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- Applications of differentiation

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Get Started Now- 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

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 $lim_{x \to \infty } f(x)$ to determine the right-end behavior; evaluate $lim_{x \to -\infty } f(x)$ 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) Compute$f' (x)$

find the critical numbers:

• use the First Derivative Test to find: intervals of increase/decrease and local extrema.

e) Compute$f'' (x)$

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

•Concavity Test:

•inflection points occur where the direction of concavity changes.

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 $lim_{x \to \infty } f(x)$ to determine the right-end behavior; evaluate $lim_{x \to -\infty } f(x)$ 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) Compute$f' (x)$

find the critical numbers:

• use the First Derivative Test to find: intervals of increase/decrease and local extrema.

e) Compute$f'' (x)$

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

•Concavity Test:

•inflection points occur where the direction of concavity changes.

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

$f(x)=\frac{x^3-8}{x^3+8}$a)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:

$f(x)=-x^3-6x^2-9x$a)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

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