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 And dx/dt (Specific values at a point in time) (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
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 We seek the rate that the area is changing at that specific time We want to know r = 8
Solution Strategy Draw a figure label with variables do NOT assign exact values unless they never change in the problem Find formulas that relate the variables A r
Solution Strategy Differentiate the equation with respect to time Substitute in the given information
Example Consider a particle traveling in a circular pattern
Example Given Find when x = 3 Note: we must differentiate implicitly with respect to t
Example Now substitute in the things we know Find other values we need when x = 3, 32 + y2 = 25 and y = 4
Example Result
Particle on a Parabola Consider a particle moving on a parabola y2 = 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? •
Particle on a Parabola Differentiate the original equation implicitly with respect to t Substitute in the values known Solve for dy/dt
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?
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
Assignment Lesson 6.5 Page 409 Exercises 1 – 27 odd