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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. - PowerPoint PPT Presentation
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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 whenr = 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 = 83dr
dt
dA
dt
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
Ar
2A r 3dr
dt
Solution Strategy
3. Differentiate the equation with respect to time
4. Substitute in the given information
5
2dA dr
rdt dt
8
3
r
dr
dt
22 8 3 48 in / sec
Example
Given
Find when x = 3
Note: we must differentiate implicitly with respect to t
6
2 2 25 4dx
x ydt
dy
dt
2 2 0dx dyx ydt dt
Example
Now substitute in the things we know
• x = 3
Find other values we need• when x = 3,
32 + y2 = 25 and y = 4
7
2 2 0dx dyx ydt dt
4dx
dt
Example
Result
8
2 2 0dx dyx ydt dt
2 4 4 2 4 0
324
8
dy
dtdy
dt
Particle on a Parabola
Consider a particle movingon a parabola y2 = 4x at (1,-2)
Its horizontal velocity(rate of changeof x) is 3ft/sec
What is the vertical velocity,the rate of change of y?
9
•
3dx
dt
??dy
dt
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
21
3Volume r h
80dV
dt
??dr
dt
Draining Water Tank
At this point in timethe height is fixed
Differentiate implicitly with respect to t, Substitute in known valuesSolve for dr/dt
12
2112
3Volume r
12 12
3
dV drr
dt dt
Assignment
Lesson 6.5
Page 409
Exercises 1 – 27 odd
13