2023
AP
®
Calculus AB
Sample Student Responses
and Scoring Commentary
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Inside:
Free-Response Question 6
Scoring Guidelines
Student Samples
Scoring Commentary
AP® Calculus AB/BC 2023 Scoring Guidelines
© 2023 College Board
Part B (AB): Graphing calculator not allowed
Question 6 9 points
General Scoring Notes
The model solution is presented using standard mathematical notation.
Answers (numeric or algebraic) need not be simplified. Answers given as a decimal approximation should be
correct to three places after the decimal point. Within each individual free-response question, at most one
point is not earned for inappropriate rounding.
Consider the curve given by the equation
3
.
62xy y
= +
Model Solution Scoring
(a)
Show that
2
2
.
2
dy y
dx
yx
=
−
( )
( )
3 2
66 362
dy dy
yx y
dx dx
dd
xy y
dx dx
⇒+ == +
Implicit
differentiation
1 point
( )
2
2
2
22
2
dy dy y
y yx
dx dx
yx
⇒= − ⇒=
−
Verification 1 point
Scoring notes:
• The first point is earned only for the correct implicit differentiation of
3
.62xy y= +
Responses
may use alternative notations for
,
dy
dx
such as
.y
′
• The second point cannot be earned without the first point.
• It is sufficient to present
to earn the second point, provided there are no
subsequent errors.
Total for part (a) 2 points
(b) Find the coordinates of a point on the curve at which the line tangent to the curve is horizontal, or
explain why no such point exists.
For the line tangent to the curve to be horizontal, it is necessary
that
20y =
(so
0y =
) and that
2
2 0.yx−≠
Sets
20y =
1 point
Substituting
0y =
into
3
62xy y= +
yields the equation
6 0 2,x ⋅=
which has no solution.
Therefore, there is no point on the curve at which the line tangent
to the curve is horizontal.
Answer with reason 1 point
AP® Calculus AB/BC 2023 Scoring Guidelines
© 2023 College Board
Scoring notes:
• The first point is earned with any of
2 0,y =
0,y =
0,
dy
dx
=
0,dy =
0,
y′=
or
2
2
0.
2
y
yx
=
−
• A response need not state that at a horizontal tangent,
2
2 0.yx−≠
Total for part (b) 2 points
(c) Find the coordinates of a point on the curve at which the line tangent to the curve is vertical, or
explain why no such point exists.
For a line tangent to this curve to be vertical, it is necessary that
20y ≠
and that
2
20yx−=
(so
2
2
y
x =
).
Sets
2
20yx−=
1 point
Substituting
2
2
y
x =
into
3
62xy y= +
yields the equation
2 33
3 2 2 2 1.
yy y y y=+ ⇒ =⇒=
⋅
Substitutes
2
2
y
x =
into
3
62xy y= +
1 point
Substituting
1y
=
in
3
62xy y= +
yields
6 2 1,x = +
or
1
.
2
x =
The tangent line to the curve is vertical at the point
( )
1
,1 .
2
Answer 1 point
Scoring notes:
• The first point can be earned by presenting
2
2y x=
or
2.y
x
=
• The second point can be earned for the substitution of
2yx=
into
3
6 2,xy y
= +
or for
substituting
3
2
6
x
y
y
=
+
into
2
2 0.
y x− =
• A response earns all three points by setting
2
2 0,y
x−=
declaring the point
( )
1
,1 ,
2
and
verifying that this point is on the curve
3
6 2.xy y= +
• A response that identifies the point
( )
1
,1
2
but does not verify that the point is on the curve, does
not earn the second or the third point.
• To earn the third point the response must present both coordinates of the point
( )
1
,1 .
2
The
coordinates need not appear as an ordered pair as long as they are labeled.
Total for part (c) 3 points
(d)
A particle is moving along the curve. At the instant when the particle is at the point
( )
1
, 2,
2
−
its
horizontal position is increasing at a rate of
2
3
dx
dt
=
unit per second. What is the value of
,
dy
dt
the
rate of change of the particle’s vertical position, at that instant?
AP® Calculus AB/BC 2023 Scoring Guidelines
© 2023 College Board
2
6 6 03
dy dy
dx
yx y
dt dt dt
+=+
Uses implicit
differentiation with
respect to
t
1 point
At the point
( )
(
)
1
, , 2,
2
xy = −
( )
( )
( )
( )
2
21
62 6 32
32
dy dy
dt dt
−+ =−
8 3 12
dy dy
dt dt
⇒− + =
8
9
dy
dt
⇒=−
unit per second
Answer 1 point
Scoring notes:
• The first point is earned by presenting one or more of the terms
6 ,y
dx
dt
6,
dy
x
dt
or
2
3 .y
dy
dt
• Units will not affect scoring in this part.
• An unsupported response of
8
9
−
earns no points.
• Alternate solution:
t
dy dy
dt d d
x
dx
= ⋅
( )
( )
( )
( )
( )
, 1 2, 2
2
22
4
3
2212
xy
dy
dx
= −
−
= = −
−−
( ) ( ) ( ) ( )
, 1 2, 2 , 1 2, 2
42 8
33 9
xy xy
dx
d
d
t
y dy
dt dx
=−=−
⋅= = ⋅=
−−
unit per second
o The first point is earned for the statement
t
dy dy
dt d dx
dx
= ⋅
or equivalent.
o A numerical expression, such as
42
33
− ⋅
or
( )
( )
( )
2
22
2
,
13
22
2
−
⋅
−−
earns both points.
Total for part (d) 2 points
Total for question 6 9 points
1 of 2
Sample 6A
2 of 2
Sample 6A
1 of 2
Sample 6B
2 of 2
Sample 6B
1 of 2
Sample 6C
2 of 2
Sample 6C
AP
®
Calculus AB 2023 Scoring Commentary
© 2023 College Board.
Visit College Board on the web: collegeboard.org.
Question 6
Note: Student samples are quoted verbatim and may contain spelling and grammatical errors.
Overview
This problem asked students to consider the curve defined by the equation
3
6 2.xy y= +
In part (a) students were asked show that
2
2
.
2
dy y
dx
yx
=
−
A correct response will implicitly differentiate the
equation
3
62xy y= +
with respect to
,x
then solve the resulting equation for
.
dy
dx
In part (b) students were asked to find the coordinates of a point on the curve at which the tangent line is horizontal,
or to explain why no such point exists. A correct response will note that a horizontal tangent line must have
0,
dy
dx
=
which requires
20y =
and, therefore,
0.y
=
But if
0,y =
using the given equation
3
62xy y= +
yields
6 0 2,x ⋅ =
which has no solution. Therefore, there is no point on this curve at which the tangent line is horizontal.
In part (c) students were asked to find the coordinates of a point on the curve at which the tangent line is vertical,
or to explain why no such point exists. A correct response will begin by noting that such a point requires
2
2
20 .
2
y
yx x− =⇒=
Substituting into the equation
3
62
xy y= +
yields
1y
=
and then
1
,
2
x =
resulting in
a vertical tangent line at the point
(
)
1
,1 .
2
In part (d) students were asked to find the value of
dy
dt
at the instant when the particle is at the point
( )
1
, 2,
2
−
given
that at that instant the particle’s horizontal position is increasing at a rate of
2
.
3
dx
dt
=
A correct response will
implicitly differentiate the equation
3
62
xy y= +
with respect to
t
and then solve the resulting equation for
dy
dt
using
1
, 2,
2
xy
= = −
and
2
.
3
dx
dt
=
Sample: 6A
Score: 9
The response earned 9 points: 2 points in part (a), 2 points in part (b), 3 points in part (c), and 2 points in part (d).
In part (a) the response earned the first point on the second line with the equation
2
66 3 ,
dy dy
yx y
dx dx
+=
the correct
implicit differentiation for the given curve. The response correctly solves for
dy
dx
on the third line and then earned
the second point on the last line with the boxed equation.
AP
®
Calculus AB 2023 Scoring Commentary
© 2023 College Board.
Visit College Board on the web: collegeboard.org.
Question 6 (continued)
In part (b) the response earned the first point on the second line with the equation
2 0.y =
The response earned the
second point with the correct answer that “there are no points with a horizontal tangent,” together with reason that
“there are no
x
-values at
0.
y =
”
In part (c) the response earned the first point on the second line with the equation
2
2 0.yx−=
The response then
earned the second point on the fourth line with the correct substitution. The response would have earned the third
point on the seventh line with the statements
1y =
and
1
;
2
x =
however, the response restates the answer correctly
as an ordered pair and earned the point with the boxed section on the last line.
In part (d) the response earned the first point on the second line with the correct implicit differentiation of the curve
with respect to
.t
The response would have earned the second point with the middle expression on the fourth line;
however, the response presents two correct simplifications of this numerical answer and earned the point with the
boxed answer.
Sample: 6B
Score: 4
The response earned 4 points: 2 points in part (a), 1 point in part (b), no points in part (c), and 1 point in part (d).
In part (a) the response earned the first point with the equation on the second line. The response correctly solves for
dy
dx
on the subsequent three lines and earned the second point on the last line with the final simplification.
In part (b) the response earned the first point on the first line with the equation
2
2
0.
2
y
yx
=
−
The response then
concludes that
0,y =
which would also have earned the first point. The response did not earn the second point as
there is no conclusion stating that no point exists.
In part (c) the response did not earn the first point as there is no evidence that the denominator of our presented
dy
dx
has been set equal to
0.
The response does not indicate any substitution, so it did not earn the second point. Finally,
as the correct point is not presented, the response did not earn the third point.
In part (d) the response states that
2
23 6 6
dy dy
dx
y yx
dy dt dt
+=+
on the third line. While this is not the correct
implicit differentiation of the given curve with respect to
,t
because at least one of the three terms involving the
rates
dx
dt
or
dy
dt
is correct, the response earned the first point on this line. The response did not earn the second point
because the answer presented is not correct.
Sample: 6C
Score: 2
The response earned 2 points: no points in part (a), 1 point in part (b), 1 point in part (c), and no points in part (d).
In part (a) the response did not earn the first point as no correct implicit differentiation of the given curve is
presented, except for the one that was given in the stem of the problem. Because the first point was not earned, the
response did not earn the second point.
AP
®
Calculus AB 2023 Scoring Commentary
© 2023 College Board.
Visit College Board on the web: collegeboard.org.
Question 6 (continued)
In part (b) the response earned the first point with the equation
2 0.y =
Note that the next equation
0y =
would
also have earned this point. As the response never concludes that no such point exists, the response did not earn the
second point.
In part (c) the response earned the first point on the first line with the equation
2
2 0.yx−=
Note that any of the
first three lines would have earned this first point. As there is no substitution presented and the correct answer is not
stated, the response earned neither the second point nor the third point.
In part (d) the response states on the first line that
2
2
2
dy y
dt
yx
=
−
and then uses this expression as the basis for
substitution with the given values for
x
and
.y
If this expression is viewed as the implicit differentiation of the
curve with respect to
,t
then the response does not present at least one correct term with a rate
dx
dt
or
.
dy
dt
If, on the
other hand, this expression is meant to be
,
dy
dx
then the solution presented never makes use of
.
dx
dt
In either case,
the response did not earn the first point. As the stated answer is incorrect, the response did not earn the second point.