Instantaneous Rate of Change
What is Instantaneous Rate of Change?
We need to shift our thinking from “average rate of
change” to “instantaneous rate of change”.
Average rate of change is calculated over an interval,
whereas an instantaneous rate of change is found for a
particular point.
 For example: A car is traveling on a 75 mile trip for 3
hours. What is the average rate of change? What is the
instantaneous rate of change at t = 2?
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Instantaneous Rate of Change
 For example: A car is traveling on a 75 mile trip for 3 hours. The
speed doesn’t remain constant.
2
Average Rate of Change
vs.
Instantaneous Rate of Change
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Average Rate of Change
(Difference Quotient)
We Need
Some
Notation
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Procedure
1. Move x2 Closer to x1, such that Δx gets small.
(if x1 is the point you’re interested in)
2. Continue to take the slopes of the secants over
smaller intervals.
3. The limit of the slopes of the secants become the
slope of the tangent line at x1, such that Δx
becomes zero.
Limit of
Secants Demo
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Average Rate of Change
(Difference Quotient)
msec
y f ( x2 )  f ( x1 ) f ( x1  x)  f ( x1 )



x
x2  x1
x  x  x
msec
f ( x1  x)  f ( x1 ) f ( x1  h)  f ( x1 )


x
h
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Instantaneous Rate of Change
(Limit of Difference Quotient)
mtan  lim msec
x 0
 lim
x 0
f ( x2 )  f ( x1 )
y
 lim
 lim
x
x 0 x
x 0
f ( x1  x)  f ( x1 )
f ( x  h)  f ( x )
 lim
x
h
h 0
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Example
Suppose that a ball is dropped from a tower. By Galileo's law
the distance fallen by any freely falling body is expressed by
the equation s(t) =16t2 where s(t) is in feet and t is in
seconds.
(a) Find the average velocity between t = 1and t = 2.
(b) Find the instantaneous velocity at time t = 1 and t = 2.
Something to think about:
How long will it take a free
falling object to reach a
velocity of 200 ft / sec?
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Approximation
Suppose you want to find the instantaneous rate of change of
f(x) = 1.2x + 2x at x = 4
•
Over short intervals of time, the average rate of
change is approximately equal to the instantaneous
rate of change.
•
You could use a very small interval and then calculate
the average rate of change.
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