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2012 AP® CALCULUS AB FREE-RESPONSE QUESTIONS
CALCULUS AB
SECTION II, Part B
Time— 60 minutes
Number of problems—4
No calculator is allowed for these problems.
3. Let f be the continuous function defined on
4, 3 whose graph, consisting of three line segments and a
semicircle centered at the origin, is given above. Let g be the function given by g x
x
1
f t dt.
(a) Find the values of g 2 and g 2 .
(b) For each of g
3 and g
3 , find the value or state that it does not exist.
(c) Find the x-coordinate of each point at which the graph of g has a horizontal tangent line. For each of these
points, determine whether g has a relative minimum, relative maximum, or neither a minimum nor a
maximum at the point. Justify your answers.
(d) For 4 x
reasoning.
3, find all values of x for which the graph of g has a point of inflection. Explain your
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2013 AP® CALCULUS AB FREE-RESPONSE QUESTIONS
4. The figure above shows the graph of f „, the derivative of a twice-differentiable function f, on the closed
interval 0 … x … 8. The graph of f „ has horizontal tangent lines at x 1, x 3, and x 5. The areas of the
regions between the graph of f „ and the x-axis are labeled in the figure. The function f is defined for all real
numbers and satisfies f 8 4.
(a) Find all values of x on the open interval 0 x 8 for which the function f has a local minimum. Justify
your answer.
(b) Determine the absolute minimum value of f on the closed interval 0 … x … 8. Justify your answer.
(c) On what open intervals contained in 0 x 8 is the graph of f both concave down and increasing?
Explain your reasoning.
5
3
(d) The function g is defined by g x f x . If f 3 , find the slope of the line tangent to the graph
2
of g at x 3.
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2011 AP® CALCULUS AB FREE-RESPONSE QUESTIONS
4. The continuous function f is defined on the interval
4
x
and one line segment, as shown in the figure above. Let g x
(a) Find g 3 . Find g x and evaluate g
3. The graph of f consists of two quarter circles
2x
x
0
f t dt.
3.
(b) Determine the x-coordinate of the point at which g has an absolute maximum on the interval
Justify your answer.
(c) Find all values of x on the interval
reason for your answer.
4
x
4
x
3.
3 for which the graph of g has a point of inflection. Give a
(d) Find the average rate of change of f on the interval 4 x 3. There is no point c, 4 c 3, for
which f c is equal to that average rate of change. Explain why this statement does not contradict the
Mean Value Theorem.
WRITE ALL WORK IN THE EXAM BOOKLET.
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2011 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B)
CALCULUS AB
SECTION II, Part B
Time—60 minutes
Number of problems— 4
No calculator is allowed for these problems.
x and g x
3. The functions f and g are given by f x
and the graphs of f and g, as shown in the figure above.
6
x. Let R be the region bounded by the x-axis
(a) Find the area of R.
(b) The region R is the base of a solid. For each y, where 0 y 2, the cross section of the solid taken
perpendicular to the y-axis is a rectangle whose base lies in R and whose height is 2y. Write, but do not
evaluate, an integral expression that gives the volume of the solid.
(c) There is a point P on the graph of f at which the line tangent to the graph of f is perpendicular to the graph
of g. Find the coordinates of point P.
4. Consider a differentiable function f having domain all positive real numbers, and for which it is known that
f x
4 x x 3 for x 0.
(a) Find the x-coordinate of the critical point of f. Determine whether the point is a relative maximum, a relative
minimum, or neither for the function f. Justify your answer.
(b) Find all intervals on which the graph of f is concave down. Justify your answer.
(c) Given that f 1
2, determine the function f.
WRITE ALL WORK IN THE EXAM BOOKLET.
© 2011 The College Board.
Visit the College Board on the Web: www.collegeboard.org.
GO ON TO THE NEXT PAGE.
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