Related Rates ES: Explicitly assessing information and drawing conclusions.

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

Implicit Differentiation

Related Rates One of the applications of mathematical modeling with calculus involves related rates word problems. One of the applications of mathematical modeling with calculus involves related rates word problems. Related rates problems involve finding a rate at which a quantity changes, by relating that quantity to another quantity whose rate of change is known. Related rates problems involve finding a rate at which a quantity changes, by relating that quantity to another quantity whose rate of change is known. The rate of change is usually with respect to time. The rate of change is usually with respect to time.

Related Rates Procedure for Solving Related Rates Problems 1.Draw a diagram, if applicable, to visualize the problem. 2.Organize all information into a table. 3.Write an equation that relates the variables to one another. If possible solve for unknown values of the variables. 4.Use implicit differentiation to differentiate each side of the equation with respect to time. 5.Substitute known values and solve for the unknown.

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

r Models Original Shape/Scenario Original is moving over time dA/dt is how fast area is changing over time. dr/dt is how fast the radius is changing over time dA/dt? r5 m dr/dt2 m/s

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

Models Original Shape/Scenario Original is moving over time da/dt is how fast the base is moving away. db/dt is how fast the top is falling down. a12 ft b5 ft da/dt6 ft/s db/dt? 13ft a b

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

Models Original Shape/Scenario Original is moving over time da/dt is how fast the distance to first is decreasing. db/dt is how fast the distance to second is decreasing. a45 ft b45√5 ft da/dt-24 ft/s db/dt? 90ft a b

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

Models Original Shape/Scenario Original is moving over time da/dt is how fast the base is moving away. db/dt is how fast the top is falling down. a6 ft b8 ft db/dt-4 ft/min da/dt? 10ft a b