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E. Implicit Differentiation
dy
dy
or
at some point ( x, y ) . You perform implicit
dx
dx
differentiation when the function is not given explicitly in the form “y =” or when it is difficult or impossible
to put it in that form.
!
! term!in the given expression,
How to find it: Take the derivative of every
remembering that the chain rule is
dy
dy
used and the derivative of y is not 1 but
. If you are asked to find
at a given point, it is best to plug
dx
dx
dy
dy
the point in, once you have taken the derivative and solve for
. If the
cancels out of the equation, the
dx
dx
tangent line is vertical at that point. Implicit examples show up in related rates problems when you are doing
!
!
implicit differentiation with respect
for time t.
dy
27. If sin("x ) cos("y ) = y, find
at (1,0)
!
!
dx
What you are finding: You are asked to either find
A. 1+ "
B. 1" #
C. -1
D. -π
E. π
!
!
!
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28. Find the equation of the tangent line to x 3 + y 3 = 3xy + 5x " 3y at (2,2) .
A. x + 9y = 20
"1
D. y =
9
B. 13x " 3y = 20
E. y = 2
!
!
C. 9y " x = 16
!
!
!
!
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29. If x 2 + 2y 2 = 34 , find the behavior of the curve at (-4, 3).
!
A. Increasing, concave up
D. Decreasing, concave down
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B. Increasing, concave down
E. Decreasing, inflection point
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C. Decreasing, concave up
Demystifying the MC AB Calculus Exam
30. Find the y-intercept of the tangent line to 4 x + 2 y = x + y + 3 at the point (9, 4).
A. -2
B. 10
C.
!
5
2
D. 15
E.
!
4
9
!
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d2y
31. If x 2 + y 2 = a 2 , where a is a constant, find
.
dx 2
A.
!
"y 2 " x 2
y2
!
B.
x2 " y2
y3
C.
"1
y2
!
!
D.
"a 2
y3
E.
1
a2
!
!
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32. Let x and y be functions of time t that are related by the equation y 2 = xy + e . At time t = 1, the value
dy
dx
of y is e and
when t = 1.
= 2 . Find
dt
dt
A.
!
2e " 2
e
!
B.
!
2e + 2
e
!
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C.
!
2
e
!
!
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D.
e+2
e
E.
e"2
e
!
Demystifying the MC AB Calculus Exam