Related Rates Lesson 6.5

General vs. Specific Note the contrast … General situation properties true at every instant of time Specific situation properties true only at a particular instant of time We will consider a rock dropped into a pond … generating an expanding ripple 2

Expanding Ripple At the point in time when r = 8 radius is increasing at 3 in/sec That is we are given We seek the rate that the area is changing at that specific time We want to know 3 r = 8

Solution Strategy 1. Draw a figure label with variables do NOT assign exact values unless they never change in the problem 2. Find formulas that relate the variables 4 A r

Solution Strategy 3. Differentiate the equation with respect to time 4. Substitute in the given information 5

Example Given Find when x = 3 Note: we must differentiate implicitly with respect to t 6

Example Now substitute in the things we know x = 3 Find other values we need when x = 3, y 2 = 25 and y = 4 7

Example Result 8

Particle on a Parabola Consider a particle moving on a parabola y 2 = 4x at (1,-2) Its horizontal velocity (rate of change of x) is 3ft/sec What is the vertical velocity, the rate of change of y? 9

Particle on a Parabola Differentiate the original equation implicitly with respect to t Substitute in the values known Solve for dy/dt 10

Draining Water Tank Radius = 20, Height = 40 The flow rate = 80 gallons/min What is the rate of change of the radius when the height = 12? 11

Draining Water Tank At this point in time the height is fixed Differentiate implicitly with respect to t, Substitute in known values Solve for dr/dt 12

Assignment Lesson 6.5 Page 409 Exercises 1 – 27 odd 13