CALCULUS READINESS TEST
SOLUTIONS
1
PART I
(In all graphing problems, assume the usual coordinate system.)
r
1. The right circular cylinder shown
on the right has a circular base and a
closed top. Its surface area in terms of
r and h is
ANSWER: (A) 2πr2 + 2πrh
πr2
h
2πr
r
2πrh
πr2
A = 2πr2 + 2πrh
y
2. If f is a function whose graph is the
parabola sketched to the right, then
f (x) > 0 whenever
−2
ANSWER: (E) −2 < x < 1
1
x
3. Money in a bank quadruples every 8 years. If $100 is deposited
today, what will its value be after 24 years?
ANSWER: (B) $6400
SOLUTION: 100 → 400 → 1600 → 6400
8 yrs 8 yrs 8yrs
2
4. The y-coordinate of the point of intersection of the graphs of
x − 2y = −18 and x + y = −9 is
ANSWER: (C) 3
SOLUTION:
x − 2y = −18
x + y = −9
x = 2y − 18
x = −9 − y
2y − 18 = −9 − y
⇒ 3y = 9
y=3
5. Definition: A function f is odd if f (−x) = −f (x) for each x in
the domain of f . Of the following, which best represents the graph
of an odd function?
ANSWER: (E)
6. (25)1/2 (27)−1/3 = (25)1/2 ·
ANSWER: (B)
1
1
5
=5· =
1/3
(27)
3
3
5
3
3
7. Of the following, which best represents the graph of y = x2 − 2x − 2?
ANSWER: (A)
SOLUTION: Parabola, opens up because coefficient of x2 is positive.
8. If log4 (x + 9) = 3 then x =
ANSWER: (D) 55
SOLUTION:
log4 (x + 9) = 3
x + 9 = 43 = 64
x + 9 = 64
x = 55
9. Of the following, which best represents the graph of y = 5x ?
ANSWER: (A)
SOLUTION: 5x monotone increasing, 5x > 0 ∀ x
10. If
(2x + 1)(x − 5)
= 0 then x =
x+3
ANSWER: (C) 5 or −
SOLUTION: If
1
2
AB
= 0 ⇒ A = 0 or B = 0
C
x−5 =0
x =5
2x + 1 = 0
x = − 12
4
11. Since 37 is approximately equal to 2000, of the following, which
best approximates 314 ?
ANSWER: (D) 4,000,000
SOLUTION:
37 ≈ 2000
314 = (37 )2 ≈ (2000)2 = 4, 000, 000
12. In the figure shown to the right, what is the distance between
the points P and Q?
ANSWER: (B) 5
SOLUTION:
q
d =
q
(x2 − x1 )2 + (y2 − y1 )2
d = (3 − (−1))2 + (6 − 3)2
√
= √42 + 32 √
= 16 + 9 = 25
=5
13. If f (x) =
5c + 13
c+7
5x + 3
5(c + 2) + 3
5c + 10 + 3
then f (c + 2) =
=
=
x+5
(c + 2) + 5
c+7
ANSWER: (E)
5c + 13
c+7
14. In a standard coordinate system, the graph of the equation y =
−3x + 6 is
ANSWER: (E) a line falling to the right.
SOLUTION: Slope < 0 ⇒ falls left to right
5
15. What is the area of the rectangle
shown in the figure to the right?
f (x) = x2 − 3x + 5
Note: The figure is not drawn to scale.
0.4
0.1
x
ANSWER: (D) 1.188
SOLUTION:
f (.4) = (.4)2 − 3(.4) + 5 = .16 − 1.2 + 5 = −1.04 + 5 = 3.96
A = l · w = (.4 − .1)(f (.4)) = (.3)(3.96) = 1.188
16. The quantity p − q is a factor of how many of the following?
p2 − q 2
p2 + q 2
p3 − q 3
p3 + q 3
ANSWER: (C) two only
SOLUTION:
p3 − q 3 = (p − q)(p2 + pq + q 2 )
p3 + q 2 = (p + q)(p2 − pq + q 2 )
p2 − q 2 = (p + q)(p − q)
DOES NOT FACTOR
p2 + q 2
6
17. A rectangle R has width x and
length y. A rectangle S is formed from
R by multiplying each of the sides of
the rectangle R by 3 as shown in the
figure to the right. What is the area
of the portion of S lying outside R?
S
R
Note: The figure is not drawn to scale.
ANSWER: (B) 8xy
SOLUTION:
A(R) = xy,
A(S) = 9xy
A(S\R) = A(S) − A(R) = 9xy − xy = 8xy
18. The inequality |x − 7| ≤ 2 is equivalent to
ANSWER: (A) 5 ≤ x ≤ 9
SOLUTION:
|f (x)| ≤ A ⇒ −A ≤ f (x) ≤ A
−2 ≤ x − 7 ≤ 2
5 ≤x≤9
19. The length of a certain rectangle
is 3 meters more than twice its width.
What is the width of the rectangle if
the perimeter of the rectangle is 90 meters?
w
l
ANSWER: (C) 14m
SOLUTION:
l
P
90
90
= 2w + 3
= 2l + 2w
= 2(2w + 3) + 2w
= 4w + 6 + 2w ⇒ 84 = 6w ⇒ 14 = w
7
20. Definition: A function f has a maximum value at c if f (c) ≤ f (x)
for every x in the domain of f .
The domain of the function whose graph is shown to the right is [0,5].
At which of the following numbers does the function appear to have
a maximum value?
ANSWER: (D) 4
SOLUTION: MAX = “HIGHEST” POINT OF f (x) ⇒ x = 4
END OF PART I
8
PART II
(Assume radian measure in all trigonometric expressions.)
21. Of the following, which best represents the graph of y = cos x for
π
π
x between − and ?
2
2
ANSWER: (C)
y = cos(x)
y
y
(C)
1
−π/2
π/2
x
−π/2
π/2
−1
22. cos2 θ − 1 =
ANSWER: (C) − sin2 θ
SOLUTION:
sin2 θ + cos2 θ = 1
cos2 θ = 1 − sin2 θ
cos2 θ − 1 = 1 − sin2 θ − 1
= − sin2 θ
µ ¶
π
23. If f (x) = cos(3x) then f
6
µ
π
= cos 3 ·
6
ANSWER: (E) 0
9
¶
= cos
π
=0
2
x
24. For which of the following values of x is csc x not defined?
y
ANSWER: (A) −π
SOLUTION:
y = sin(x)
π
y = sin(x)
1
csc x =
⇒ csc x NOT DEFINED if sin x = 0
sin x
sin(−π) = − sin π = (−1)(0) = 0
25. sec2 θ cot θ cos θ =
1
cos θ cos θ
1
·
·
=
= csc θ
cos2 θ sin θ
1
sin θ
ANSWER: (B) csc θ
END OF PART II
10
2π
x