Related RatesRelated Rates

TS: Explicitly assessing information and TS: Explicitly assessing information and drawing conclusions.drawing conclusions.

ObjectiveObjective

To solve related rate problems To solve related rate problems using implicit differentiation. using implicit differentiation.

Implicit DifferentiationImplicit Differentiation2 3Find for 2 3dy

dx x y x y 2x 23 dy

dxy 3y (2 )x

2 23 dydxx y 32xy

2 2(3 1)dydx x y

dydx

2 dydx 0

dydx 2

32 2xy

32( 1)xy 2 23 1x y

Related RatesRelated Rates

One of the applications of mathematical One of the applications of mathematical modeling with calculus involves related rates modeling with calculus involves related rates word problems.word problems.

Related rates problems involve finding a rate Related rates problems involve finding a rate at which a quantity changes, by relating that at which a quantity changes, by relating that quantity to another quantity whose rate of quantity to another quantity whose rate of change is known.change is known.

The rate of change is usually with respect to The rate of change is usually with respect to time.time.

Related RatesRelated Rates

Procedure for Solving Related Rates ProblemsProcedure for Solving Related Rates Problems1.1. Draw a diagram, if applicable, to visualize the Draw a diagram, if applicable, to visualize the

problem. problem. 2.2. Organize all information into a table. Organize all information into a table. 3.3. Write an equation that relates the variables Write an equation that relates the variables

to one another. If possible solve for unknown to one another. If possible solve for unknown values of the variables. values of the variables.

4.4. Use implicit differentiation to differentiate Use implicit differentiation to differentiate each side of the equation with respect to each side of the equation with respect to time. time.

5.5. Substitute known values and solve for the Substitute known values and solve for the unknown. unknown.

Ripple ProblemRipple Problem

A stone is dropped into Lake Erie, causing A stone is dropped into Lake Erie, causing circular ripples whose radii increase by 2 circular ripples whose radii increase by 2 meters/second. At what rate is the disturbed meters/second. At what rate is the disturbed area growing when the outer ripple has area growing when the outer ripple has radius 5 meters?radius 5 meters?

Ladder ProblemLadder Problem

A 13-foot ladder is leaning against a house. A 13-foot ladder is leaning against a house. The bottom of the ladder is pulled away from The bottom of the ladder is pulled away from the house at a rate of 6 feet per second. the house at a rate of 6 feet per second. How fast is the top of the ladder falling down How fast is the top of the ladder falling down the wall when the bottom of the ladder is 12 the wall when the bottom of the ladder is 12 feet from the house? feet from the house?

Baseball ProblemBaseball Problem

After hitting a After hitting a baseball, a batter runs baseball, a batter runs toward first base at a toward first base at a rate of 24 feet per rate of 24 feet per second. How fast is second. How fast is the distance between the distance between second base and the second base and the batter changing at the batter changing at the instant that the batter instant that the batter is midway between is midway between home and first base?home and first base?

Ladder ProblemLadder Problem

Joey is perched at the Joey is perched at the top of a 10-foot ladder top of a 10-foot ladder leaning against the back leaning against the back wall of an apartment wall of an apartment building (spying on an building (spying on an enemy of his) when it enemy of his) when it starts to slide down the starts to slide down the wall at a rate of 4 ft per wall at a rate of 4 ft per minute. Joey's minute. Joey's accomplice, Lou, is accomplice, Lou, is standing on the ground standing on the ground 6 ft. away from the wall. 6 ft. away from the wall. How fast is the base of How fast is the base of the ladder moving when the ladder moving when it hits Lou? it hits Lou?