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Graphs of Functions and Derivatives Practice Problems for AP Calculus

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By — McGraw-Hill Professional
Updated on Mar 10, 2014

Review the following concepts if needed:

Part A—The use of a calculator is not allowed.

  1. If f(x )=x3x2 – 2x, show that the hypotheses of Rolle's Theorem are satisfied on the interval [–1, 2] and find all values of c that satisfy the conclusion of the theorem.
  2. Let f (x )=ex. Show that the hypotheses of the Mean Value Theorem are satisfied on [0, 1] and find all values of c that satisfy the conclusion of the theorem.
  3. Determine the intervals in which the graph of is concave upward or downward.
  4. Given f (x)=x + sin x 0 ≤ x ≥ 2π, find all points of inflection of f.
  5. Show that the absolute minimum of on [–5, 5] is 0 and the absolute maximum is 5.
  6. Given the function f in Figure 7.7-1, identify the points where:
    1. f '< 0 and f'' > 0,
    2. f ' < 0 and f '' < 0,
    3. f ' =0,
    4. f '' does not exist.
  7. Practice Problems

  8. Given the graph of f ''in Figure 7.7-2, determine the values of x at which the function f has a point of inflection. (See Figure 7.7-2.)
  9. Practice Problems

  10. If f ''(x)=x2(x +3)(x – 5), find the values of x at which the graph of f has a change of concavity.
  11. The graph of f ' on [–3, 3] is shown in Figure 7.7-3. Find the values of x on [–3, 3] such that (a) f is increasing and (b) f is concave downward.
  12. Practice Problems

  13. The graph of f is shown in Figure 7.7-4 and f is twice differentiable. Which of the following has the largest value:
    1. f (–1)
    2. f '(–1)
    3. f ''(–1)
    4. f (–1) and f '(–1)
    5. f '(–1) and f ''(–1)
  14. Sketch the graphs of the following functions indicating any relative and absolute extrema, points of inflection, intervals on which the function is increasing, decreasing, concave upward or concave downward.

    Practice Problems

  15. f (x)=x4x2

Part B—Calculators are allowed.

  1. Given the graph of f ' in Figure 7.7-5, determine at which of the four values of x (x1, x2, x3, x4) f has:
    1. the largest value,
    2. the smallest value,
    3. a point of inflection,
    4. and at which of the four values of x does f '' have the largest value.
    5. Practice Problems

  2. Given the graph of f in Figure 7.7-6, determine at which values of x is
  3. Practice Problems

    1. f ''(x )=0
    2. f ''(x )=0
    3. f '' a decreasing function.
  4. A function f is continuous on the interval [–2, 5] with f (–2)=10 and f (5)=6 and the following properties:
  5. Practice Problems

    1. Find the intervals on which f is increasing or decreasing.
    2. Find where f has its absolute extrema.
    3. Find where f has points of inflection.
    4. Find the intervals where f is concave upward or downward.
    5. Sketch a possible graph of f.
  6. Given the graph of f ' in Figure 7.7-7, find where the function f
    1. Has its relative extrema.
    2. Is increasing or decreasing.
    3. Has its point(s) of inflection.
    4. Is concave upward or downward.
    5. If f (0)=1 and f (6)=5, draw a sketch of f.
    6. Practice Problems

  7. If f (x )=|x2 – 6x –7|, which of the following statements about f are true?
    1. f has a relative maximum at x =3.
    2. f is differentiable at x =7.
    3. f has a point of inflection at x = – 1.
  8. How many points of inflection does the graph of y = cos(x2) have on the interval [–π, π]?
  9. Sketch the graphs of the following functions indicating any relative extrema, points of inflection, asymptotes, and intervals where the function is increasing, decreasing, concave upward or concave downward.

  10. f (x )= cos x sin2 x [0, 2π]
  11. Find the Cartesian equation of the curve defined by , y =t2 – 4t +1.
  12. Find the polar equation of the line with Cartesian equation y =3x – 5.
  13. Identify the type of graph defined by the equation r =2 – sin θ and determine its symmetry, if any.
  14. Find the value of k so that the vectors 3, –2 and 1, k are orthogonal.
  15. Determine whether the vectors 5, –3 and 5, 3 are orthogonal. If not, find the angle between the vectors.

 

(Calculator) indicates that calculators are permitted.

  1. Find
  2. Evaluate
  3. Find
  4. (Calculator) Determine the value of k such that the function
  5. A function f is continuous on the interval [–1, 4] with f (–1)=0 and f (4)=2 and the following properties:
  6. Practice Problems

    1. Find the intervals on which f is increasing or decreasing.
    2. Find where f has its absolute extrema.
    3. Find where f has points of inflection.
    4. Find intervals on which f is concave upward or downward.
    5. Sketch a possible graph of f.
  7. Evaluate
  8. Evaluate
  9. Find the polar equation of the ellipse x2 +4y2 =4.

 Solutions for these practice problems can be found at:  Solutions to Graphs of Functions and Derivatives Practice Problems for AP Calculus

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