# Math 122: Chapter 9.1-10.3 Study Guide

1. Separate and solve the differential equation. Look at problems 9.1.15-24.
2. Solve the differential equation using the method of integrating factors. Look at problems 9.1.9-14.
3. Solve the initial value problem using any method. Look at problems 9.1.27-32.
4. Find all values of r that yield solutions to the differential equation. Example, let y=xr. Find the solutions to x2y"+4xy'+2y=0. Begin by finding y' and y" and substituting into the equation. Then solve for r.
5. Find the solution to the second order differential equation. Look at problems 9.4.3-22. Three parts: find the general solution for two of them and solve the initial value problem for the third.
6. Given a direction field and an initial value, Sketch the solution to the problem. Look at problems 9.2.3, 5.
7. Match the differential equation with the direction field. Six parts. Look at problem 9.2.9.
8. The first four derivatives of a function, evaluated at x=0, are given. Write a 4th order Maclaurin series for the function. You do not know what the function is. Sort of look at problems 10.1.7-16.
9. The first four derivatives of a function, evaluate at a point, are given. Write a 4th order Taylor series for the function. Use your approximation to approximate the function at a point. Look at a graph and state an interval for which the approximation will be good. You do not know what the function is. Sort of look at problems 10.1.17-24.
10. Write the first five terms of the sequence and determine whether or not the sequence converges. If it converges, find its limit. Two parts. Look at problems 10.2.5-22.
11. Find the general term of the sequence, beginning with n=1. Determine whether or not the sequence converges. If it converges, find its limit. Two parts. Look at problems 10.2.23-30.
12. Use the difference of consecutive terms to show that a sequence is strictly increasing or strictly decreasing. Look at problems 10.3.1-6.
13. Use the ratio of consecutive terms to show that a sequence is strictly increasing or strictly decreasing. Look at problems 10.3.7-12.
14. Use differentiation to show that a sequence is strictly increasing or strictly decreasing. Look at problems 10.3.13-18.

## Notes

• The exam is scheduled for 2 days.
• There is a take home portion of the exam worth 30% of the test grade. A copy of the take home exam is available online.

## Points per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Take Home Total 5 5 5 4 9 3 6 3 6 6 6 4 4 4 30 100