# Math 121 - Exam 2 Study Guide

1. The graph of a function and its derivative are shown, decide which is which. Three parts.
2. Sketch the graph of the derivative of the function whose graph is shown. Two parts. Look at problems 3.2.25-26.
3. A function is graphed. Decide where it is increasing, decreasing, concave up, concave down, give the x-values of any relative maximums, minimums, or inflection points.
4. The derivative of a function is graphed. Decide where the original function is increasing, decreasing, concave up, concave down, give the x-values of any relative maximums, minimums, or inflection points. Sketch a graph of the original function. Look at problems 4.2.15-18.
5. 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.
6. Use implicit differentiation to find dy/dx. Look at problems 3.7.11-20
7. 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. Seven parts. Look at problems 3.4.19-22; 3.6.5-6, 71
8. Find and simplify 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
9. Find the second derivative of a function
10. A polynomial 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, 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. 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.
12. 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.
13. 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.
14. Use a local linear approximation to estimate the value given. Look at problems 3.9.19-27.
15. 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. Nine parts.
16. Evaluate the limit. Some require work to be shown and some don't.

## Take Home Exam

1. Newton's method problem -- find the zeros of a function.
2. Applied maximization / minimization problem
3. Related rate problem
4. Related rate problem

## 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 16 Take home Total 6 8 7 9 6 5 21 24 5 8 7 10 8 5 18 21 32 200