Related Rates Lesson 6 5 Related Rates Consider

Related Rates Consider a formula in x and y Suppose both x and y

Related Rates We seek the rate of change of y with respect to time

General vs. Specific Note the contrast … General situation • properties true at every

Example We will consider a rock dropped into a pond … generating an expanding

Expanding Ripple At the point in time when r=8 • radius is increasing at

Solution Strategy 1. Draw a figure l label with variables l do NOT assign

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

Example Consider a particle traveling in a circular pattern

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

Example Now substitute in the things we know x=3 • Find other values we

Particle on a Parabola Consider a particle moving on a parabola y 2 =

Particle on a Parabola Differentiate the original equation implicitly with respect to t Substitute

Draining Water Tank Radius = 20, Height = 40 The flow rate = 80

Draining Water Tank At this point in time the height is fixed Differentiate implicitly

Assignment Lesson 6. 5 Page 409 Exercises 1 – 27 odd 17

Slides: 17

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Related Rates Lesson 6. 5

Related Rates Consider a formula in x and y Suppose both x and y are functions of time, t Then it is possible to use implicit differentiation to take the derivative with respect to t

Related Rates We seek the rate of change of y with respect to time dy/dt for a particular x So we need to know • x • y (Specific values at a point in time) • And dx/dt (A general quality )

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 4

Example We will consider a rock dropped into a pond … generating an expanding ripple

Expanding Ripple At the point in time when r=8 • radius is increasing at 3 in/sec • That is we are given r=8 We seek the rate that the area is changing at that specific time • We want to know 6

Solution Strategy 1. Draw a figure l label with variables l do NOT assign exact values A r unless they never change in the problem 2. Find formulas that relate the variables 7

Solution Strategy 3. Differentiate the equation with respect to time 4. Substitute in the given information 8

Example Consider a particle traveling in a circular pattern

Example Given Find when x = 3 Note: we must differentiate implicitly with respect to t 10

Example Now substitute in the things we know x=3 • Find other values we need • when x = 3, 32 + y 2 = 25 y=4 and 11

Example Result 12

Particle on a Parabola Consider a particle moving on a parabola y 2 = 4 x at (1, -2) Its horizontal velocity (rate of change of x) is 3 ft/sec What is the vertical velocity, the rate of change of y? • 13

Particle on a Parabola Differentiate the original equation implicitly with respect to t Substitute in the values known Solve for dy/dt 14

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? 15

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 16

Assignment Lesson 6. 5 Page 409 Exercises 1 – 27 odd 17