# Math 121 - Exam 2 Study Guide

1. Sketch the graph of the derivative of the function whose graph is shown. Two parts. Look at problems 3.2.25-26.
2. Sketch a continuous curve with the stated properties. Two parts. Look at problems 4.1.9-10, 29-30.
3. Find the derivative of functions. You need to know the power, product, quotient rules, and chain rules. Seven parts. You may need to multiply or divide before differentiating. Look at problems 3.3.1-20, 3.5.1-18, and 3.6.7-40
4. The values of x, f(x), f'(x), g(x), and g'(x) are given in a table. Use the rules of derivatives to find the derivatives as the specified point. Six parts. Look at problems 3.4.19-22; 3.6.5-6, 71
5. The graph of a derivative of a function is given. Use the graph to approximate the x-coordinates of any horizontal tangents, relative maximums, relative minimums, and inflection points. Sketch a function that might be the original curve. Look at problems 4.2.15-18.
6. A function and its first two derivatives are given in factored form. Make a sign chart for f' and f". Identify the x-intercepts, y-intercepts, vertical asymptotes, and any critical points. Give the x-coordinates of any relative maximums, relative minimums, or inflection points. Look at the quick check exercises in section 4.3.
7. Assume that a continuous function has exactly one critical point. Use the first or second derivative tests to identify whether there is a relative minimum, relative maximum, neither a maximum nor minimum, both a maximum and minimum, the situation is impossible, or not enough information has been given. Ten parts.
8. The derivative of a function is given in factored form. Determine any critical points and identify them as relative minimums, relative maximums, or neither. Two parts.
9. Determine, by inspection, whether the function will have an absolute maximum, absolute minimum, neither a maximum nor minimum, both, or not enough information has been provided. "By inspection" means that if you have to take the derivative to answer the question, then there is not enough information given. Six parts.
10. A rational function and its first two derivatives are given in factored form. Make a sign chart for f' and f". Identify the x-intercepts, y-intercepts, vertical asymptotes, and any critical points. Give the x-coordinates of any relative maximums, relative minimums, or inflection points. Look at the quick check exercises in section 4.3.
11. Rectilinear motion. Evaluate the position, velocity, speed, and acceleration at a particular time, make a sign chart for velocity and acceleration, give the points where the partical is stopped and the intervals where the particle is speeding up and slowing down.
12. Use a local linear approximation to estimate the value given. Look at problems 3.9.19-27.
13. Find the absolute extrema of the function (if any) on the specified interval. Two parts. One on a closed interval and one on an open interval.
14. Use implicit differentiation to find dy/dx. Look at problems 3.7.11-20
15. Use the limit definition of a derivative to find the derivative. Look at problems 3.2.9-14 (only the part about finding the derivative) or 3.2.15-20.

## Take Home Exam

1. Newton's method problem -- find the zeros of a function.
2. Applied maximization / minimization problems.
3. Mean value theorem -- find all values of c guaranteed by the theorem; annotate the graph to illustrate the mean value theorem.
4. Given the values of two trigonometric functions (example: cos α and sin β), find the exact values of other trigonometric expressions involving α and β. You will need to know many of the formulas from appendix A. α and β are acute angles. Six parts.
5. Evaluate the limits. Six parts.

## Notes

• Most problems are similar to examples we've worked out in class.
• Problems from the book may be similar to the problems on the test, but you should not expect the questions on the test to be identical to those in the book.
• Show work on all problems, especially those on the take home exam so I know you're doing it and not just copying.
• The take home exam is due the day of the in-class exam.

## Point values per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Take home Total 6 6 28 24 8 8 10 6 6 7 9 5 8 5 6 58 200