Still Confused?

Try reviewing these fundamentals first.

- Home
- Calculus 1
- Applications of differentiation

Still Confused?

Try reviewing these fundamentals first.

Nope, I got it.

That's that last lesson.

Start now and get better math marks!

Get Started NowStart now and get better math marks!

Get Started NowStart now and get better math marks!

Get Started NowStart now and get better math marks!

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

We have over 170 practice questions in Calculus 1 for you to master.

Get Started Now