# Math 121 - Exam 3 Study Guide

1. Evaluate the summation. Look at problems 5.4.11-16.
2. Simplify the limit of the summation. Look at problems 5.4.25-26.
3. Use part 2 of the fundamental theorem of calculus to evaluate the derivative. Look at problems 5.6.43-46. Pay attention to those where the upper limit is a function.
4. Find the average value of a function over an interval. Look at problems 6.6.3-8.
5. Use the areas shown in the figure to find the definite integrals. Six parts. Look at problem 5.5.17-18.
6. Use a u-substitution to rewrite the definite integral and change the limits on the integral. Then use a formula from Geometry to evaluate the integral. Look at problems 5.8.13-16
7. Find a polynomial function with integer coefficients having the indicated extrema and y-intercept. Use the fact that extrema of a polynomial occur when f'(x)=0 to find the derivative function and then integrate to find the original function f. After obtaining integer coefficients (multiply by LCD), use the y-intercept as an initial value.
8. Evaluate the integrals. They may be indefinite or definite and may or may not require substitutions. Seven parts.
9. A particle moves along an s-axis. Use the given information to find the position function of the particle. Look at problems 5.7.5-8.
10. Work problem involving springs. Look at problems 6.7.6-11.
11. Work problem. Look at problems 6.7.12-21.
12. Evaluate the limits. Five parts.
13. Find all absolute extrema of the function on the given interval.
14. Find the fluid force on a vertically submerged surface. Look at problems 6.8.3-8.
15. The graph of an unknown function is shown along with key points on the curve. Use trapezoids to approximate the area under the curve. Make a table of the midpoints between the left and right endpoints of each interval, the change in x and y and the length of the secant line segment for each interval. Use the table to find the volume of the solid generated when the region is rotated about the x-axis and the y-axis. Approximate the length of the curve and the surface area of revolution when the curve is rotated about one of the coordinate axes. Look at problem 6.2.37.
16. Related rate problem. Look at problem 3.8.10
17. Find and simplify the derivative of the function. Three parts.
18. A function and its first two derivatives are given in factored form. Make sign charts, find the coordinates of the relative maximums, relative minimums, and inflection points. Sketch the graph.
19. A continuous function has exactly one critical point. Use the first and second derivative tests to tell whether those points are absolute maximums, absolute minimums, neither, both, there's not enough information to tell, or the situation is impossible.
20. Use the delta-epsilon definition of a limit to prove the given limit is correct.
21. Use the limit definition of a derivative to find a derivative.
22. A table of values is given. Use it to find the indicated derivatives.
23. The value of a trig function and the quadrant the angle lies in is given. Use it to find other trigonometric expressions.
24. Sketch the region enclosed by two curves. Write definite integrals that can be used to find the volume of rotation about each of the coordinate axes and then use a computer algebra system (CAS) like Derive, or the TI89 or TI92 to evaluate the integrals. Look at problems 6.2.7-11, 15-20 and 6.3.5-14.
25. A function is given. Find the arc length and surface area of revolution. In each case, setup a definite integral and then use a computer algebra system (TI89, TI92, or Derive) to evaluate or approximate the integrals. Look at problems 6.4.9-12 and 6.5.27-30..

## Notes

• Questions 24 and 25 are to be worked using WinPlot and Derive.
• On many of the application problems from chapter 6, the instructions are to setup the integral that is needed to find the answer but to not evaluate it. This will help with the time issue and hopefully avoid arithmetic errors.
• Some of the questions are over previous chapters.

## Point values per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Total 4 6 5 5 12 8 5 35 6 6 6 20 6 6 30 5 15 10 10 5 5 12 8 10 10 250