# Math 121 - Exam 4 Study Guide

## In-Class Exam

1. A graph of a function is given. Make a sign chart for f, f', and f". Give the intervals where f is increasing, decreasing, concave up, and concave down. Give all values of x where there is an inflection point, where f' is 0, and where f' is undefined. Look at problem 4.1.7.
2. Sketch a continuous curve with the stated properties. Three parts. Look at problems 4.1.9-10, 29-30.
3. 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.
4. 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.
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. Five parts.
6. 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 4.3.15-18.
7. The sign chart for either f' or f" is given. Only the critical points of the original function are indicated on the sign chart. Identify any relative maximums or relative minimums. Three parts.
8. 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.
9. 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.
10. Rectilinear motion -- find the velocity and acceleration, 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.

## Take Home Exam

1. 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.
2. Newton's method problem -- find the zeros of a function.
3. Mean value theorem -- find all values of c guaranteed by the theorem; annotate the graph to illustrate the mean value theorem.
4. Applied maximization / minimization problem. Several problems will be given, choose two of them to answer.

## Notes

• There is an in-class part to the test worth 70 points and a take home exam worth 30 points.
• The in-class exam is to be done without a calculator or computer. There is only one problem where you will need to find the derivative (#10); in all other problems, the derivatives are given if they are needed.
• The take-home exam is due the day of the in-class exam.

## Point values per problem

 part # Pts In-Class portion of exam Take Home 1 2 3 4 5 6 7 8 9 10 1 2 3 4 Total 7 9 10 4 5 6 6 8 8 7 8 8 4 10 100