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Graphs of Functions and Derivatives Rapid Review for AP Calculus

By — McGraw-Hill Professional
Updated on Oct 24, 2011

Review the following concepts if needed:

Review Problems and Answers

  • If f '(x) = x2 – 4, find the intervals where f is decreasing. (See Figure 7.6-1.)
  • Rapid Review

    Answer: Since f '(x) < 0 if – 2 < x < 2, f is decreasing on (–2, 2).

    • If f"(x) =2x – 6 and f ' is continuous, find the values of x where f has a point of inflection. (See Figure 7.6-2.)

    Rapid Review

    Answer: Thus, f has a point of inflection at x = 3.

    • See Figure 7.6-3. Find the values of x where f has change of concavity.

    Rapid Review

    Answer: f has a change of concavity at x = 0. See Figure 7.6-4.

    Rapid Review

    • See Figure 7.6-5. Find the values of x where f has a relative minimum.

    Rapid Review

    Answer: f has a relative minimum at x = – 2. See Figure 7.6-6.

    Rapid Review

    • See Figure 7.6-7. Given f is twice differentiable, arrange f (10), f '(10), f "(10) from smallest to largest.

       

      Rapid Review

      Answer: f(10) = 0, f '(10) > 0 since f is increasing, and f "(10) < 0 since f is concave downward. Thus the order is f "(10), f (10), f '(10).

    • See Figure 7.6-8. Find the values of x where f ' is concave up.

    Rapid Review

    Answer: f ' is concave upward on (– ∞, 0). See Figure 7.6-9.

    Rapid Review

    • The path of an object is defined by x =2t, y = t + 1. Find the location of the object when t =5.

    Answer: When t =5, x =2(5) = 10 and y = 5 + 1 = 6 so the location is (10, 6).

    • Identify the shape of each equation: (a) (b) r =6 cos 3θ

    Answer: (a) spiral, (b) rose

    • Find the magnitude of the vector and the angle it makes with the positive x-axis.

       

      Answer: .

    • If a = {4, – 2}, b = {– 3, 1}, and c = {0, 5}, find 3a – 2b + c.

    Answer: 3 {4, – 2} – 2{– 3, 1} + {0, 5} = {12, – 6} + {6, – 2} + {0, 5} = {18, – 8} + {0, 5} = {18, – 3}.

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