CALCULUS
AP Night Practice Test
SECTION I: Part A
Time-- 30 minutes
Number of questions – 12
CALCULATORS MAY NOT BE USED ON THIS PORTION
Directions: Solve each of the following problems. After examining the form of the choices, decide which is the best of the choices
given and write down the corresponding letter on your answer page. No credit will be given for answers circled on the practice test.
Do not spend too much time on any one problem.
In this practice test:
(1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from
among the choices the number that best approximates the exact numerical value.
1) The graph of which function has y = -1 as an asymptote?
A)
B)
C)
D)
Look for a function that has a limit of 1 as → ∞. Let’s look at the limit of each possible answer: y = e-x
y=
y = ln (x + 1)
y=
A. lim
B. lim
E) y =
→
→
D. lim
→
E. lim
→
0 lim
1 using L’Hospital’s Rule →
→
C. lim
lim
→
ln
1
lim
→
∞ 1 lim
using L’Hospital’s Rule using L’Hospital’s Rule →
AnswerE
Note: The lesson on this question is to first look at the rational functions, eliminate all terms except the ones with the highest exponents, and consider their limits. Doing this on Problem 1 gives a quick answer. 2) lim
A) 1
B) 10
C) -1
D)
E)
→
Do the same thing suggested in orange above in Problem 1. This gives: lim
→
10
10
10 10
10
lim
→
10
10
AnswerB
Page1
3) If
A)
B)
C)
D)
E)
3
8
, then lim
→
Notice that the limit shown is the definition of the derivative of : 10
14
20
-14
-20
2
lim
2
2 2
→
Then, 2
3
8
6
16
6∙2
16
6
16
2
12
2
AnswerB
4) Let f and g be differentiable functions such that
f (1) = 4, g (1) = 3, f ‘(3) = -5
f ’(1) = -4, g’(1) = -3, g’(4) = 2
If h(x) = g (f (x)), then h’(1) =
A) -8
B) 12
C) -15
D) -5
E) -12
Use the chain rule in Lagrange (Prime) Notation. ∙ ′
∙
1
,
where:
1
∘
∙
2 ∙
Answer:A
5) If e y = x, then
A) 1
B)
C)
D) ln x
E) ln y
implies:
Then, AnswerB
Page2
6) If the line y = 4x + 3 is tangent to the curve y = x2 + c, then c is
A)
B)
C)
D)
E)
2
4
7
11
15
Now, since the slope at the point of intersection is , we have: So,
Then, the point of tangency is on both curves, so: 4
3
4∙
3
11
4
and so AnswerC
7) Find the second derivative of f (x) if f (x) = (2x + 3)4.
A)
B)
C)
D)
E)
4(2x + 3)3
8(2x + 3)3
12(2x + 3)2
24(2x + 3)2
48(2x + 3)2
2
′
3 4∙ 2
3
8∙ 2
3 3∙8∙ 2
∙ 2
3
∙
∙ 2
AnswerE
8) If the nth derivative of y is denoted as y(n) and y = -sin x, then y(7) is the same as
A)
B)
C)
D)
E)
dy/dx
d2y/dx2
y
d3y/dx3
none of these
sin sin cos sin sin cos cos cos Notice the cyclical nature of the derivatives of the sine and cosine functions (reminiscent of the cyclical nature of powers of – note: this is not a coincidence because the two are related through: cos
sin ). AnswerD
Page3
9) Evaluate the limit, if possible: lim
A)
B)
C)
D)
E)
¼
-¼
1
0
DNE
√
→
First, we need to see if the limit is indeterminate: √
5 2
9
√9 5 2
9 9
9
0

0
Since the limit is indeterminate, we can apply L’Hospital’s Rule. 5
lim
2
9
→
1
lim 2
5
1
→
lim
→
1
2
1
5
2 ∙ √9
5
AnswerA
10) A particle moves along the x-axis so that its position at any time t (in seconds) is s (t )  2t 2  3t  5 . The
acceleration of the object at t = 2 seconds is
(A) 19 units/s 2
(B) 11 units/s 2
(C) 16 units/s 2
(D) 4 units/s 2
(E) 0 units/s 2
2
3
5 ′
4
3
′
is velocity
′′
4
′′
is acceleration
′′ 2
4
11)
AnswerD
If f ( x)  16 x then f "(4) is equal to
1
2
(B) -32
(C) -4
(D) -2
(E) -16
(A) -
16√
16
1
∙ 16
2
′
′ 4
is position
8
1
∙ 8
2
4
√4
4
√
AnswerA
Page4
12) If g ( x )  f ( x )  h( x ) , lim g ( x)  4 , and lim h( x)  4 then
x 3
x 3
(A) f ( x) is continuous at x  3
(B) f (3)  4
(C) lim f ( x)  4
x 3
(D) g (3)  4
(E) none of these
Squeeze Theorem: If 
, and

lim
→
lim
→
Then

lim
→
4 and so Implication: →
AnswerC
13) A point moves along the curve y = x2 + 1 in such a way that when x = 3, the x-coordinate is increasing at
the rate of 5 ft/sec. At what rate is the y-coordinate changing at that time?
A) 80 ft/sec
B) 45 ft/sec
C) 30 ft/sec
D) 85 ft/sec
E) 40 ft/sec
1 2 and so 2∙3
∙
∙
ft/sec AnswerC
Page5
14) Let f be defined as follows, where a
, for
0.
,
0, for
.
Which of the following are true about f ?
I. lim →
exists.
II. f (a) exists.
III. f (x) is continuous at x = a.
A)
B)
C)
D)
E)
None
I only
II only
I and II only
I, II, and III only
Consider these one at a time: I.
The limit exists if it is the same from the left and right. lim
→
lim
2 fromboththeleftandtheright. →
Therefore, the limit exists and it is equal to 2 . IisTRUE II.
0 and so it exists. IIisTRUE Continuity: A function, , is continuous at
III.
iff:
a.
is defined,
b. lim →
exists, and
c. lim →
lim
, so the third condition in the definition is violated. →
IIIisFALSE
AnswerD Page6
15) The slope of the tangent line to the curve y3x + y2x2 = 6 at (2, 1) is
A)
B) -1
C)
∙ ∙
∙
6
∙
D)
E) 0
∙
∙
∙
∙
∙
∙
0 0 ∙
2 and Then, substitute ∙
∙
∙
∙
∙
1 to get: 5
∙
6
8
AnswerC
16) Find dy/dx for y = 4sin2 (3x).
A)
B)
C)
D)
E)
8sin(3x)
24sin(3x)
8sin(3x)cos(3x)
12sin(3x)cos(3x)
24sin(3x)cos(3x)
∙
∙
8 sin 3
∙
∙
AnswerE
18)
=
First, simplify the given function: A)
B)
C) x3
D) 3x2
E) 3
Then, take the derivative: 3
AnswerD Page7
CALCULUS
SECTION II, PART A
Calculators may not be used for any questions on this section.
1) Consider the curve given by y2 = 2 + xy.
(a) Show that
.
(b) Find all points (x, y) on the curve where the line tangent to the curve has slope .
(c) Show that there are no points (x, y) on the curve where the line tangent to the curve is
horizontal.
(d) Let x and y be functions of time t that are related by the equation y2 = 2 + xy. At time t = 5, the
value of y is 3 and dy/dt = 6. Find the value of dx/dt at time t = 5. 2
(a) 2
0
∙
2
(b) ∙ which implies Going back to the original function, 2
(c) ∙ which implies The points, then are 2
(d) and Then, Going back to the original function, Then, note that: ∙ which implies 2
2∙
. Also we are given ∙
,
2
, then If the slope is 0, then the numerator of the fraction is 0, so . Since this is not possible, cannot be . ,
√2 2
If 2
∙
∙
⇒ Page8
2)
(a) Use the Pythagorean Theorem to get the value of the hypotenuse, . ⇒ 3
4
⇒ (b) 2
Note that is negative since the movement is 2
2 ∙
towards the lighthouse, whereas, is positive since movement is away from the lighthouse. Substitute in values: ,
,
Note that cos ∙ ∙
2 ∙
∙
2 ∙
∙
/
, ,
(c) sin
2∙
10 ∙
60
⇒
Next, take a derivative with respect to ∙
∙
⇒ ∙ ∙
∙
⇒ rad/sec Page9
CALCULUS
SECTION II, PART B
Time: 15 minutes
Number of Problems: 1
Calculators are required for some portions of this section.
3) A 12,000-liter tank of water is filled to capacity. At time t = 0, water begins to drain out of the tank at a rate
that is modeled by r(t), measured in liters per hour, where r is given by the piecewise function below.
a) Is r continuous at t = 5? Show analytical work that leads to your answer.
b) Find r’(3). Using correct units, explain the meaning of that value in the context of the this problem.
(a) Continuity: A function, , is continuous at
is defined,
d.
e. lim →
exists, and
f. lim →
lim
lim
lim
1000
→
→
→
(b) 600
3
lim
3,000
8
375
1000
.
367.879
→
′
′ 3
at Continuity requires the limits from the left and right be equal, which they are not. Therefore, the function is not continuous at .
3 ∙
∙
∙
∙ ∙
Note: the units of iff:
,
liters/hour2 are liters/hour, so the units of are liters/hour/hour or liters/hour2. Meaning: ′
liters/hour2 means that the rate of flow from the tank is increasing at a rate of liters/hour2. Page10