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Curve sketching

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Now Playing:Curve sketching – Example 1a
Examples
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  1. Use the guidelines to sketch the graph of:

    f(x)=x3−8x3+8f(x)=\frac{x^3-8}{x^3+8}
    1. Domain

    2. Intercepts

    3. Asymptotes

    4. Compute f′(x):f' (x):
      - critical numbers
      - intervals of increase/decrease
      - local extrema

    5. Compute f′′(x):f'' (x):
      - possible inflection points
      - intervals of concavity
      - verify inflection points

Curve sketching
Jump to:Notes
Notes
In this section we will expand our knowledge on the connection between derivatives and the shape of a graph. By following the "5-Steps Approach", we will quantify the characteristics of the function with application of derivatives, which will enable us to sketch the graph of a function.
Guidelines for Curve Sketching
a) domain

b) Intercepts
y-intercept: set x=0 and evaluate y.
x-intercept: set y=0 and solve for x. (skip this step if the equation is difficult to solve)

c) Asymptotes
vertical asymptotes:
for rational functions, vertical asymptotes can be located by equating the denominator to 0 after canceling any common factors.

horizontal asymptotes:
evaluate limx→∞f(x)lim_{x \to \infty } f(x) to determine the right-end behavior; evaluate limx→−∞f(x)lim_{x \to -\infty } f(x) to determine the left-end behavior.

slant asymptotes:
for rational functions, slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator.

d) Computef′(x) f' (x)
find the critical numbers: critical numbers
• use the First Derivative Test to find: intervals of increase/decrease and local extrema.

e) Computef′′(x) f'' (x)
• inflection points occur where the direction of concavity changes. find possible inflection points by equating thef′′(x) f'' (x) to 0.

•Concavity Test:concavity test

•inflection points occur where the direction of concavity changes.