1. AP CALCULUS RELATED RATES
Related rate problems involve finding the ________ at which some variable changes.
rate

2. AP CALCULUS RELATED RATES
For example, when a balloon is being blown up with air, both the _____________ and the ________________ of the balloon are changing.
radius
volume

3. AP CALCULUS RELATED RATES
In each case the rate is a ___________ that has to be computed given the rate at which some other variable, like time, is known to change.
derivative

4. AP CALCULUS RELATED RATES
To find this derivative we write an equation that relates the two variables. We then ____________
both sides of the equation with respect to ________ to express the derivative we SEEK in terms
of the derivative we KNOW.
differentiate
time

5. AP CALCULUS RELATED RATES
Often the key to relating the variables in this type of problem is DRAWING A PICTURE that shows the geometric relationships between the variables.

6. Procedure:
Identify and LABEL all the given info and what you are asked to find. Draw a picture if appropriate.
Write an EQUATION relating the variables.
Differentiate both sides of the equation with respect to TIME.
Substitute and Solve. Sometimes you will need to use the original equation or other equations to solve for missing parts.

7. BASIC SKILL: Draw a sketch and differentiate basic geometry formulas with respect to time.
Let A be the Area of a circle of radius r. How is dA/dt related to dr/dt?

8. BASIC SKILL: Draw a sketch and differentiate basic geometry formulas with respect to time.
Let V be the Volume of a cube of side length x. How is dV/dt related to dx/dt?

9. BASIC SKILL: Draw a sketch and differentiate basic geometry formulas with respect to time.
Let V be the Volume of a sphere of radius r. How is dV/dt related to dr/dt?

10. Obtain the related-rate equation

11. Two Rates That Are Related
Suppose x and y are both differentiable functions of t and are related by the equation Find dy/dt when x=1, given that dx/dt=2.

12. Ripples in a Pond
A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 ft/sec. When the radius is 4 ft, at what rate is the total area A of the disturbed water changing?

13. An Inflating Balloon
Air is being pumped into a spherical balloon at a rate of 4.5 cubic ft/min. Find the rate of change of the radius when the radius is 2 ft.