CALCULUS BC
TEST ON 4.6, 5.1 – 5.2, & FUNCTIONS
DEFINED BY INTEGRALS
NAME_____________________________
PERIOD____DATE__________________
1. Suppose that g  is an even continuous function and that g  2   4 and 2 g '  x  dx  14
5
2
Find g  5 .
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2. A bowl of soup is placed on the stovetop. The temperature of the soup at
selected times is given in the table below.
x (min.)
T  x  (° F)
0
94
4
112
(a) Use data from the table to find
7
122
12
154
 0 T   x  dx .
7
(b) What is the average temperature of the soup in the 12 minute interval. Use a right side
Riemann sum to calculate.
(c) Estimate the value of T   5 Explain what this value represents and include units.
(d) (Extra Credit) Estimate the value of (T 1 ) '(100)
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Multiple Choice. Write your answer in the blank.
_____ 3. Let g be the function given by g  x    100  t 2  5t  4  e  t dt . Which of the
x
2
1
following statements about g must be true?
I. g is decreasing on (1,4).
II. g is increasing on (,1) U (4, ) .
III. g 1  0  1
(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III
dy
.
dx
4. x  tan  xy   y 3  5
Find
 cos x 
5. y  ln 

 x2 
6.
 [tan  x   sec( x) ]dx
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Evaluate.
4 4x  4
dx
7. 
0 x2  2x
8.
e3
1
 ln x  dx
x
Multiple Choice. Show your work, and write your answer in the blank.
_____ 9. If F  x   
(A) 1
(B)
1
2
 
t 2 dt , find the value of F    .
4
3
1
(C)
(D)
(E) 1
2
2
sin x
0
(F)
3
4
ln( x  3)
. Find the x-coordinate of the critical point(s)
x3
of f, and determine whether this point is a relative minimum, a relative maximum, or neither for
the function f. Justify your answer.
10. Let f be the function given by f  x  
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11. The graph of the function f shown in the figure consists
of two line segments. Let g be the function
x
given by g  x    f  t  dt . Justify in a sentence.
3
(a) Find g  4 , g   4 , and g   4 .
f (x)
(b) For what value(s) of x in the open interval  2, 5
does the graph of g have a point of inflection? Justify.
(c) For what value(s) of x in the open interval
Justify.
 2, 5 does g
have a relative min?
(d) Find the absolute maximum for the function g. Justify
(e) Find
d g  x
(
) at x = 1
dx
x