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 criteria for evaluating free-response questions. These guidelines assist educators in assessing student performance on the exam, focusing on key calculus concepts such as derivatives, integrals, and the Fundamental Theorem of Calculus. Designed for AP Calculus instructors, the guidelines outline scoring rubrics and sample student responses to enhance grading consistency. This resource is essential for teachers preparing students for the AP exam and ensuring they understand the expectations for free-response answers.

Key Points

  • Includes scoring criteria for AP Calculus AB free-response questions.
  • Features sample student responses to illustrate grading standards.
  • Covers essential calculus concepts like derivatives and integrals.
  • Aids educators in preparing students for the AP Calculus exam.
, ,
  • MR ALi
200
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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
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End of Document
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Faqs of AP Calculus AB 2018 International Practice Exam Scoring Guidelines
What are the key components of the AP Calculus AB scoring guidelines?
The AP Calculus AB scoring guidelines detail the evaluation criteria for free-response questions, emphasizing clarity, correctness, and mathematical reasoning. Each question is broken down into specific components that educators must consider when grading, such as the accuracy of solutions and the justification of answers. The guidelines also provide a rubric that assigns points based on the completeness and correctness of student responses, ensuring a standardized grading process.
How do the scoring guidelines assist teachers in grading?
The scoring guidelines serve as a comprehensive framework that helps teachers evaluate student responses consistently. By outlining specific expectations for each free-response question, educators can more easily identify strengths and weaknesses in student work. The inclusion of sample responses further aids in illustrating how points are awarded, allowing teachers to calibrate their grading and provide constructive feedback to students.
What topics are 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 demonstrate their understanding of these concepts through both multiple-choice and free-response questions. The exam assesses not only computational skills but also the ability to apply calculus concepts to real-world problems, making a solid grasp of these topics essential for success.
What is the importance of understanding the scoring guidelines for students?
Understanding the scoring guidelines is crucial for students as it helps them recognize what is expected in their responses. By familiarizing themselves with the criteria used to evaluate their work, students can tailor their study and practice efforts to meet these expectations. This knowledge can lead to improved performance on the exam, as students learn to articulate their reasoning and present their solutions clearly.
How can educators use the scoring guidelines to improve student outcomes?
Educators can utilize the scoring guidelines to design targeted instruction that addresses common areas of difficulty identified in previous exams. By aligning their teaching strategies with the scoring criteria, teachers can focus on developing students' problem-solving skills and conceptual understanding. Additionally, using the guidelines to provide feedback on practice exams can help students refine their approaches and enhance their readiness for the actual AP exam.