BC 2-3
Name ___________________________
Show enough work that I can reproduce your results. No calculators.
Skills:
Determine which of the following integrals converge and which diverge. Justify your answer clearly
using an appropriate test or method and showing ALL appropriate work.
1)




e
2)
1
IMSA
dx
x  ln  x  
2
8z5  z
2 z 6  5z 4  1
dz
F12
Determine which of the following integrals converge and which diverge. Justify your answer clearly
using an appropriate test or method and showing ALL appropriate work.
3)

1 2z


e
0
4)
1
IMSA
z2
dz
5  cos w
w4  1
dw
F12
Show all appropriate work clearly for full credit. Calculators are allowed.
5.
The graph of  is shown at
the right. Use it to answer
each of the following.
a.
Sketch the midpoint
Riemann approximation
with N = 5 equal
subintervals that
approximates

4
0
b.
f  x  dx . (3 pts)
Write out the sum that represents the midpoint Riemann approximation with N = 5
equal subintervals for the integral

4
0
c.
Will the midpoint Riemann sum approximation, with N = 5 equal subintervals overestimate
the value of

4
0
d.
f  x  dx . DO NOT simplify this sum.
f  x  dx ? Explain your reasoning.
Will the trapezoidal approximation with N = 5 overestimate the value of

4
0
f  x  dx ?
Explain your reasoning.
IMSA
F12
6. This problem is an AP Problem (2008 #2).
t (hours)
0
1
3
4
7
8
9
L(t)
(people)
120
156
176
126
150
80
0
Concert tickets went on sale at noon (t = 0) and were sold out within 9 hours. The number of
people waiting in line to purchase tickets at time t is modeled by a twice differentiable function L
for 0  t  9 . Values of L(t) at various times t are shown in the table above.
a. Use the data in the table to estimate the rate at which the number of people waiting in line
was changing at 5:30 P.M. (t = 5.5). Show the computations that lead to your answer.
Indicate units of measure.
b. Use a trapezoidal sum with three subintervals to estimate the average number of people
waiting in line during the first 4 hours that tickets were on sale. Write the sum out clearly;
you need not calculate the final answer.
c. For 0  t  9 , what is the fewest number of times at which L(t ) must equal 0? Give a
reason for your answer.
d. The rate at which tickets were sold for 0  t  9 is modeled by r (t )  550  t  et / 2 tickets
per hour. Based on this model, set up an integral that gives how many tickets were sold by
3:00 P.M. (t = 3)?
IMSA
F12
Concepts:
7. Determine whether the following integral converges or diverges. Justify your answer clearly using
an appropriate test or method and showing ALL appropriate work.

1
0
dx
1 x
8. Evaluate:
3

1
0
IMSA
dx
x3x
F12
8. For which values of p will the integral

1
0
IMSA
ln x
xp
dx converge? Explain carefully.
F12