# Math 121 - Exam 4 Study Guide

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. 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. The graph of a polynomial function is shown. Use the graph to approximate the x-values where f(x) = 0, f'(x) = 0, and f"(x) = 0. Then use your knowledge of college algebra and differential calculus to write a possible function f.
6. The graph of a rational function is shown. Use the graph to approximate the x-values where f(x) = 0, f'(x) = 0, and f"(x) = 0. Then use your knowledge of college algebra and differential calculus to write a possible function f.
7. 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.
8. The graph of a derivative of a function is given. Use the graph to approximate the x-coordinates of any relative maximums, relative minimums, and inflection points. Sketch a function that might be the original curve. Look at problems 4.3.15-18.
9. 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.
10. A function and its first two derivatives are given in factored form. Identify the x-intercepts, y-intercepts, and given any critical points. Give the x-coordinates of any relative maximums, relative minimums, or inflection points.
11. 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.
12. One of the following three problems. You won't know which problem you're going to get until you get it. Each member of the group will get a different problem.
1. Newton's method problem -- find the zero of a function
2. Mean value theorem -- show that the conditions are satisified and find all values of c guaranteed by the theorem; annotate the graph to illustrate the mean value theorem.
3. Rectilinear motion -- find the velocity and acceleration, determine the position, velocity, speed, and acceleration at a particular point in time, make a sign chart for velocity and acceleration, give the intervals where the particle is speeding up and where it is slowing down.
13. Applied maximization / minimization problem. Several problems will be given, choose one of them to answer. The group must decide on which problem to work and submit.

## Notes

• The exam is split into several parts. Part of the test will be individual and part of the exam will be group.
• Groups will consist of three people and will be assigned at random. You will know before the exam who your group is so that you can make efforts to study together with them and bring them up to speed. Some class time will be given on review day to work with your group, but you will probably want to find additional time.
• You must work in a group, part of the grade is an assessment of how well you are able to work together with others and explain the material to them so that they understand.
• Questions 1-10 are to be worked individually and without a calculator.
• Question 11 is to be worked individually with a calculator.
• Question 12 is a group question to be worked individually.
• There are three different questions, each worth 6 points. Each member of the group will be given one question to answer. The points each person receives are found by adding the points for the question [s]he worked and one-third of the points each of the partners obtained on their questions. Here is an example: John gets 5 points on his question, Sally gets 4 points on hers, and Tracy gets 6 points on her question. John gets 5 + (4+6)/3 = 25/3 = 8 points, Sally gets 4 + (5+6)/3 = 23/3 = 8 points, and Tracy gets 6 + (5+4)/3 = 27/3 = 9 points.
• The topics include Newton's Method, the Mean Value Theorem, and Rectilinear Motion.
• You will want to work together to make sure that each of the three members of the group understands each of the three topics since your grade will in part (up to 4 points) depend on their understanding of the topic.
• Question 13 is a group question to be worked as a group. Each member of the group will get the same score. The group must decide which of the three papers is the best response turn in that one sheet to be graded. Several problems will be given, the group is to choose one of them and turn it in.

## Point values per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 13 Total 8 9 10 6 8 8 5 8 6 6 10 10 6 100