1. Related Rates
Lesson 6.5

2. 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

3. 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 )

4. 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

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

6. 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

7. 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

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

9. Example
Consider a particle traveling in a circular pattern

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

11. Example Now substitute in the things we know Find other values we need
when x = 3, y2 = and y = 4

12. Example
Result

13. 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? •

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

15. 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?

16. 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

17. Assignment
Lesson 6.5
Page 409
Exercises 1 – 27 odd