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CALCULUS BC
WORKSHEET ON INTEGRATION BY PARTS AND REVIEW
Work these on notebook paper. No calculator except on problem 15.
Evaluate.
6.
e
x dx
7.

3.
2
 x sin x dx
8.
4.
 e3x dx
5.
 arctan  2x  dx
1.
 xe
2.
 x sec
2x
dx
2
x
9.
1
0
4x
sin x dx
xe 5x dx
e
 e x ln x dx
x e
3 2x
dx
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2 x  5 for x  3
8
Find  f  x  dx.
10. Let f  x   
0
 x  1 for x  3.
11. Find
dy
in terms of x and y, given cos x  3sin  2 y   5 .
dx
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Use your calculator on problem 15.
12. A tank contains 120 gallons of oil at time t = 0 hours. Oil is being pumped into the tank at a
rate R  t  , where R  t  is measured in gallons per hour and t is measured in hours. Selected
values of R  t  are given in the table below.
t (hours)
R  t  (gallons per hour)
0
8.9
3
6.8
5
6.4
9
5.9
12
5.7
(a) Estimate the number of gallons of oil in the tank at t = 12 hours by using a trapezoidal
approximation with four subintervals and values from the table. Show the computations
that lead to your answer.
(b) Estimate the number of gallons of oil in the tank at t = 12 hours by using a right Riemann sum
with four subintervals and values from the table. Show the computations
that lead to your answer.
(c) A model for the rate at which oil is being pumped into the tank is given by the function
G t   3 
10
, where G  t  is is measured in gallons per hour and t is measured in
1  ln  t  2 
hours. Use the model to find the number of gallons of oil in the tank at t = 12 hours.
CALCULUS BC
Mixed Integration Worksheet
Work the following on notebook paper. No calculator.
5
dx
1. 
 2
2 x  1
2. arctan  5x  dx

5 x
dx
 2x  x 1
3. 

4.
e
2
x
cos  2 x  dx
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 x
2
10. Given the region bounded by the graphs of y  cos   , y  0, x  0, and x   .
Find the volume of the solid generated by revolving the region about the x-axis.
CALCULUS BC
WORKSHEET 1 ON LOGISTIC GROWTH
Work the following on notebook paper. Do not use your calculator.
1. Suppose the population of bears in a national park grows according to the logistic differential
dP
 5P  0.002 P 2 , where P is the number of bears at time t in years.
dt
(a) Given P  0   100.
equation
(i) Find lim P  t  .
t 
(ii) What is the range of the solution curve?
(iii) For what values of P is the solution curve increasing? Decreasing? Justify your answer.
d 2P
(iv) Find
and use it to find the values of P for which the solution curve is concave up and concave
dt 2
down.
Justify your answer.
(v) Does the solution curve have an inflection point? Justify your answer.
(vi) Use the information you found to sketch the graph of P  t  .
(b) Given P  0   1500.
(i) Find lim P  t  .
t 
(ii) What is the range of the solution curve?
(iii) For what values of P is the solution curve increasing? Decreasing? Justify your answer.
(iv) For what values of P is the solution curve concave up? Concave down? Justify your answer.
(v) Does the solution curve have an inflection point? Justify your answer.
(vi) Use the information you found to sketch the graph of P  t  .
(c) Given P  0  3000.
(i) Find lim P  t  .
t 
(ii) What is the range of the solution curve?
(iii) For what values of P is the solution curve increasing? Decreasing? Justify your answer.
(iv) For what values of P is the solution curve concave up? Concave down? Justify your answer.
(v) Does the solution curve have an inflection point? Justify your answer.
(vi) Use the information you found to sketch the graph of P  t  .
(d) How many bears are in the park when the population of bears is growing the fastest?
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2. Suppose a rumor is spreading through a dance at a rate modeled by the logistic differential
equation
dP
P 

 P3
 . What is lim P  t  ? What does this number represent in
dt
2000 
t 

the context of this problem?
TURN->>>
3. (From the 1998 BC Multiple Choice)
The population P  t  of a species satisfies the logistic differential equation
dP
P 

 P2 
,
dt
5000 

where the initial population is P  0  3000 and t is the time in years. What is lim P  t  ?
t 
(A) 2500
(B) 3000
(C) 4200
(D) 5000
(E) 10,000
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_
4. Suppose a population of wolves grows according to the logistic differential equation
dP
 3P  0.01P 2 ,
dt
where P is the number of wolves at time t in years. Which of the following statements are true?
I. lim P  t   300
t 
II. The growth rate of the wolf population is greatest at P = 150.
III. If P > 300, the population of wolves is increasing.
(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) I, II, and III
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_
5. Suppose that a population develops according to the logistic equation
dP
 0.05P  0.0005P 2
dt
where t is measured in weeks.
(a) What is the carrying capacity?
(b) A slope field for this equation is shown at the right.
Where are the slopes close to 0?
Where are they largest?
Which solutions are increasing?
Which solutions are decreasing?
(c) Use the slope field to sketch solutions for initial
populations of 20, 60, and 120.
What do these solutions have in common?
How do they differ?
Which solutions have inflection points?
At what population level do they occur?
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6. (a) On the slope field shown on the right
for
dP
 3P  3P 2 , sketch three
dt
solution curves showing different types
of behavior for the population P.
(b) Describe the meaning of the shape of
the solution curves for the population.
Where is P increasing?
Decreasing?
What happens in the long run?
Are there any inflection points?
Where?
What do they mean for the population?
CALCULUS BC
WORKSHEET ON L’HOPITAL’S RULE
Work the following on notebook paper. No calculator.
1. lim
x 3
2x  6
x2  9
2. lim
x 3
x 1  2
x 3
3. lim
x 
5 x 2  3x  1
3x 2  5
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__
Evaluate by using L’Hopital’s Rule, if possible.
x2  x  2
4. lim
x 2
x2
5. lim
x 0
4  x2  2
x
e x  1  x 
6. lim
x 0
x
7. lim
sin  2 x 
13.
 ln x 
lim
x 
x
14. lim   x ln x 
x 0


15. lim  x sin
x 
16. lim x
1
x 0
sin  3 x 
8. lim
arcsin x
x
17.
9. lim
3x 2  2 x  1
2 x2  3
18. lim 
x 2
10. lim
x2  2 x  1
x 1
19. lim 
x1
x 0
x 
x 
11. lim
x 
12. lim
x 
x
x2  1
ln x
x
3
1

x
x
x 
lim 1  x 
1
x
x 0
x 
 8


2
 x 4 x2
2 
 3


 ln x x  1 
20. lim
x2
e5 x
21. lim
e2 x  1
ex
x 
x 0