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Introduction to Second Derivatives And Beyond - At A Glance:

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 mix, 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 f (x) = -4x3 + 3x + 2.


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.

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