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Precalculus
Final Exam Review Sheet
Solve the equation.
2
1) x + 1 =
x
Use a method of your choice to solve the equation.
2) x2 + 3x - 28 = 0
Write the sum or difference in the standard form a + bi.
3) (3 + 9i) - (-8 + i)
Solve the equation.
x+4 x+5
4)
=
5
6
Find an equation for the circle.
5) Center (2, 1), radius 2
Solve the equation algebraically.
6) x - 10x - 25 = 0
Find the zeros of the function.
7) f x = 9x2 - 21x + 10
Use limits to describe the behavior of the rational function near the indicated asymptote.
8
8) f(x) = x+2
Describe the behavior of the function near its vertical asymptote.
Find the inverse of the function.
9) f(x) = 5x + 2
Write the product in standard form.
10) (6 + 3i)(5 + 3i)
Find the asymptote(s) of the given function.
(x - 3)(x + 3)
11) h(x) =
vertical asymptotes(s)
x2 -9
Find the range of the function.
4
12) f(x) =
3-x
Solve the equation by changing it to exponential form.
13) log x = 4
1
14) log x = 4
3
Rewrite the expression as a sum or difference or multiple of logarithms.
10 m
15) log7
n
16) log2
x2 y4
6
Find the exact solution to the equation.
1
17) 3 (6 - 3x) =
27
Solve the problem.
18) Find the present value of a loan with an annual interest rate of 6% and periodic payments of $792.09 for a term
of 6 years, with payments made and interest charged 12 times per year.
19) Find the periodic payment of a loan with present value $20,000 and an annual interest rate 6.3% for a term of 3
years, with payments made and interest charged 12 times per year.
20) At what interest rate must $5400 be compounded annually to equal $13,598.12 after 12 yr? (Round to the nearest
percent.)
21) Matthew obtains a 25-year $142,000 house loan with an APR of 8.02%. What is his monthly payment?
Convert the angle to degrees, minutes, and seconds.
22) 51.82°
23) 248.56°
Use the arc length formula and the given information to find the indicated quantity.
π
24) s = 3.7 ft, θ = rad; find r
6
25) s = 18 cm, θ = 54°; find r
Solve the problem.
26) From a distance of 45 feet from the base of a building, the angle of elevation to the top of the building is 66°.
Estimate the height of the building to the nearest foot.
27) A police helicopter is monitoring the speed of two cars on a straight road. The helicopter is at an altitude of 4200
feet directly above the road. At one instant, the angle of elevation from the first car to the helicopter is 21°, and
the angle of elevation from the second car to the helicopter is 17°. How far apart are the two cars to the nearest
foot?
2
Evaluate without using a calculator.
8
28) sin θ, if cos θ =
and tan θ < 0
9
Solve the problem.
29) From a boat on the lake, the angle of elevation to the top of a cliff is 28°24'. If the base of the cliff is 2291 feet from
the boat, how high is the cliff (to the nearest foot)?
30) When sitting atop a tree and looking down at his pal Joey, the angle of depression of Mack's line of sight is 59°
46'. If Joey is known to be standing 10 feet from the base of the tree, how tall is the tree (to the nearest foot)?
31) A building has a ramp to its front doors to accommodate the handicapped. If the distance from the building to
the end of the ramp is 23 feet and the height from the ground to the front doors is 4 feet, how long is the ramp?
(Round to the nearest tenth.)
Find the exact value of the composition.
32) arctan[sin(π/2)]
π
33) cos-1 cos 4
Describe the transformation required to obtain the graph of the given function from the basic trigonometric graph.
34) y = -cot x
35) y = csc x - 8
36) y = 8 cot πx
The given measurements may or may not determine a triangle. If not, then state that no triangle is formed. If a triangle is
formed, then use the Law of Sines to solve the triangle, if it is possible, or state that the Law of Sines cannot be used.
37) B = 108°, c = 4, b = 8
Solve the triangle.
38) B = 36°, a = 40, c = 19
39) a = 5, b = 12, c = 9
Use the fundamental identities to find the value of the trigonometric function.
1
40) Find tan θ if cos θ = and sin θ < 0.
6
41) Find csc θ if cot θ = -
35 and cos θ < 0 .
Use basic identities to simplify the expression.
1
42)
+ sec θ cos θ
cot2θ
3
43)
tan θ
cot θ
Find an exact value.
44) cos 15°
45) cos 165°
46) cos
π
12
47) cos
19π
12
Answer the question.
48) In how many ways can you answer the questions on an exam that consists of 11 true-false questions?
49) How many automobile license plates can be made involving 2 letters followed by 3 digits?
Solve the problem.
50) There are 14 people in a club. A committee of 6 persons is to be chosen to represent the club at a conference. In
how many ways can the committee be chosen?
51) There are 12 essay questions on a test. Students must select 3 questions to write essays on. In how many ways
can the 3 questions be chosen?
Solve.
52) How many ways can a president, vice-president, and secretary be chosen from a club with 12 members?
53) There are 6 women running in a race. How many first, second, and third place possibilities can occur?
Expand the binomial.
54) (5x + 2)3
55) (-4x - 2y) 4
Find the coefficient of the given term in the binomial expansion.
56) x7 term, (x - 2)14
57) x7y2 term, (x + y)9
Find the probability of the event.
58) Give the probability that the roll of a die will show 4 or 6.
59) Two 6-sided dice are rolled. What is the probability the sum of the two numbers on the dice will be 4?
4
Find the probability.
60) A 5-card hand is dealt from a deck of 52 cards. What is the probability that
a) all are from the same suit?
b) all are hearts?
c) exactly 2 are spades?
61) A 7-card hand is dealt from a deck of 52 cards. What is the probability that all 7 cards are hearts?
Write out the first five terms of the sequence.
62) a n = 3n - 2
Find the first six terms of the sequence.
63) a 1 = 1, an = 5 · an-1
Determine whether the sequence converges or diverges. If it converges, give the limit.
5 5
64) 60, 10, ,
, ...
3 18
Find an explicit rule for the nth term of the arithmetic sequence.
65) 1, -1, -3, -5, ...
Find an explicit rule for the nth term of the sequence.
66) 2, 10, 50, 250, ...
67) The second and fifth terms of a geometric sequence are -28 and 1792, respectively.
Find a recursive rule for the nth term of the sequence.
68) 3, 11, 19, 27, ...
69) The second and fifth terms of an arithmetic sequence are -12 and -3, respectively.
Write the series using summation notation.
70) 2 - 10 + 50 - 250 + ...
Find the sum of the arithmetic sequence.
71) 29, 31, 33, 35, ..., 47
Find the sum of the geometric sequence.
72) 5, 15, 45, 135, 405
Solve.
73) An auditorium has 30 rows with 10 seats in the first row, 12 in the second row, 14 in the third row, and so forth.
How many seats are in the auditorium?
Find the sum of the first n terms of the sequence.
74) 30, 38, 46, 54, ... ; n = 8
5
75) 4, -8, 16, ... ; n = 11
Determine whether the infinite geometric series converges. If the series converges, determine the limit.
76) 7 + 21 + 63 + 189 + ...
77) 36 - 12 + 4 -
4
+ ...
3
6
Answer Key
Testname: FINAL EXAM REVIEW SHEET
1)
2)
3)
4)
x = -2 or x = 1
x = 4 or x = -7
11 + 8i
x=1
5) (x - 2)2 + (y - 1)2 = 4
6) 5
5
2
7) and
3
3
8)
lim f(x) = ∞,
lim f(x) = -∞
x→-2x→-2 +
x2 - 2
9) f-1(x) =
for x ≥ 0
5
10) 21 + 33i
11) x = 1, x = -1
12) (-∞, 0) ∪ (0, ∞)
13) x = 104
14) x = 3 4
15) log7 10 +
1
log7 m - log7 n
2
16) 2 log2 x + 4 log2 y - log2 6
17) x = 3
18) $47,794.33
19) $611.16
20) 8%
21) $1097.86
22) 51°49′12′′
23) 248°33′56′′
22.2
24)
ft
π
25)
60
cm
π
26) 101 feet
27) 2796 feet
17
28) 9
29) 1239 ft
30) 17 ft
31) 23.3 ft
π
32)
4
33)
π
4
34) Reflection across the x-axis
35) Vertical translation down 8 units
36) Vertical stretch by a factor of 8 and a horizontal shrink by a factor of
7
1
π
Answer Key
Testname: FINAL EXAM REVIEW SHEET
37) C = 28.4°, A = 43.6°, a ≈ 5.8
38) b ≈ 27, C ≈ 24.2, A ≈ 119.8
39) A ≈ 22.2°, B ≈ 115°, C ≈ 42.8°
40) - 35
41) 6
42) sec2 θ
43) tan2 θ
6+ 2
44)
4
45)
-
64
2
46)
6+
4
2
47)
64
2
48) 2048
49) 676,000
50) 3003
51) 220
52) 1320
53) 120
54) 125x3 + 150x2 + 60x + 8
55) 256x4 + 512x3 y + 384x2 y2 + 128xy3 + 16y4
56) -439,296
57) 36
1
58)
3
59)
1
12
60) a) 0.00198 b) 0.000495 c) 0.27428
61) 0.0000513
62) 1, 4, 7, 10, 13
63) 1, 5, 25, 125, 625, 3125
64) Converges; 0
65) a n = 1 + -2(n-1)
66) a n = 2 · 5 n-1
67) a n = 7 · (-4)n-1
68) a n = a n-1 + 8
69) a n = a n-1 + 3
∞
70)
∑
2(-5)n
n=0
71) 380
72) 605
73) 1170
74) 464
8
Answer Key
Testname: FINAL EXAM REVIEW SHEET
75) 2732
76) Diverges
77) Converges; 27
9