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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.

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Average Rate of Change vs.

Instantaneous Rate of Change

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Average Rate of Change(Difference Quotient)

We Need We Need Some Some NotationNotation

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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)

2 1 1 1sec

2 1

( ) ( ) ( ) ( )f x f x f x x f xym

x x x x x x

1 1 1 1sec

( ) ( ) ( ) ( )f x x f x f x h f xm

x h

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Instantaneous Rate of Change(Limit of Difference Quotient)

2 1tan sec

0 0 0

( ) ( ) lim lim lim

x x x

f x f xym m

x x

1 1

0 0

( ) ( ) ( ) ( )lim limx h

f x x f x f x h f x

x h

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ExampleSuppose 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:Something to think about:

How long will it take a free How long will it take a free falling object to reach a falling object to reach a velocity of 200 ft / sec?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.