Math 110 Final Exam
May 8, 2006
Show all work of course.
In 1-3, find the integral.
1.
 xe
2.
x
 xe dx
2x
2
dx
Name
3.
x  12
 3x  x
2
dx
n!( n  1)!
converges or diverges.
( 2n)!
n 1

4. Use the ratio test to determine if

5. In biology class, Pat was studying a bacteria population. The population was growing
at a rate proportional to the size of the population, and she was supposed to determine the
time it took the population to double. Pat started at 9 am with 1000 bacteria. The
population had not doubled by the time class got out. Pat had another class, and then had
to go to work. By the time she got back to the lab at 7 pm, the population had grown to
6000. Fortunately, Pat had studied calculus with differential equations, and was able to
use the above information to determine the doubling time. Show how she might have
done this.
dy 1
 , with initial condition y (1)  0
dx x
a) Sketch the slope field for the differential equation, as well as the solution satisfying the
given initial condition.
6. Consider the differential equation
y




x













b) Use Euler’s method, two steps, to estimate y(2), the value of the solution when x = 2.
Do the numerical work here, by hand, and sketch your results in the slope field above.
c) What is the exact value of y(2)? How do you know?
7. a) Write down (or derive) the Maclaurin series for f ( x)  sin( x)
1
b) Use the first two nonzero terms to approximate sin   . What is the upper bound on
2
the error, using the error analysis for alternating series? Use your calculator to find the
actual error.
c) Use the result of 7 (a) to write the Maclaurin series for g ( x) 
sin( x)
x
1
d) Use at least three nonzero terms in your series above to estimate
sin( x)
dx
x
0

1
1
dx . Explain why this integral is improper. Determine the
p
0 x
values of p for which this integral converges. Be careful. This is not the family of
integrals we did in class.
8. Consider the integral 
9. Suppose the region bounded by y  x 2  1 , y = 2, and the y-axis is revolved about the
x-axis. Find the volume of the resulting solid of revolution.
10. Give an example of a separable differential equation, with an initial condition. Solve
your equation, and check to make sure your solution satisfies both the differential
equation and the initial condition.