Department of Mathematics
Math 112 Spring 2006
Final Exam
NAME: _________________________________
Instructor: _______________________________
Instructions:
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There are 20 problems on the next 11 pages, each problem is weighted 5 points. Every
problem will be graded so be sure to complete the entire exam.
Show all work clearly to receive full credit. Use principles of calculus to support all answers.
For example: definite integral solutions must include steps of evaluation.
If you need more space for your solutions, use the back of one of the pages. Clearly indicate
where your extra work can be found.
Cross out all work you do not want considered in grading. Otherwise, it will be graded
accordingly and could reduce your score.
Using a TI-89 or a TI-92 is not permitted.
Formulas:
sin 2 x  cos 2 x  1
sin 2 x 
1
1  cos 2 x 
2
sin 2x  2 sin x cos x
1  tan 2 x  sec 2 x
cos 2 x 
1
1  cos 2 x 
2
cos2 x   cos 2 x  sin 2 x
1. Find the area between the curves y  x and y  5 x  x 2 .
2. Given the curves, y = x2 + 1, y = x + 3 shown below:
(a) Suppose the area between these curves
is rotated about the x-axis. Write an expression for
A(x), the area of a cross section of the volume
perpendicular to the x-axis.
(b) Find the volume of the solid of revolution.
Evaluate each integral below. Show each step and clearly indicate any substitutions used.

3.
2
 35sin
4
x cos3 x dx
0
4.

x2  4
dx
x
Evaluate each integral below. Show each step and clearly indicate any substitutions used.
ln x
5.  2 dx
x
6.

6 x 2  3x  1
dx
4 x  1 x 2  1


1
7. Evaluate the improper integral or show that it is divergent.
e
2x
dx

8. The tank shown is full of water. Given that water weighs 62.5 lb ft 3 and R = 7 ft, find the work required
to pump the water out of the tank.
9. The Gateway Arch in St.Louis closely resembles a parabola that is 630 ft. tall. It can be modeled by the
equation: y  630  .00635 x 2 . Set up, BUT DO NOT SOLVE, the integral needed to compute the length of
the outside of this arch.
1
10. Use the following two methods to approximate

1  x 4 dx . Use four subintervals for your
0
approximations and round your final answers to 6 decimal places.
(a) Trapezoid Rule
(b) Simpson’s Rule
(c) Which of the two approximation methods above is expected to be more accurate?
11. Use Euler’s Method with step size 0.1 to estimate y0.3 , where yx  is the solution to the initial value
problem: y   x  2 y
y0  1
12. The Clausius-Clapeyron equation determines the vapor pressure of a liquid as a function of temperature.
For instance, the vapor pressure of water at 100 C is 1 atmosphere because 100 C is the boiling point of
water at sea-level.
dp
p

The differential equation that determines pT  is
, where R is the ideal gas constant and  is
dT RT 2
the “heat of vaporization.” Take  as a constant and solve this differential equation.
13. Biologists stocked a lake with 450 fish and estimated the carrying capacity of the fish to be 11,000. The
number of fish tripled in the first year. Assume that the size of the fish population satisfies the logistic
equation. Find an expression for the size of the population after t years.
14. Answer “yes” or “no” to each of the following questions.
________ (a) If an  is a sequence and lim an  0 , then do we know the sequence oonverges?
n 

________ (b) If
 a n is a series and lim a n  0 , then do we know the series
n 1
n

________ (c) If
 a n is a series and lim a n  0 , then do we know the series
n 1

________ (d) If
 a n is a series and
n 1

________ (e) If
 a n is a series and
n 1
n

a
n 1
converges?
n

a
n 1
diverges?
n


n 1

a n converges, then do we know the series


n 1
a
n 1
n
converges?

a n diverges, then do we know the series
a
n 1
n
diverges?
15. Determine whether each series converges or diverges. If convergent, find the sum. Give reasons for your
conclusions, naming the test(s) used and showing every step.

(a)

(b)

16. Determine if the series
 1 



n 0  3 

 n 
ln 

 3n  7 
n 1

 1n
n
is absolutely convergent, conditionally convergent, or divergent.
n 1
Give reasons for your conclusions, naming the test(s) used and showing every step.
n 1

cos n
is absolutely convergent, conditionally convergent, or divergent. Give
2
4
n 1
reasons for your conclusions, naming the test(s) used and showing every step.
17. Determine if the series
n
18. Determine the radius of convergence and the interval of convergence of the power series.
n

x  8
  1 n

n
n 1
n6

x2 x3 x4


 ...
n
2
3
4
n 1
x2
Use the series above to derive the Maclaurin series for f  x  
.
1 x
19. The Maclaurin series for ln 1  x  is
x
  1n1
n
x
20. Find the Taylor series for f x   cos x about a   .