Curve sketching

Curve sketching

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.

Lessons

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.
  • 1.
    Use the guidelines to sketch the graph of:

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

    b)
    Intercepts

    c)
    Asymptotes

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

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

  • 2.
    Use the guidelines to sketch the graph of:
    f(x)=−x3−6x2−9xf(x)=-x^3-6x^2-9x
    a)
    Domain

    b)
    Intercepts

    c)
    Asymptotes

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

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

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3.
Applications of differentiation
3.1
Position velocity acceleration
3.2
Critical number & maximum and minimum values
3.3
l'Hospital's rule
3.4
Curve sketching
3.5
Optimization
3.6
Related rates
3.7
Mean value theorem
3.8
Linear approximation
Calculus 1
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