UNIT 6 — RELATED RATES (Section 6.4)

Related rates are about things that change together.

For example:

• A balloon gets bigger as air enters.

• A ladder slides down a wall.

• A shadow changes length as a person walks.

• A circle grows as its radius increases.

1. The Big Idea

Two or more quantities are changing with time.

We use derivatives to connect their rates of change.

2. Drawing a Diagram

Always draw a picture when possible.

Label:

• known quantities

• unknown quantities

• changing quantities

This makes the problem much easier.

3. Common Notation

If x changes with time:

dx/dt

means:

"how fast x is changing."

If y changes with time:

dy/dt

means:

"how fast y is changing."

4. Related Rates Steps

Step 1

Draw a picture.

Step 2

Identify what you know.

Step 3

Identify what you want.

Step 4

Write an equation connecting the variables.

Step 5

Differentiate both sides with respect to time.

Step 6

Plug in the known values.

Step 7

Solve for the unknown rate.

5. Example: Growing Circle

Area of a circle:

A = pi r^2

Suppose the radius grows at:

dr/dt = 2 cm/s

Find how quickly the area changes when:

r = 3 cm

Step 1

Start with:

A = pi r^2

Step 2

Differentiate with respect to time:

dA/dt = 2pi r dr/dt

Step 3

Plug in:

r = 3

dr/dt = 2

Therefore:

dA/dt = 2pi(3)(2)

= 12pi

Answer:

dA/dt = 12pi cm^2/s

6. Why Do We Not Plug In First?

If we plug in the values before differentiating, we may accidentally turn changing variables into constants.

Differentiate first.

Then substitute.

This is one of the most important habits in related rates.