Math 121 - Chapter 4 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. Look at a graph and identify all relative extrema and all absolute extrema. Also make a sign chart for the derivative.
  3. You are given a curve and a point on the curve. Demonstrate Newton's method by finding the next two points towards the approximation of the zero.
  4. Sketch a continuous curve that has the stated properties. Three parts. Look at problem 4.1.29-30.
  5. 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.
  6. 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.
  7. 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.
  8. 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.
  9. 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.
  10. 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?
  11. Find all absolute extrema, if any, on the stated interval. Two parts. Look at problems 4.5.5-22.
  12. The graph of y=f'(x) is given. Identify where the original function has a relative maximum, relative minimum, and point of inflection. Make a sketch that could be the graph of the original function. Look at problem 4.2.13-16.

Notes:

Point values for each problem

# 1 2 3 4 5 6 7 8 9 10 11 12 Total
Pts 5 5 3 9 7 6 4 4 5 5 8 4 65

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Last updated March 29, 2003 8:41 AM