## Math 122 - Final Exam Study Guide

1. Sketch the region enclosed by the curves and find its area.
2. Sketch the region enclosed by the curves and find the volume of the solid generated when the region is rotated about an axis.
3. Find the exact arc length of the curve on the stated interval.
4. Find the area of the surface generated by revolving the given curve about an axis.
5. Find the fluid force against a surface submerged vertically in a liquid.
6. Find the work required to stretch a spring.
7. Integrate using integration by parts.
8. Integrate using a trigonometric substitution.
9. Integrate using partial fractions.
10. Evaluate an improper integral.
11. Values of a function are given at certain points. Use the trapezoid method and Simpson's method to find the area under the curve. Two parts.
12. Solve the differential equation. Two parts. One is an initial value problem.
13. Draw a direction field for the differential equation.
14. Use Euler's method to approximate the initial-value problem over the stated interval. Present your answer as a table and a graph.
15. Determine whether the sequence is eventually strictly increasing or eventually strictly decreasing and for what values of n this holds.
16. Determine whether the series converges, and if so, find its sum. Two parts.
17. Identify each series as convergent, conditionally convergent, or divergent. Five parts.
18. Find the radius of convergence and the interval of convergence.
19. Find the fourth degree Maclauring polynomial for the given function. Then write the Maclaurin series in sigma notation.
20. Obtain the first four non-zero terms of a Maclaurin series by making an appropriate substitution in a known Maclaurin series and performing any algebraic operations required. State the radius of convergence.
21. Sketch the curve in polar coordinates. Two parts.
22. Calculate the arc length of the polar curve.
23. Sketch the region described in polar form and find its area.
24. Find the equation of the ellipse that satisfies the given conditions.
25. Find the polar equation for the hyperbola that satisfies the given conditions.
26. Write the equation of the conic section (given in polar form) in rectangular coordinates.
27. Identify each statement as true or false. Fifteen parts. Concentrate on properties of hyperbolic trig functions, integration by parts, exponential and logistic growth models, convergence and divergence of tests, monotonic series, convergence of a power series, derivatives of a power series, the relationship between the derivatives of trig functions and their cofunctions, the relationship between the derivatives of hyperbolic trig functions and their reciprocal functions, the convergence of a p-series, and the convergence of a finite series.

### Notes:

• The final exam is open notebook.
• The exam is in the chapter order.
• The exam is based primarily off of old exams, although there may be some material on the final that isn't from an old exam.
• You may copy tables from the book and put in your notebook. In particular, make sure you get the derivatives and integrals of trig, inverse trig, hyperbolic trig, and inverse hyperbolic trig functions.
• Make sure this study guide is in your notebook, along with any notes about where to find information.
• If there are any areas of the test where your notes are weak, supplement them before taking the exam.
• Have your notes extremely organized before you arrive to take the test. If you have to spend a lot of time during the test looking for the material on the test, you will run over the allotted time.
• Each part on the exam is worth 5 points except for the last problem where each answer is worth 2 points.