AP Calculus BC Scoring Guidelines from the 2019 Exam Administration AP ® Calculus BC Scoring Guidelines 2019 © 2019 The College Board College Board, Advanced Placement, AP, AP Central, and the acorn l[.]
Trang 1Calculus BC
Scoring Guidelines
Trang 2AP® CALCULUS AB/CALCULUS BC
2019 SCORING GUIDELINES
© 2019 The College Board
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Question 1
(a) 5
0E t dt 153.457690
To the nearest whole number, 153 fish enter the lake from midnight
to 5 A.M
1 : integral
2 :
1 : answer
0
5 0 L t dt
The average number of fish that leave the lake per hour from
midnight to 5 A.M is 6.059 fish per hour
1 : integral
2 :
1 : answer
(c) The rate of change in the number of fish in the lake at time t is
given by E t L t
E t L t t
E t L t for 0 t 6.20356, and E t L t for 0
6.20356 Therefore the greatest number of fish in the lake is t 8
at time t 6.204 (or 6.203)
— OR —
Let A t be the change in the number of fish in the lake from
midnight to t hours after midnight
0
t
A t E s L s ds
A t E t L t t C
Therefore the greatest number of fish in the lake is at time
6.204 (or 6.203)
3 : 1 : answer
1 : justification
E t L t
(d) E 5 L 5 10.7228 0
Because E 5 L 5 0, the rate of change in the number of fish
is decreasing at time t 5
1 : considers 5 and 5
2 :
1 : answer with explanation
Trang 3(a) 2
0
The area of S is 3.534
1 : integral
2 :
1 : answer
0
The average distance from the origin to a point on the curve
r r for 0 is 1.580 (or 1.579)
1 : integral
2 :
1 : answer
x
1
m
1 : equates polar areas
1 : inverse trigonometric function
3 : applied to as limit of
integration
1 : equation
m
(d) As k , the circle r kcos grows to enclose all points to
the right of the y -axis
0
0
1 lim
2
2
d
1 : limits of integration
2 :
1 : answer with integral
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2019 SCORING GUIDELINES
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Question 3
2 6 2
6
9
4
f x dx
f x dx
5 2
1 :
3 :
1 :
1 : answer
f x dx
— OR —
3
0 20
x x
1 : Fundamental Theorem of Calculus
2 :
1 : answer
2
g x f x x x x
x g x
2
1
2 1
2
1 4
4
On the interval 2 x 5, the absolute maximum value
of g is 5 11 9
4
1 :
3 : 1 : identifies 1 as a candidate
1 : answer with justification
x
1
1
lim
1 arctan 1
2 1 4
x
x
1 : answer
Trang 5(a) V r h2 12h h
1 :
2 :
1 : answer with units
dt dt
2
2
dt
200
d h
dt for h 0, the rate of change of the
height is increasing when the height of the water is 3 feet
2 2
1 :
20
1 : answer with explanation
dt h
(c)
1
10
1 10 1
2
10
1
2 5
10
1
10
20
h
h
C
1 : separation of variables
1 : antiderivatives
1 : constant of integration
4 :
and uses initial condition
1 : h t
Note: 0 4 if no separation of variables Note: max 2 4 [1-1-0-0] if no constant
of integration
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2019 SCORING GUIDELINES
© 2019 The College Board
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Question 5
2 2
2
x
f x
2
k
1 : denominator of
3 : 1 :
1 : answer
f x
f x
2
1 1
0
0
1
0
1ln 3 1ln 3 1ln 4 1ln 2 1ln 2
x x
1 : partial fraction decomposition
3 :
1 : antiderivatives
1 : answer
(c)
2
2
2
0
b
Because lim1 1
1
b b does not exist, the integral diverges
1 : improper integral
3 : 1 : antiderivative
1 : answer with reason
Trang 7(a) f 0 and 3 f 0 2
The third-degree Taylor polynomial for f about x is 0
23
1 : two terms
2 :
1 : remaining terms
(b) The first three nonzero terms of the Maclaurin series for e x are
2
1
2!
The second-degree Taylor polynomial for e f x about x x is 0
2
2
x x
1 : three terms for
2 :
1 : three terms for
x x
e
e f x
1
0
1
0
1
0
1
3 2
3
3 1
t t
2 :
1 : answer
(d) The alternating series error bound is the absolute value of the first
omitted term of the series for h 1
1
t t
9
20
1 : uses fourth-degree term
of Maclaurin series for
1 : uses first omitted term
3 :
of series for 1
1 : error bound
f h