AP Calculus AB
Review of Tangent Line Problems
Multiple Choice
= class problem
= homework problem
1969 #20
An equation for a tangent to the graph of y  arcsin 2x at the origin is
a) x – 2y = 0 b) x – y = 0
c) x = 0
d) y = 0
e) πx – 2y = 0
1969 #36
The approximate value of y  4  sin x at x = 0.12, obtained from the tangent to the graph at
x = 0, is
a) 2.00
b) 2.03
c) 2.06
d) 2.12
e) 2.24
1973 #3
The slope of the line tangent to the graph of 𝑦 = ln(𝑥 2 ) at x = e2 is
a)
1
e2
b)
2
e2
c)
4
e2
d)
1
e4
e)
4
e4
1973 #11
If the line 3x – 4y = 0 is tangent in the first quadrant to the curve y = x3 + k, then k is
a)
1
2
b)
1
4
c) 0
d) 
1
8
e) 
1
2
1985 #8
𝑥
The slope of the line tangent to the graph of 𝑦 = 𝑙𝑛 (2) at x = 4 is
a)
1
8
b)
1
4
c)
1
2
d) 1
e) 4
1985 #43
An equation of the line tangent to y  x 3  3x 2  2 at its point of inflection is
a) y = -6x – 6
b) y = -3x + 1
c) y = 2x + 10
d) y = 3x – 1
e) y = 4x + 1
1988 #11
An equation of the line tangent to the graph of f ( x )  x (1  2 x ) 3 at the point (1, -1) is
a) y = -7x + 6
b) y = -6x + 5
1993 #7 (calculator question)
An equation of the line tangent to the graph of y 
a) 13x – y = 8
d) y = 2x – 3
c) y = -2x + 1
2x  3
at the point (1, 5) is
3x  2
c) x – 13y = 64
b) 13x + y = 18
d) x + 13y = 66
1997 #10
An equation of the line tangent to the graph y  cos(2 x ) at x 
𝜋
a) 𝑦 − 1 = − (𝑥 − 4 )
e) y = 7x – 8

is
4
𝜋
𝜋
b) 𝑦 − 1 = −2 (𝑥 − 4 )
𝜋
d) 𝑦 = − (𝑥 − 4 )
e) -2x + 3y = 13
c) 𝑦 = 2 (𝑥 − 4 )
𝜋
e) 𝑦 = −2 (𝑥 − 4 )
1997 #12
At what point on the graph of y  21 x 2 is the tangent line parallel to 2x – 4y = 3?
1
1
a) (2 , − 2)
1 1
b) (2 , 8)
1
c) (1, − 4)
1
d) (1, 2)
e) (2, 2)
1997 #14
Let f be a differentiable function such that f (3) = 2 and f '(3) = 5. If the tangent line to the graph
of f at x = 3 is used to find an approximation to a zero of f, that approximation is
a) 0.4
b) 0.5
c) 2.6
d) 3.4
e) 5.5
1997 #80 (calculator question)
2
Let f be the function given by f ( x )  2e4 x . For what value of x is the slope of the line tangent
to the graph of f at (x, f (x)) equal to 3?
a) 0.168
b) 0.276
c) 0.318
d) 0.342
e) 0.551
1998 #18
An equation of the line tangent to the graph of y  x  cos x at the point (0, 1) is
a) y = 2x + 1
b) y = x + 1
c) y = x
d) y = x – 1
e) y = 0
1998 #77 (calculator question)
Let f be the function given by f ( x )  3e2 x and let g be the function given by g( x )  6x 3 . At
what value of x do the graphs of f and g have parallel tangent lines?
a) -0.701
b) -0.567
c) -0.391
d) -0.302
e) -0.258
1998 #87 (calculator question)
Which of the following is an equation of the line tangent to the graph of f ( x )  x 4  2 x 2 at the
point where f '(x) = 1?
a) y = 8x – 5
b) y = x + 7
d) y = x – 0.122
c) y = x + 0.763
e) y = x – 2.146
2003 #24
Let f be the function defined by f (x) = 4x3 – 5x + 3. Which of the following is an equation of the
line tangent to the graph of f at the point where x = -1?
a) y = 7x – 3
b) y = 7x + 7
d) y = -5x – 1
c) y = 7x + 11
e) y = -5x – 5
2003 #26
What is the slope of the line tangent to the curve 3y2 – 2x2 = 6 – 2xy at the point (3, 2)?
a) 0
b)
4
9
c)
7
9
d)
6
7
e)
5
3
2003 #89 (calculator question)
Let f be a differentiable function with f (2) = 3 and f '(2) = -5, and let g be the function defined by
g(x) = xf (x). Which of the following is an equation of the line tangent to the graph of g at the
point where x = 2?
a) y = 3x
b) y – 3 = -5(x – 2)
d) y – 6 = -7(x – 2)
c) y – 6 = -5(x – 2)
e) y – 6 = -10(x – 2)
AP Calculus AB
Review of Tangent Line Problems
Free Response
1989 #1
Let f be the function given by f (x) = x3 – 7x + 6.
a) Find the zeros of f.
b) Write an equation of the line tangent to the graph of f at x = -1.
c) Find the number c that satisfies the conclusion of the Mean Value Theorem for f on the
closed interval [1, 3].
1991 #1
Let f be the function that is defined for all real numbers x and that has the following properties.
i)
f ''(x) = 24x – 18
ii)
f '(1) = -6
iii)
f (2) = 0
a) Find each x such that the line tangent to the graph of f at (x, f (x)) is horizontal.
b) Write an expression for f (x).
c) Find the average value of f on the interval 1 ≤ x ≤ 3.
1991 #3
3
Let f be the function defined by f ( x)  (1  tan x) 2 for  4  x  2 .
a) Write an equation for the line tangent to the graph of f at the point where x = 0.
b) Using the equation found in part a, approximate f (0.02).
c) Let f -1 denote the inverse function of f. Write an expression that gives f -1(x) for all x in
the domain of f -1.
1992 #4
Consider the curve defined by the equation y + cosy = x + 1 for 0 ≤ y ≤ 2π.
dy
a) Find
in terms of y.
dx
b) Write an equation for each vertical tangent to the curve.
d2y
c) Find
in terms of y.
dx 2
1994 #3
Consider the curve defined by x2 + xy + y2 = 27.
a) Write an expression for the slope of the curve at any point (x, y).
b) Determine whether the lines tangent to the curve at the x-intercepts of the curve are
parallel. Show the analysis that leads to your conclusion.
c) Find the points on the curve where the lines tangent to the curve are vertical.
1995 #3 (calculator question)
Consider the curve defined by -8x2 + 5xy + y3 = -149.
dy
a) Find
.
dx
b) Write an equation for the line tangent to the curve at the point (4, -1).
c) There is a number k such that the point (4.2, k) is on the curve. Using the tangent line
found in part b, approximate the value of k.
d) Write an equation that can be solved to find the actual value of k so that the point (4.2, k)
is on the curve.
e) Solve the equation found in part d for the value of k.
l
1996 #6 (calculator question)
Line l is tangent to the graph of
x2
y x
at the point Q, as shown in
500
the figure to the right.
x2
y x
500
Q
(0, 20)
500
P O
a) Find the x-coordinate of the point Q.
b) Write an equation for the line l.
x2
c) Suppose the graph of y  x 
shown in the figure, where x and y are measured in
500
feet, represents a hill. There is a 50-foot tree growing vertically at the top of the hill.
Does a spotlight at point P directed along line l shine on any part of the tree? Show the
work that leads to your conclusion.
1997 #2 (calculator question)
P (0, 3)
Let f be the function given by f ( x )  3 cos x .
As shown to the right, the graph of f crosses the
y-axis at point P and the x-axis at point Q.
y  f ( x)
O
𝜋
Q ( 2 , 0)
a) Write an equation for the line passing through the points P and Q.
b) Write an equation for the line tangent to the graph of f at point Q. Show the analysis that
leads to your conclusion.
c) Find the x-coordinate of the point on the graph of f, between points P and Q, at which the
line tangent to the graph of f is parallel to line PQ.
d) Let R be the region in the first quadrant bounded by the graph of f and line segment PQ.
Write an integral expression for the volume of the solid generated by revolving the region
R about the x-axis. Do not evaluate.
2001 #4
Let h be a function defined for all x ≠ 0 such that h(4) = -3 and the derivative of h is given by
x2  2
for all x ≠ 0.
h' ( x ) 
x
a) Find all values of x for which the graph of h has a horizontal tangent, and determine
whether h has a local maximum, a local minimum, or neither at each of these values.
Justify your answers.
b) On what intervals, if any, is the graph of h concave up? Justify your answer.
c) Write an equation for the line tangent to the graph of h at x = 4.
d) Does the line tangent to the graph of h at x = 4 lie above or below the graph of h for
x > 4? Why?
2002 #6
-1.5
-1
-7
x
f (x)
f '(x)
-1.0
-4
-5
-0.5
-6
-3
0
-7
0
0.5
-6
3
1.0
-4
5
1.5
-1
7
Let f be a function that is differentiable for all real numbers. The table above gives the values of
f and its derivative f ' for selected points x in the closed interval -1.5 ≤ x ≤ 1.5. The second
derivative of f has the property that f ''(x) > 0 for -1.5 ≤ x ≤ 1.5.
1.5
a) Evaluate ∫0 (3𝑓 ′ (𝑥) + 4)𝑑𝑥. Show the work that leads to your answer.
b) Write an equation of the line tangent to the graph of f at the point where x = 1. Use this
line to approximate the value of f (1.2). Is this approximation greater than or less than the
actual value of f (1.2)? Give a reason for your answer.
c) Find a positive real number r having the property that there must exist a value c with
0 < c < 0.5 and f ''(c) = r. Give a reason for your answer.
2
d) Let g be the function given by 𝑔(𝑥) = {2𝑥 2 − 𝑥 − 7 for 𝑥 < 0.
2𝑥 + 𝑥 − 7 for 𝑥 ≥ 0
The graph of g passes through each of the points (x, f (x)) given in the table above. Is it
possible that f and g are the same function? Give a reason for your answer.
2006 #6
The twice-differentiable function f is defined for all real numbers and satisfies the following
conditions: f (0) = 2, f '(0) = -4, and f ''(0) = 3.
a) The function g is given by g( x )  e ax  f ( x ) for all real numbers, where a is a constant.
Find g'(0) and g''(0) in terms of a. Show the work that leads to your answers.
b) The function h is given by h( x )  cos( kx ) f ( x ) for all real numbers, where k is a constant.
find h'(x) and write an equation for the line tangent to the graph of h at x = 0.