Math 121 Fall 2009, Final Exam PRACTICE
Instructions
· Closed book, closed notes, except for one 8.5”-by-11” (or A4) sheet of paper, okay to use both sides. You may be required to turn in your note sheet with the exam, so write your name on it.
· 90 minutes are allowed for this exam.
· Clearly indicate your answer.
· You must show all relevant work and justify your answers appropriately.
· Partial credit will be given, but not without sufficient support.
· No calculators that have a QWERTY-type keyboard are allowed, unless previously approved. The proctor's discretion is final.
· Answers that violate simple upper or lower bounds may result in more point deductions.
· There is a shortcut we learned about the average value of a sine or cosine; you may use that shortcut (but say you’re using it).
· THIS PRACTICE EXAM DOES NOT COVER EVERYTHING THAT MIGHT BE ON THE FINAL!
· The final is comprehensive, but more emphasis will be placed on the material from the last few weeks of the course.
· Study old tests and homeworks.
1. For each integration method listed, give an example integral that is best solved with that method. You do not have to solve the integrals.
a. By parts
b. By u-substitution
c. By partial fractions
d. By trig identities
e. By trig substitution
f. By numerical methods
2. Suppose a person (over 21 years old) has a blood-alcohol content (BAC) of 0.05%. What is their BAC after 5 hours? Their liver gets rid of alcohol at the rate of 0.01 percentage points per hour until their BAC reaches 0.02%, after which it gets rid of alcohol at this rate: f(t) = 0.01% * exp(-t/3), where t is time since hitting a BAC of 0.02%.
3. Let f(x) = sin(x) from 0 to pi.
a. Write an integral formula that will compute the average value of f(x)
b. Find a numerical value for your answer to part (a) by doing the integration.
4. Find the sum of this series, if it converges:
5. a) What is the Maclaurin series for sin(a*x) ?
.
b) show that the derivative of sin(a*x) is a*cos(a*x) using your series from part (a).
6. The following integral is important in applied probability ; evaluate the integral using symbolic methods to start with (that is, don’t just use the midpoint rule or a similar tactic). Boil your answer down to a final number in decimal form.
7. Suppose f ’’(0) > 0. If you take the Maclaurin series up to first order, will you usually get an LB or UB or neither on f(x) for x values close to 0? Explain. Assume that f is a nice function; that is, we’re talking about what will usually happen, ignoring weird cases.
8. A) Use the Trapezoid method with n=4 to integrate the function f(x) = sin(pi x) from x=0 to x=1. Also, say whether you should get an LB, UB, or neither by using the Trapezoid method in this case.
B) Use Simpson’s method with n=4 to integrate the function f(x) = sin(pi x) from x=0 to x=1.
9. Give the Maclaurin series for f(t) = t * e^t, up to 3rd order. Then give the general n’th order term.
10. Find a general formula for the volume of a paraboloid: f(x) = sqrt(a*x), spun around the x-axis, for x values between 0 and b.