BC 1-2
Quiz #6
Name: ________________
Show all appropriate work clearly for full credit. NO CALCULATORS
Skills:
1.
Express

5
1
2
e x dx as a limit of Riemann sums. DO NOT evaluate this limit.
n
2.
The limit lim
n

4k 
 cos  n   3 n     n 
4k
4
can represent an integral. What integral?
k 1
3.
The functions f and g are given by f ( x)  x and g ( x)  6  x . Let R be the region
bounded by the x-axis and the graphs of f and g.
(a) Sketch the region R.
(b) Find the area of the region R .
BC 1-2
Quiz #6
Name: ________________
2011 AP® CALCULUS BC FREE-RESPONSE QUESTIONS
(4, 1)
(3, 3)
4. The continuous function f is defined on the interval [4,3] . The graph consists of two quarter
x
circles and one line segment as shown in the figure above. Let g ( x)  2 x   f (t ) dt .
0
a. Find g ( 3) .
b. Find g ( x) , and then evaluate g (3) .
c. Determine the x-coordinate of the point(s) at which g has an absolute maximum on the interval
[ 4,3] . Justify your answer.
d. Find all values x  (4,3) for which the graph of g has an inflection point. Give a reason for
your answer.
e. Find the average rate of change of f on the interval [4,3] . 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.
BC 1-2
Quiz #6
Name: ________________

5. Evaluate:
 sin x dx
0
8
6. Evaluate the integral

3
3
f ( x) dx given that f is even,

3
8
f ( x) dx  5 , and
 f ( x) dx  2 .
8
7. Given the graph of f  below, sketch the graphs of f and f  on the same axes assuming that all
three graphs go through the origin. Clearly mark each graph.
y  f ( x )
BC 1-2
Concepts:
x

2
  sin(t  t ) dt 

8. Evaluate: lim   x
x 0 

x3




Quiz #6
Name: ________________
BC 1-2
Quiz #6
x
Name: ________________
1
dt . You may recognize L as a function that we have seen before; in
t
1
fact L( x)  ln( x) . Using only the integral definition of L ( x ) , try to state/prove some properties
of L.
10. For x  0, let L( x)  
a.
b.
c.
d.
e.
f.
g.
L(1) 
L( x ) 
Where is L positive? Negative?
Where is L increasing? Decreasing?
Where is L concave up? Down?
L( x  y ) 
L(2n ) 
h. lim  L( x)
x 