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Graphing with Calculators for AP Calculus

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

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

Example 1

Using a calculator, sketch the graph of indicating all relative extrema, points of inflection, horizontal and vertical asymptotes, intervals where f (x ) is increasing or decreasing, and intervals where f (x) is concave upward or downward.

  1. Domain: all real numbers; Range: all real numbers
  2. No symmetry
  3. Relative Minimum: (0, 0)
  4. Relative Maximum: (1.2, 2.03)

    Points of Inflection: (– 0.6, 2.56)

  5. No asymptote
  6. f (x ) is decreasing on (– ∞, 0], [1.2,∞) and increasing on (0, 1.2).
  7. Evaluating the f '' (x ) on either side of the point of inflection (–0.6, 2.56)
  8. Graphing with Calculators

Example 2

Using a calculator, sketch the graph of , indicating all relative minimum and maximum points, points of inflection, vertical and horizontal asymptotes, intervals on which f (x ) is increasing, decreasing, concave upward or concave downward.

  1. Domain: all real numbers; Range (0, 1]
  2. Symmetry: f (x ) is an even function, and thus is symmetrical with respect to the y -axis.
  3. Relative maximum: (0, 1)
  4. No relative minimum

    Points of inflection: (–1, 0.6) and (1, 0.6)

  5. y =0 is a horizontal asymptote; no vertical asymptote.
  6. f (x) is increasing on (– ∞, 0] and decreasing on [0,∞).
  7. f (x ) is concave upward on (– ∞, –1) and (1, ∞); and concave downward on (–1, 1).
  8. See Figure 7.3-3.

    Graphing with Calculators

    Graphing with Calculators

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

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