AP Calculus AB 2018 International Practice Exam Scoring Guidelines

AP Calculus AB 2018 International Practice Exam Scoring Guidelines

The AP Calculus AB 2018 International Practice Exam Scoring Guidelines provide detailed evaluation criteria for free-response questions. These guidelines assist educators in assessing student performance on the exam, focusing on key calculus concepts such as limits, derivatives, integrals, and the Fundamental Theorem of Calculus. Designed for AP Calculus students and teachers, the document outlines scoring rubrics and sample responses to help clarify expectations. It serves as a valuable resource for preparing students for the AP exam and improving their problem-solving skills in calculus.

Key Points

  • Includes scoring rubrics for free-response questions in AP Calculus AB.
  • Covers key calculus concepts such as limits, derivatives, and integrals.
  • Provides sample responses to illustrate scoring expectations.
  • Designed for educators to assess student performance effectively.
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© 2018 The College Board.
Visit the College Board on the Web: www.collegeboard.org.
(a)
( )
( ) ( )
51
20.5 15.1
3 1.35
51 4
gg
g
−
−
≈
′
= =
−
At time
3t
=
minutes, the rate at which grain is being added to the silo is
increasing at a rate of 1.35 cubic feet per minute per minute.
1 : approximation
2 :
1 : interpretation with units
(b)
The total amount of grain added to the silo from time
t 0=
to time
8t =
is
( )
8
0
g t dt
∫
cubic feet.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
8
0
110 5 51 6 65 8 86
15.1 1 20.5 4 18.3 1 22.7 2 160.8
g t dt g g g g≈ ⋅− + ⋅−+ ⋅ − + ⋅−
= ⋅+ ⋅ + ⋅+ ⋅ =
∫
1 : integral expression
3 :
1 : right Riemann sum
1 : approximation
(c)
( )
8
0
99.051497w t dt =
∫
The approximate
amount of unspoiled grain remaining in the silo at
time
8t =
is
( )
8
0
160.8 61.749w t dt−=
∫
(or 61.748) cubic feet.
1 : integral
2 :
1 : answer
(d)
( ) ( )
6 6 18.3 16.063173 2.236827 0gw−=− = >
Because
( ) ( )
6 6 0gw− ,
the amount of unspoiled grain is increasing at
>
time
6t = .
( ) ( )
1 : considers 6 6
2 :
1 : answer
gw−
AP
®
CALCULUS AB
2018 SCORING GUIDELINES
Question 1
AP
®
CALCULUS AB
2018 SCORING GUIDELINES
© 2018 The College Board.
Visit the College Board on the Web: www.collegeboard.org.
Question 2
(a)
( )
5 0.538462v
′
=
The accelerati
on of the snail at time
t = 5
minutes is 0.538 inches per
minute per minute.
1 : answer
(b)
( )
15
0
76.043074v t dt =
∫
The d
isplacement of the snail over the
interval
0 15t≤≤
minutes is
76.043 inches.
{
1 : integral
2 :
1 : answer
(c)
( )
15
0
1
5.069538
15
v t dt =
∫
( )
2
1.4ln 1 5.069538 6.031tt+ = ⇒=
minutes
{
1 : average value expression
2 :
1 : answer
(d)
The velocity of the ant at time t,
12 15t≤≤ ,
is
2dt t c= +
∫
2
inches
per minute for some constant c.
For
12 15t≤≤ ,
the displacement of the ant is
( )
( )
15
15
2
12
12
2 81 3
t
t
t c dt t ct c
=
=
+=+ =+
∫
inches.
Thus,
81 3 76.043074 1.652309c+ = ⇒=−c.
The velocity
of the ant at time
12t =
is
B =2⋅−12 1.652309 82 =24.3
(or 22.347) inches per minute.
— OR —
The velocity of the ant at time t,
12 15t≤≤ ,
is
( )
2 12t− B+
inches per
minute.
For
12 15t≤≤ ,
the displacement of the ant is
( )
( )
( )
( )
2
15
2
12
15
1
2 2 91
t
t
t B dt Bt Bt
=
=
− −+ =+=
∫
1 2 3+
inches.
9 3 76.043074 22.348BB+ = ⇒=
(or 22.347) inches per minute
1 : ant’s velocity
1 : ant’s displacement
4 :
1 : equation
1 : answer
© 2018 The College Board.
Visit the College Board on the Web: www.collegeboard.org.
AP
®
CALCULUS AB
2018 SCORING GUIDELINES
Question 3
(a)
( ) ( )
7
0
9
7 3 7 21 3 24
2
f g t dt
π
=+= =⋅ −+ −
∫
9
2
π
( ) ( )
73 733fg
′
=+ =+=6
( )
( )
1 : 7
2 :
1 : 7
f
f
′
(b)
On the interval
4 3,x−≤ ≤
( ) ( )
3f x gx
′
= + .
Because
( )
0fx
′
≥
for
4 3x−≤ ≤,
f is nondecreasing over
the entire interval, and the maximum must occur when
3x = .
2 : answer with justification
(c)
( )
0
1
lim
2
x
gx
−
→
′
= −
( )
0
lim
x
gx
+
→
′
does not exist.
{
1 : left-hand limit
2 :
1 : right-hand limit
(d)
( )
( )
( )
2
0
2
lim 7 6 7 0
x
f x g t dt
−
→−
+ =−+ + =
∫
( )
36
2
lim 1 0
x
x
e
+
→−
−=
Using L’Hospital’s Rule,
( ) ( ) ( )
36 36
22
7 32
31
lim lim
3 3
13
xx
xx
fx f x g
ee
++
→− →−
′
+ +−
+
= = = =
4
3
−
.
1 : limits equal 0
3 :
1 : applies L’Hospital’s Rule
1 : answer
Note: max
13
[1-0-0] if no limit notation
attached to a ratio of derivatives
/ 7
End of Document
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FAQs of AP Calculus AB 2018 International Practice Exam Scoring Guidelines

What are the main topics covered in the AP Calculus AB exam?
The AP Calculus AB exam covers a range of topics including limits, derivatives, integrals, and the Fundamental Theorem of Calculus. Students are expected to understand the concepts of continuity and differentiability, as well as how to apply these concepts to solve real-world problems. The exam also emphasizes the interpretation of graphical representations of functions and their derivatives, which is crucial for understanding calculus applications.
How are free-response questions scored in the AP Calculus AB exam?
Free-response questions on the AP Calculus AB exam are scored based on a rubric that evaluates the correctness of the answer, the clarity of the explanation, and the use of appropriate calculus techniques. Each question is broken down into specific scoring components, allowing for partial credit to be awarded for correct reasoning or steps, even if the final answer is incorrect. This scoring approach encourages students to show their work and reasoning.
What is the significance of the Fundamental Theorem of Calculus in AP Calculus AB?
The Fundamental Theorem of Calculus establishes the relationship between differentiation and integration, two central concepts in calculus. It states that if a function is continuous on an interval, then the integral of its derivative over that interval equals the change in the function's values. Understanding this theorem is crucial for solving problems related to area under curves and accumulation functions, which are commonly tested in the AP Calculus AB exam.
What types of problems can be found in the AP Calculus AB free-response section?
The free-response section of the AP Calculus AB exam includes a variety of problem types, such as finding derivatives, evaluating integrals, and applying calculus concepts to real-world scenarios. Problems may require students to analyze functions graphically, compute limits, or solve differential equations. This section tests not only mathematical skills but also the ability to communicate reasoning clearly and effectively.
How can educators use the scoring guidelines to improve student performance?
Educators can utilize the scoring guidelines to provide targeted feedback to students on their free-response answers. By understanding the specific criteria used to evaluate responses, teachers can identify areas where students struggle and offer additional support or resources. Furthermore, discussing sample responses helps students learn how to articulate their reasoning and approach problems more effectively, ultimately enhancing their performance on the AP exam.

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