Math 121 - Chapters 5-6 Study Guide

1. Use the graph of y=f(x) to make a sign chart and then 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 4.1.7.
2. Sketch a continuous curve that has the stated properties. Three parts. Look at problem 4.1.29-30.
3. Classify each critical point as a relative maximum, relative minimum, neither, impossible, or not enough information given. Seven 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. Example: If f'(3)=0 and f"(3)=-2, then there is a relative maximum at x=3.
4. Determine by inspection whether each of the functions will have an absolute minimum, absolute maximum, both, neither, or not enough information given. Six parts. Example: If f(x)=-x⁶ over all real numbers, then there will be an absolute maximum.
5. 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.
6. The graph of a rational 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.
7. 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 4.3.1-10.
8. Given a polynomial function in factored form, answer the following questions. Look on pages 259-260.
   1. What is the right hand behavior of the graph?
   2. What is the left hand behavior of the graph?
   3. Where will the graph cross the x-axis?
   4. Where will the graph touch the x-axis?
   5. Where will the graph be tangent to the x-axis?
   6. Where will the graph have an inflection point on the x-axis?
9. Find all absolute extrema, if any, on the stated interval. Three parts. Look at problems 4.5.5-22.
10. 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 4.4.11-14
11. 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 4.8.3-8
12. 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 4.8.11-16

Notes:

• There is a take home portion worth 18 points. It is due the day of the in-class exam.