AP
®Calculus AB Practice Exam
Instructions:Write your solution using dark pencil or ink on white paper. Label each part of each question. Write clearly and legibly. Cross out any errors you make; erased or crossed-out work will not be scored.
Manage your time carefully. During the timed portion for Part A, work only on the questions in Part A. Calculators are permitted but are not required for any part of any question. If using a calculator, you must clearly indicate your setup, and you must show the mathematical steps necessary to produce your results. Your work must be expressed in standard mathematical notation rather than calculator syntax. For example,∫5
1 x
2dxmay not be written as fnInt(X2,X,1,5).
Show all of your work, even though a question may not explicitly remind you to do so. Justifications require that you give mathematical reasons, and that you verify the needed conditions under which relevant theorems, properties, definitions, or tests are applied. Your work will be scored on the correctness and completeness of your methods as well as your answers. Answers without supporting work will usually not receive credit.
Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If you use decimal approximations in calculations, your work will be scored on accuracy. Unless otherwise specified, your final answers should be accurate to three places after the decimal point.
CALCULUS AB Part A Time—25 minutes Number of problems—1
A GRAPHING CALCULATOR IS NOT REQUIRED FOR THESE PROBLEMS.
t (days) 0 3 5 7 9 12 14
h(t)(# of butterflies) 1 2 4 7 11 16 22
1. The number of butterflies spotted in Amelie’s garden each afternoon is modeled by a differen-tiable and increasing functionhfor 0 ≤ t ≤ 14, wheret is measured in days. Selected values ofh(t)are given in the table above.
(a) Use the data in the table to find an approximation forh0(8). Using correct units, interpret the meaning of this value in the context of the problem.
Solution:
h0(8) ≈ 11−7
9−7 = 2
On day 8, the number of butterflies spotted in Amelie’s garden is increasing at a rate of appproximately 2 butterflies per day.
Score:
1: approximation
1: interpretation with correct units
(b) Explain why there must be at least one timet, for 0< t < 14, such thath0(t)=1.5.
Solution:
Sinceh(t)is continuous and differentiable on[0,14], the Mean Value Theorem guar-antees a value 0< c <14 such that h0(c)= h(1414)−−0h(0) = 2114 = 1.5.
Score:
1: h(t)continuous on[0,14]and differentiable on(0,14)
Note: accept differentiable on[0,14]since it is given 1: average rate of change on[0,14]=1.5
(c) Approximate the value of∫14
0 h(t)dtusing a left Riemann sum on the subintervals[0,5], [5,9], and [9,14]. Is this approximation greater than or less than ∫14
0 h(t)dt? Give a reason for your answer.
Solution:
∫14
0 h(t)dt ≈ 5(1)+4(4)+5(11)=76
This approximation is less than∫08h(t)dt sinceh(t)is increasing and we are using a left Riemann sum.
Score:
1: left Riemann sum 1: approximation
1: underestimate with reason
(d) The number of bees spotted in Amelie’s garden during the same 14-day period is modeled by the function g(t)=7e17(x−1) +cos(2x−16) for 0 ≤ t ≤ 14, wheret is measured in days. At timet = 8, is the number of bees increasing at an increasing rate, increasing at a decreasing rate, or neither? Based on your answer, what can you conclude about the graph ofg(t)att = 8? Justify your answer.
Solution:
g00(t)= 17e17(t−1)−4 cos(2t −16) g00(8)= 1
7e−4 < 0
g0(8) > 0 and g00(8) < 0, therefore at t = 8, the number of bees is increasing at a decreasing rate.
g(t)is concave down att =8 sinceg00(8)< 0.
Score:
1: increasing at a decreasing rate sinceg0(8) > 0 andg00(8)< 0 1: gis concave down att =8, with reason
(e) The number of bees spotted in Amelie’s garden can also be modeled by the function B(x)=50√k+2xwherexis the daily high temperature, in degrees Farenheit, andk is a positive constant. When the number of bees spotted is 100, the daily high temperature is increasing at a rate of 2◦F per day. According to this model, how quickly is the number of bees changing with respect to time when 100 bees are spotted?
Solution:
dx dt
B(x)=100
dB
dt =
dB
dx ·
dx dt dB
dx =
50
√
k+2x
B(x)=100 whenk =4−2x dB
dx = 25 whenB(x)= 100 dB
dt = 25(2)=50 bees per day
The number of bees spotted in Amelie’s garden is increasing at a rate of 50 bees per day when 100 bees are spotted.
Score:
1: use of chain rule to find dBdt 1: dBdx = 25
CALCULUS AB Part B Time—15 minutes Number of problems—1
A GRAPHING CALCULATOR IS NOT REQUIRED FOR THESE PROBLEMS.
2. The function f is defined on the closed interval[−4,9]. The graph of f consists of a semicircle, a quarter circle, and three linear segments, as shown in the figure above. Letgbe the function defined byg(x)=3x+∫x
2 f(t)dt. (a) Findg(8)andg0(8).
Solution:
g(8)= 24+
∫ 8
2
f(t)dt
= 22.5+ π 4
g0(x)= 3+ f(x)
g0(8)= 3+ f(8)
= 3
Score: 1: g(8)
(b) Find the value of x in the closed interval [−4,9] at which g attains its maximum value. Justify your answer.
Solution:
g0(x)= 3+ f(x)
0= 3+ f(x)
f(x)= −3
x = {−1,7}
g00(x)= f0(x)
g00(−1)= f0(−1)=0
The absolute maximum ofgwill occur at either x =−4, x =−1, x =7, orx =9.
x g(x)
−4 −12+∫−4
2 f(t)dt =−12+
9π 2
−1 −3+∫2−1 f(t)dt =−3+ 94π
7 21+∫7
2 f(t)dt =21+
π
4
9 27+∫9
2 f(t)dt =27+
π
4
Of these possibilities,g(9)has the largest value, thereforeghas an absolute maximum atx = 9.
Score:
1: critical values ofgat x ={−1,7}
1: considers endpoints 1: answer with justification
(c) Find lim
x→2f
0(
x), or state that it does not exist. Justify your answer.
Solution: lim
x→2− f
0(x)=∞
lim
x→2+ f
0
(x)=∞
lim
x→2f
0
Score:
1: answer with justification
(d) Find lim
x→3
8g(2x) −2π −160 ex−3−1
.
Solution: lim
x→3
(8g(2x) −2π−160)= 0
lim
x→3
(ex−3−1)=0
Therefore, L’Hôpital’s Rule may be applied.
lim
x→3
8g(2x) −2π−160
ex−3−1
= lim
x→3
16g0(2x) ex−3
= 16g0(6) 1
=16(3+ f(6))
=32
Score:
1: Conditions for L’Hôpital’s rule verified 1: correct derivatives
1: answer
Suggested Scoring:
Raw Score: Exam Score:
14-23 5
12-13 4
9-11 3
6-8 2
