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- Calculus 1
- Applications of differentiation
Curve sketching
- Lesson: 1a2:08
- Lesson: 1b4:06
- Lesson: 1c17:43
- Lesson: 1d29:52
- Lesson: 1e43:53
- Lesson: 2a1:34
- Lesson: 2b4:37
- Lesson: 2c3:21
- Lesson: 2d11:24
- Lesson: 2e22:30
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) to determine the right-end behavior;
evaluate limx→−∞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)
find the critical numbers:
• use the First Derivative Test to find: intervals of increase/decrease and local extrema.
e) Computef′′(x) • inflection points occur where the direction of concavity changes.
find possible inflection points by equating thef′′(x) to 0.
•Concavity Test:
•inflection points occur where the direction of concavity changes.


- 1.Use the guidelines to sketch the graph of:
f(x)=x3+8x3−8a)Domainb)Interceptsc)Asymptotesd)Compute f′(x): - critical numbers - intervals of increase/decrease - local extremae)Compute f′′(x): - possible inflection points - intervals of concavity - verify inflection points - 2.Use the guidelines to sketch the graph of: f(x)=−x3−6x2−9xa)Domainb)Interceptsc)Asymptotesd)Compute f′(x): - critical numbers - intervals of increase/decrease - local extremae)Compute f′′(x): - possible inflection points - intervals of concavity - verify inflection points