Graphing using Calculus

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Graphing using Calculus

• Mat 231

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Steps to graphing

• Find the domain of the function.
• Find the x- and y-intercepts.
• Find the vertical and horizontal asymptotes, if
  any.
• Determine where the function is increasing and
  decreasing.
• Determine any local extreme values.
• Determine where the function is concave up and
  down.
• Find all the inflection points.

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Example 1

• Graph f(x) -2(x1)2(x-3)3
• Domain (-8, 8)
• x-intercept(s) -1, 3
• y-intercept 54
• Intervals of Decrease (-8, -1), (3/5, 8)
• Intervals of Increase (-1, 3/5)
• Local Minimum(s) f(-1) 0
• Local Maximum(s) f(3/5) 70.8
• Intervals where f is concave up (-8, -0.4),
  (1.6, 3)
• Intervals where f is concave down (-0.4, 1.6),
  (3, 8)
• Inflection point(s) (-.4, 29.7) (1.6,38.1)
  (3, 0)

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Graph
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Example 2

• Graph the function f(x) x-sinx on 0, 2p
• Domain 0, 2p
• x-intercept(s) 0
• y-intercept 0
• Intervals of Increase (0, 2p)
• Intervals of Decrease N/A
• Local Minimum(s) N/A
• Local Maximum(s) N/A
• Intervals where f is concave up (0, p)
• Intervals where f is concave down (p, 2p)
• Inflection point(s) (p, p)

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Graph
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Example 3

• Let f(x) x(5-x)1/2
• Domain (-8, 5
• x-intercept(s) 0, 5
• y-intercept 0
• Intervals of Increase (-8, 10/3)
• Intervals of Decrease (10/3, 5)
• Local Minimum(s) N/A
• Local Maximum(s) (10/3, 4.3)
• Intervals where f is concave up N/A
• Intervals where f is concave down (-8, 5)
• Inflection point(s) N/A

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Graph
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Example 4

• Graph f(x) 1/(1x2 ).
• Domain (-8, 8)
• x-intercept(s) None
• y-intercept 1
• Intervals of Increase (-8, 0)
• Intervals of Decrease (0, 8)
• Local Minimum(s) N/A
• Local Maximum(s) (0, 1)
• Intervals where f is concave up (-8, -v3/3),
  (v3/3, 8)
• Intervals where f is concave down (-v3/3, v3/3)
• Inflection point(s) (-v3/3, 3/4) (v3/3, 3/4)

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Graph