## Math 121 - Chapters 5-6 Study Guide

1. Use the graph of y=f(x) to find the intervals on which f is increasing, decreasing, concave up, and concave down. Also give the x-coordinates of all points of inflection. Look at problem 5.1.7.
2. Sketch a continuous curve that has the stated properties. Two parts. Look at problem 5.1.31.
3. Classify each critical point as a relative maximum, relative minimum, neither, impossible, or not enough information given. Five parts. Know the first and second derivative tests to answer these. Note: There is a difference between you not knowing how to do it and not enough information being given.
4. Give a complete graph of the polynomial and label the coordinates of the intercepts, stationary points, and inflection points. Check your work with a graphing utility. Look at problems 5.3.1-10*.
5. Given a polynomial function in factored form, answer the following questions. Nine parts. Look on pages 308-310.
1. When roots are counted according to their multiplicity, how many real or complex zeros are there?
2. What is the right hand behavior of the graph?
3. What is the left hand behavior of the graph?
4. What is the maximum number of relative extrema possible?
5. What are the x-intercepts of the function?
6. Where will the graph cross the x-axis?
7. Where will the graph touch the x-axis?
8. Where will the graph be tangent to the x-axis?
9. Where will the graph have an inflection point on the x-axis?
6. Find the absolute maximum and minimum values of a function on a closed interval and state where those values occur. Look at problems 6.1.5-14*
7. Find the absolute maximum and minimum values, if any, of the function, over the set of all real numbers. Look at problems 6.1.15-20*
8. Find the absolute maximum and minimum values, if any, of the function, over the given interval. Two parts. Look at problems 6.1.23-34*
9. The position function of a particle is given. Find the velocity and acceleration functions; the position, velocity, speed, and acceleration at a specified time; when the particle is stopped; when the particle is speeding up and slowing down; and the total distance traveled over a time interval. Look at problems 6.3.11-14*
10. Applied minimum or maximum problem. Look at problems 6.2.42-45, 56-57.
11. Identify by inspection whether each of the functions will have an absolute minimum, absolute maximum, both, neither, or not enough information given. Six parts.
12. Newton's method. Look at problems 6.4.5-14*
13. Verify the hypotheses of Rolle's Theorem are satisfied on the given interval and then find all the values that are guaranteed by the conclusion of the theorem. Look at problems 6.5.3-8*
14. Verify the hypotheses of the Mean Value Theorem are satisfied on the given interval and then find all the values that are guaranteed by the conclusion of the theorem. Look at problems 6.5.11-16*
15. The graph of a polynomial function is given. Tell where f(x)=0, f'(x)=0, and f"(x)=0. Using your knowledge from college algebra and differential calculus, write a function whose graph could be that shown.
16. The graph of a rational function is given. Tell where f(x)=0, f'(x)=0 or undefined, and f"(x)=0. Using your knowledge from college algebra and differential calculus, write a function whose graph could be that shown.

### Notes:

• *'d problems are directly from the text.
 # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Total Pts 10 8 5 5 9 3 3 6 12 4 6 7 3 3 8 8 100