UNIT 3 — APPLICATIONS OF DERIVATIVES (Sections 5.1–5.4)

Now we use derivatives to understand graphs.

The derivative tells us what direction a graph is moving.

1. Increasing and Decreasing

Imagine walking up or down a hill.

If you are walking uphill:

the function is increasing.

If you are walking downhill:

the function is decreasing.

Using the First Derivative

If:

f'(x) > 0

the function is increasing.

If:

f'(x) < 0

the function is decreasing.

Example

Suppose:

f(x) = x^2

Derivative:

f'(x) = 2x

When x > 0:

2x > 0

So the function is increasing.

When x < 0:

2x < 0

So the function is decreasing.

Therefore:

Decreasing on (-infinity, 0)

Increasing on (0, infinity)

2. Critical Numbers

A critical number occurs when:

f'(x) = 0

or when:

f'(x) does not exist

provided x is in the domain of the function.

Critical numbers are important because maximums and minimums can occur there.

3. Relative Maximum

A relative maximum is like the top of a hill.

The function increases before the point and decreases afterward.

Derivative changes:

positive -> negative

4. Relative Minimum

A relative minimum is like the bottom of a valley.

The function decreases before the point and increases afterward.

Derivative changes:

negative -> positive

First Derivative Test

To find relative extrema:

Step 1

Find f'(x).

Step 2

Set:

f'(x) = 0

Step 3

Find critical numbers.

Step 4

Test the sign of f'(x) on both sides.

If:

positive -> negative

you have a relative maximum.

If:

negative -> positive

you have a relative minimum.

5. Concavity

Concavity tells us how the graph bends.

Imagine holding a smile:

:)

That graph shape is concave up.

Imagine holding a frown:

:(

That graph shape is concave down.

Concave Up

If:

f''(x) > 0

the graph is concave up.

Think:

"smile."

Concave Down

If:

f''(x) < 0

the graph is concave down.

Think:

"frown."

6. Inflection Points

An inflection point is where the graph changes concavity.

For example:

concave up -> concave down

or:

concave down -> concave up

A possible inflection point occurs when:

f''(x) = 0

But simply getting f''(x) = 0 is not enough.

The concavity must actually change.

7. Second Derivative Test

The second derivative can help classify a critical point.

Suppose:

f'(c) = 0

Then:

If f''(c) > 0:

relative minimum

If f''(c) < 0:

relative maximum

If f''(c) = 0:

the test does not tell us.

Example

f(x) = x^2

First derivative:

f'(x) = 2x

Set equal to zero:

2x = 0

x = 0

Second derivative:

f''(x) = 2

Since:

f''(0) = 2 > 0

there is a relative minimum at x = 0.

8. Curve Sketching

When sketching a graph, derivatives give us clues.

You can think like a detective.

Step 1

Find important points.

Step 2

Find where f'(x) = 0 or undefined.

Step 3

Determine increasing/decreasing behavior.

Step 4

Find f''(x).

Step 5

Determine concavity.

Step 6

Find possible inflection points.

Step 7

Plot important points.

Step 8

Connect everything using the increasing/decreasing and concavity information.

9. What Derivatives Tell Us

First derivative:

f'(x)

tells us:

• slope

• increasing/decreasing

• possible maximum/minimum

Second derivative:

f''(x)

tells us:

• concavity

• possible inflection points

• helps classify extrema