Final Exam Review - MATH 2412 Precalculus
Instructor: David Katz
Summer 2006
The following questions are the most frequently missed exam questions from this semester. Questions like these will
comprise the majority of your final exam.
Review your homework and list similar problems (number and page) to study. Record the homework problems to study in
the comment line after each question.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) A farmer has 1400 yards of fencing to enclose a rectangular garden. Express the area A of the rectangle as a
function of the width x of the rectangle. What is the domain of A?
A) A(x) = -x2 + 700x; {x|0 < x < 700}
B) A(x) = -x2 + 700x; {x|0 < x < 1400}
C) A(x) = x2 + 700x; {x|0 < x < 700}
D) A(x) = -x2 + 1400x; {x|0 < x < 1400}
Answer: A
Comment:
Use the given zero to find the remaining zeros of the function.
2) f(x) = x4 - 32x2 - 144; zero: -2i
A) 2i, 12i, -12i
B) 2i, 6i, -6i
C) 2i, 6, -6
D) 2i, 12, -12
3
C) f-1 (x) = x + 10
3
D) f-1 (x) = x + 6
Answer: C
Comment:
The function f is one-to-one. Find its inverse.
3) f(x) = (x + 2)3 - 8.
3
A) f-1 (x) = x - 2 + 8
3
B) f-1 (x) = x + 8 - 2
Answer: B
Comment:
Find all zeros of the function and write the polynomial as a product of linear factors.
4) f(x) = x3 + 12x2 + 46x + 52
A) f(x) = (x + 1)(x + 5 + i 3)(x - 2 - i 3)
B) f(x) = (x - 1)(x + 5 + i 3)(x + 5 - i 3)
C) f(x) = (x + 2)(x + 5 + i)(x + 5 - i)
D) f(x) = (x + 2)(x + 5 + i)(x - 5 - i)
Answer: C
Comment:
1
Final Exam Review - MATH 2412 Precalculus
Summer 2006
Change the exponential expression to an equivalent expression involving a logarithm.
5) x
5
= π
A)
5 = logπ x
B)
5 = logx π
C) x = logπ
5
D) x = log
B) x < -2 and x > 2
C) -4 < x < 4
D) -2 ≤ x ≤ 2
C) loga 2(x + 1)
D) loga (2x + 3)
C) No real solutions
D) {-3, 27}
5
π
Answer: B
Comment:
Find the domain of the function.
6) f(x) = log9 (4 - x2 )
A) -2 < x < 2
Answer: A
Comment:
Express as a single logarithm.
7) 3 loga (2x + 1) - 2 loga (2x + 1) + 2
a 2 (2x + 1)3
A) loga B) loga (2x + 1) + 2
(2x - 1)2
Answer: B
Comment:
Solve the equation.
8) log3 x + log3 (x - 24) = 4
A) {27}
B) {53}
Answer: A
Comment:
Solve the problem.
9) An airplane is sighted at the same time by two ground observers who are 5 miles apart and both directly west
of the airplane. They report the angles of elevation as 12° and 23°. How high is the airplane?
A) 4 mi
B) 1.04 mi
C) 1.95 mi
Answer: D
Comment:
2
D) 2.13 mi
Final Exam Review - MATH 2412 Precalculus
Summer 2006
Solve the equation. Give a general formula for all the solutions. Note: k represents any integer.
3
10) sin θ = 2
π
2π
A) θ = + kπ, θ = + kπ
3
3
π
5π
B) θ = + kπ, θ = + kπ
6
6
5π
π
+ 2kπ
C) θ = + 2kπ, θ = 6
6
π
2π
D) θ = + 2kπ, θ = + 2kπ
3
3
Answer: D
Comment:
Solve the equation on the interval [0, 2 π).
11) Suppose f(x) = 2 cos θ + 1. Solve f(x) = 0.
2π 4π
π 5π
A)
, B) , 3
3
3 3
C)
3π
2
D)
π 3π
, 2 2
C)
π π
, 4 6
D)
π π
, 4 3
π
5
D)
4π
5
Answer: A
Comment:
Solve the equation on the interval 0 ≤ θ < 2π.
12) sin2 θ - cos2 θ = 0
π 3π 5π 7π
, , A) , 4
4
4 4
B)
π
4
Answer: A
Comment:
Find the exact value of the expression. Do not use a calculator.
4π
13) tan-1 tan 5
A) - 4π
5
B)
π
5
C) - Answer: C
Comment:
3
Final Exam Review - MATH 2412 Precalculus
Summer 2006
Solve the problem. Leave your answer in polar form.
14) z = 1 - i
w = 1 - 3i
z
Find .
w
A)
C)
2
cos 75° + i sin 75°
2
B)
1
cos 15° + i sin 15°
2
D)
2
cos 15° + i sin 15°
2
1
cos 75° + i sin 75°
2
Answer: B
Comment:
Write the expression in the standard form a + bi.
5π
5π 4
+ i sin 15)
3 cos 6
6
A) - 9 3 9
- i
2
2
B) - 9 3 9
+ i
2
2
9 9 3
C) - + i
2
2
9 9 3
D) - - i
2
2
Answer: D
Comment: Cite similar homework problems ________________________________________________________
Find the exact value under the given conditions.
4 3π
21
3π
16) sin α = - , < α < 2π; cos β = - , π < β < 5 2
2
5
A)
-8 + 3 21
25
B)
6 + 4 21
25
Find sin (α - β).
C)
Answer: B
Comment:
4
-8 - 3 21
25
D)
-6 + 4 21
25