Download CHAPTER 6

Precalc B Final Exam Review
You may use your formula sheet from chapter 5 and a 4”x 6” index card for the exam.
CHAPTER 5
1
1. cos x   and sin x  0 , find the values of the remaining trigonometric functions.
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2. Simplify
sec x  cos x
.
sin 2 x
3. Prove the identity.
a. cos x  sin x tan x  sec x
b.
sec x  1
 sec x
1  cos x
4. Find all solutions of the equation in the interval 0,2  .
b. 2 sin 2 x  3 sin x  1  0
a. 2 sin x  1  0
d. cos 2 x  
3
2
e. tan x  3
c. sin
x
3

2
2
f. 2 cos x  sec x  1
5. Find the exact values of sin 2u, cos 2u, and tan 2u under the given conditions.
4
3 
sin u      u 

5
2 
6. Find the exact values of sin
sin u 
u
u
u
, cos , and tan under the given conditions.
2
2
2
12 

0  u  
13 
2
7. If u and v are in the second quadrant, and cos u  
sin(u-v), cos(u-v), and tan(u-v).
12
4
and sin v  , find the exact values of
13
5
CHAPTER 6
For problems 1 & 2, solve the triangle ABC.
1. a = 7 cm, b = 8 cm, A = 40 
2. a = 14 in, b = 12 in, c = 17 in
For problems 3 & 4, find the area of the triangle ABC.
3. a = 12 cm, b = 14 cm, C = 120 
4. a = 28 ft, b = 55 ft, c = 63 ft
5. Two ships leave a port on courses that differ by 50 and each travel at 25 knots (nautical miles
per hour). In terms of nautical miles, how far apart are the ships after 3 hours?
6. Find the component form and magnitude of the vector v: initial point (-5,4) and terminal
point (2, -1).
7. Find the component form of the vector given u  3,2 and v   2,1 .
a. u + v
b. 2u – 3v
8. Find the magnitude and direction angle of the vector v.
a. v  3(cos150 i  sin 150 j)
b. v  4i  7 j
9. Find the dot product for u  6,7 and v   3,9 .
10. Find the angle between u and v: u   3,2 v  4,0
CHAPTER 7
Solve the system of equations.
1.
x  y 1
5 x  4 y  23
2 x  6 z  9
4. 3x  2 y  11z  16
3x  y  7 z  11
2.
2x  5 y  0
3 x  2 y  19
2x  4 y  z  1
3. x  2 y  3 z  2
x  y  z  1
CHAPTER 8
1. If possible, find A+B, A-B, 3A, 3A-2B, and AB
  3 4
B= 

 4 2
1 2 
A

2 1 
2. Find the inverse of the matrix if it exists.
5 
 2
 7 33 

a. 
b.  6  15

 4  19
 0
1 
3. Find the determinant.
a.
3
5 9
6
b.
1 4
1
6
5
3
2
0 1
4. Use a determinant to find the area of the triangle with vertices (-3,2), (5,0), and (3,9)
5. Solve for x:
x  2 1
3
x
0
CHAPTER 9
1. Write the first five terms of the sequence.
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a. an  2 
b. an  n(n  1)
n
2. Simplify the factorial expressions.
7!6!
a.
6!8!
b.
n!
(n  2)!
b.
 2k
3. Find the sum.
5
a.
 4k
k 2
3
3
k 1
4. Determine whether the sequence is arithmetic, geometric, or neither. Write the formula
for the nth term if it is arithmetic or geometric.
a. 5, 3, 1, -1, -3
b. 0, 1, 3, 6, 10
c. 3, 6, 12, 24, …
400
5. Find the sum for the arithmetic sequence
 (2n  1)
n 1
6. Find a formula for the nth term of the arithmetic sequence where a5  190 and a10  115
7. Find the first term of the geometric sequence where a2  3 and a5 
1
8. Find the sum of the finite geometric series  32 
4
n 1
6

3
.
64
n 1
2
9. Find the sum of the infinite geometric series  2 
n 1  3 
.
n 1
.
10. Use the Binomial Theorem to expand the binomial expression.
a.
5  2 x 4
b. 4  5 x 
3
11. How many permutations are there of the letters in the word MATHEMATICS?
12. How many different outfits are possible if you can choose from 4 shirts, 3 pairs of pants,
and two pairs of shoes?
13. How many was can 6 people stand in line?
14. Six runners are entered in a race. In how many ways can they come in first, second, and
third place?
15. In how many ways can a group of 5 people be selected from a group of 8 people?
16. A box holds 10 white, 5 red, and 4 black marbles. If 2 marbles are selected at random,
without replacement, what is the probability that they will both be red?
17. A rummy had consists of seven cards dealt from a deck of 52 cards. How many different
rummy hands are possible?
CHAPTER 10
Find the standard form of the equation of the parabola.
1. Vertex (4, 2)
Focus (4, 0)
2. Vertex (0, 2)
Directrix x = -3
Identify the vertex, focus and directrix of the parabola. Then sketch the graph.
3. ( y  3) 2  4( x  1)
4. ( x  2) 2  8( y  4)
Find the standard form of the equation of the ellipse.
5. Vertices (2, 0), (2, 6)
Foci (2, 2), (2, 4)
6. Foci (3, 0), (-3, 0)
Major axis length: 10
Find the center, vertices and foci of the ellipse. Then sketch the graph.
7.
( x  2) 2 ( y  1) 2

1
16
25
8.
( x  5) 2 ( y  3) 2

1
36
4
Find the center, vertices, foci and asymptotes of the hyperbola. Then sketch the graph.
9.
( x  3) 2 ( y  5) 2

1
16
4
10. ( y  1) 2 
x2
1
4
LIMITS AND DERIVATIVES
1. lim (2 x  1)
x 3
4. lim
x 
x4
x3  1
x2  x  2
2. lim
x 1
x2  1
5. lim
x 
6. Use the difference quotient: lim
h 0
9x4  x
3. lim 4
x 2 x  5 x 2  x  6
3x  7
x2  2
f ( x  h)  f ( x )
to find the derivative of 2 x 2  x  3 .
h