Two kinds of rate of change

Q: A car travels 110 miles in 2 hours. What’s its average rate of change (speed)?

A: 110/2 = 55 mi/hr. That is, if we drive 55 miles in an hour, then in 2 hours, we will have driven 110 miles.

Q: If you are driving and suddenly look at your odometer, which says 60 mi/hr, what kind of rate of change is that?

A: Instantaneous R.O.C. That is, the rate at that particular time instance.

Average R.O.C. is over a period of timeInstantaneous R.O.C. is at a given point of time.

Average rate of change

Height (feet)

A rocket is shot straight up, given bythe function f(x) = -16x^2+128x, where x = time, f(x) = height at time x.

8

P

Q

Time (seconds)0 1 74 5

Q: What is the average speed between P (at x = 4) and Q (at x = 7)?

1. Find the slope of the secant line betweenP and Q.

2. Find the coordinates of P and Q. For P, f(4) = -16(4)^2+128(4) = 256, P(4,256). For Q, f(7) = -16(7)^2+128(7) = 112, so Q(7,112).3. Slope = (112-256)/(7-4) = -48 ft/sec

Instantaneous rate of change

Height (feet)

Q: What is the average R.O.C as x changes from 4 to 6?

Q: What is the average R.O.C as x changes from 4 to 5?

Q: What is the average R.O.C as x changes from 4 to 4.001?

P

Time (seconds)0 81 64 5

What do you think the instantaneous R.O.C would be at x = 4?

How fast is the rocket moving at preciselyX = 4 seconds?

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46)4()6(

ff

161256240

45)4()5(

ff

016.0001.0

256999984.2554001.4

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DefinitionAverage Rate of Change of a quantity over a period of time is the amount of change divided by the time required for the change.

Example12

10

8

6

4

2

5 10

time in hours

dist

ance

in

mile

s

0 7

250

Average speed is 35.7 mph measured over 7 hours

Important IdeaThe average rate of change (speed) over a time period is the slope of the secant line connecting the beginning and end of the time period. y

t

Average Rate of Change

2 1

2 1

y yt t

Try ThisDescribe in words how you could find the speed at exactly the 5th hour.

12

10

8

6

4

2

5 10

time in hours

dist

ance

in

mile

s0 5

SolutionThe instantaneous velocity at exactly the 5th hour is the slope of the line tangent to the velocity function at t=5.

Important IdeaThe instantaneous velocity at a point, or any other rate of change, is the slope of the tangent line at the point

212 o os gt v t s

The derivative can be used to determine the rate of change of one variable with respect to another.

Ex: Population growth, production rates, rate of water flow, velocity and acceleration.

Ex: Free fall Position function. A function, s, that gives position (relative to the origin) of an object as a function of time.

A ball dropped from a 160 foot building:

Find average velocity over each time interval.

Therefore, the average velocity is

tanchange in dis ce ychange in time x

[1, 2] [1,1.5] [1,1.1]

Negative velocity indicates object is falling

Find instantaneous velocity when t=1.1 sec

Generally if, s = s(t) is the position for an object moving in a straight line, then the velocity of the object at time t is:

0

( ) ( )( ) lim '( )t

s t t s tv t s tt

'( ) ( ) 32 os t v t t v

2( ) 16 o os t t v t s

Position Function

Velocity Function

Acceleration Function''( ) '( ) ( ) 32s t v t a t