2.6 Related Rates

𝑦=𝑥2+1

Application of Implicit Differentiation

Suppose x and y are both changing with respect to time. Take derivative with respect to .

𝑑𝑑𝑡

(𝑦 )=𝑑𝑑𝑡

(𝑥¿¿2+1)¿

Related Rate Problems

• Cool application of calculus!

• Problems in which function is changing with respect to time

• Recall: A rate of change is a derivative! • Example: If we want to find the rate at which

height is changing, let represent this rate.

General Steps for solving a Related Rate problem

Set up: Draw picture/ Label now – what values do we know are fixed? ind – what are we looking for and when? quation – relate the variables ifferentiate – both sides w.r.t. independent variable (time) ubstitute/Solve

KFED

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Example 1: Volume of sphere

Suppose a “spherical” balloon is being inflated so that its radius is changing at 3 cm/sec. How fast is the volume changing when the radius is 20 cm?

Example 2: “Triangular” motion

Suppose a 25 ft. ladder is leaning against a wall. The foot of the ladder is being pulled away from the bottom of the wall at a rate of 14 ft. sec. At what rate is the top of the ladder moving down the wall when it is 7 ft. above the ground?

Example 3: Distance= (Rate)(Time)

Two cars leave an intersection at the same time, one traveling north at 30 miles/hr. and the other going east at 40 miles/hr. How fast is the distance between them changing 30 minutes later?

Example 4: Right Circular Cone

Gravel is being deposited from a conveyor onto a pile and forms a right circular cone. The ratio of height to radius is 3:2. The conveyor delivers 20 to the pile. a) How fast is the height of the pile increasing when its radius is 10 ft? b) How fast is the radius increasing 15 min. after the operation begins?

Example 5: Angular Motion

A winch is pulling a boat into the dock that is 20 feet above the boat’s bow. If the rope is hauled in at a rate of 2 ft/sec, how fast is the boat approaching the dock when there is 22 feet of rope still out?

Example 5: Angular Motion

b) How fast is the angle between the rope and the water changing when the rope reaches 20 feet?

Example 6: filling a cone

A right circular cone with radius 4 cm and height 16 cm is being filled at 2 Find the rate of change of the height when the water reaches 10 cm?

2.6p. 153 #5, 7, 11, 13, 15, 21, 25