- Topics At a Glance
- Second Derivatives via Formulas
- Third Derivatives and Beyond
- Concavity
- Concave Up
- Concave Down
- No Concavity
- Special Points
- Critical Points
- Points of Inflection
- Extreme Points and How to Find Them
- Finding & Classifying Extreme Points
- First Derivative Test
- Second Derivative Test
- Local vs. Global Points
**Using Derivatives to Draw Graphs**- Finding Points
**Finding Shapes**- Connecting the Dots
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

We talked earlier about how we can tell from the derivative whether the original function is increasing or decreasing. We also talked about concavity), and how if we toss the second derivative into the mi*x*, we can also tell what shape the original function should have.

We need to go over the shapes again here, because the more fluent you are at translating between signs of derivatives and shapes of graphs, the easier this whole business will be.

There are four possible shapes a piece of graph can have. Actually, this is a lie - there are seven possible shapes, but three of them don't need calculus.

Here are the ones that do need calculus:

• *increasing and concave up*

If *f* is increasing and looks like part of a right-side up bowl, it looks like this:

• *increasing and concave down*

If *f* is increasing and looks like part of a upside-down bowl, it looks like this:

• *decreasing and concave up*

If *f* is decreasing and looks like part of a right-side up bowl, it looks like this:

• *decreasing and concave down*

If *f* is decreasing and looks like part of a upside-down bowl, it looks like this:

The shapes that don't need calculus are the ones where *f* has no concavity, but is a straight line. If *f* is a line, it can have three possible shapes:

• *increasing:*

• *decreasing:*

• *constant:*

Example 1

Find the shapes of the function |

Exercise 1

Given the signs of *f*' and *f*", sketch the shape of the function *f*.

The sign of *f*' is - .

The sign of *f*" is + .

Exercise 2

Given the signs of *f*' and *f*", sketch the shape of the function *f*.

The sign of *f*' is 0 .

The sign of *f*" is 0 .

Exercise 3

Given the signs of *f*' and *f*", sketch the shape of the function *f*.

The sign of *f*' is + .

The sign of *f*" is 0 .

Exercise 4

Given the signs of *f*' and *f*", sketch the shape of the function *f*.

The sign of *f*' is + .

The sign of *f*" is - .

Exercise 5

Given the signs of *f*' and *f*", sketch the shape of the function *f*.

The sign of *f*' is + .

The sign of *f*" is + .

Exercise 6

Given the signs of *f*' and *f*", sketch the shape of the function *f*.

The sign of *f*' is - .

The sign of *f*" is 0 .

Exercise 7

Given the signs of *f*' and *f*", sketch the shape of the function *f*.

The sign of *f*' is - .

The sign of *f*" is - .

Exercise 8

Given the signs of *f*' and *f*", sketch the shape of the function *f*.

The sign of *f*' is 0.

The sign of *f*" is 0.