  csc 
sin 
cos y
1  sin y
b.
sec y  tan y 
d.
csc x  sin x  cos x cot x
29. Solve the following trigonometric equations.
cot x cos 2 x  2 cot x
a.
c.
cos x(2 cos x  1)  0
b.
sin x  2   sin x
d.
2sin 2 x  3sin x  1  0
30. In how many ways can 6 books be arranged on a shelf?
31. In how many ways can a committee of 3 boys and 2 girls be formed if there are 15 boys and 12 girls eligible
to serve on the committee?
32. A shipment of 15 cell phones contains 3 defective units. The sales staff chooses 5 of these phones.
a. What is the probability that exactly two are defective?
b. What is the probability that at least one is defective?
33. Find the limit if (it exists):
a.
lim
6 x  6
x
b.
1
1

5 x 5
lim
x
x 0
lim x
x7
2
 49
d.
lim x  3
x 0
c.
x 7
e.
lim
x  0
2x
x
x 3
x 3