Functions Defined By Integrals WS
AP Calculus Name ____________________________ Period _____
Functions Defined by Integrals
The graph below is, the derivative of. The graph consists of two semicircles and one line segment. Horizontal tangents are located at and and a vertical tangent is located at.
1. On what interval is increasing? Justify your answer.
2. For what values of x does have a relative minimum? Justify.
3. On what intervals is concave up? Justify
4. For what values of x is undefined?
5. Identify the x-coordinates for all point of inflection of . Justify.
6. For what value of x does reach its maximum value on the closed interval [0, 17]? Justify
7. For what value of x does reach its minimum value on the closed interval [0, 17]? Justify
8. If , find .
9. Let . The graph of f is comprised of line segments and a quarter of a circle on [-2, 6].
a) Find and
b) Determine the open intervals where is increasing. Justify your answer.
c) Determine the intervals where is concave down. Justify your answer.
d) Sketch .
e) Determine the absolute maximum and minimum of on [-2, 6].
10. Let , where the graph of f , defined on [-5, 5], is comprised of line segments.
a) Determine the domain of .
b) Determine the range of .
c) Determine the relative extrema. Justify your answer.
11. The graph of consists of a semicircle and two line segments as shown.
a) Determine the values of x on the open interval (-2, 7) at which has a relative maximum. Justify your answer.
b) Determine the values of x on the open interval (-2, 7) at which has point of inflection. Justify your answer.
c) If , then determine .
12. The graph of the function f, consisting of three line segments, is given. Let .
(a) Compute and .
(b) Find the instantaneous rate of change of g, with respect to x, at .
(c) Find the absolute minimum value of g on the closed interval . Justify your answer.
(d) The second derivative of g is
Functions Defined by Integrals
The graph below is, the derivative of. The graph consists of two semicircles and one line segment. Horizontal tangents are located at and and a vertical tangent is located at.
1. On what interval is increasing? Justify your answer.
2. For what values of x does have a relative minimum? Justify.
3. On what intervals is concave up? Justify
4. For what values of x is undefined?
5. Identify the x-coordinates for all point of inflection of . Justify.
6. For what value of x does reach its maximum value on the closed interval [0, 17]? Justify
7. For what value of x does reach its minimum value on the closed interval [0, 17]? Justify
8. If , find .
9. Let . The graph of f is comprised of line segments and a quarter of a circle on [-2, 6].
a) Find and
b) Determine the open intervals where is increasing. Justify your answer.
c) Determine the intervals where is concave down. Justify your answer.
d) Sketch .
e) Determine the absolute maximum and minimum of on [-2, 6].
10. Let , where the graph of f , defined on [-5, 5], is comprised of line segments.
a) Determine the domain of .
b) Determine the range of .
c) Determine the relative extrema. Justify your answer.
11. The graph of consists of a semicircle and two line segments as shown.
a) Determine the values of x on the open interval (-2, 7) at which has a relative maximum. Justify your answer.
b) Determine the values of x on the open interval (-2, 7) at which has point of inflection. Justify your answer.
c) If , then determine .
12. The graph of the function f, consisting of three line segments, is given. Let .
(a) Compute and .
(b) Find the instantaneous rate of change of g, with respect to x, at .
(c) Find the absolute minimum value of g on the closed interval . Justify your answer.
(d) The second derivative of g is