MODELING WITH RATES OF CHANGE

Section 2.4b

The “Do Now”Find the slope of the given curve at x = a.

3

5

xy

x

0limh

y a h y a

h

Slope:

0

3 3

5 5limh

a h a

a h ah

0

3 5 3 5 1lim

5 5h

a h a a a h

a h a h

2 2

0

5 5 3 15 5 3 3 15lim

5 5h

a a ah h a a ah a a h

h a h a

The “Do Now”Find the slope of the given curve at x = a.

3

5

xy

x

0

8lim

5 5h

h

h a h a

0

8lim

5 5h a h a

28

5a

Bonus Question: What is the equationof the tangent to this curve at x = –2?

8 1

9 9y x

From physics, with an object in free-fall on Earth, theposition function is given as 216y t

(y is feet fallen after t seconds)

A body’s average speed along a given axis for a givenperiod of time is the average rate of change of thisposition function.

Its instantaneous speed at any time t is theinstantaneous rate of change of its position with respectto time at time t :

0

limh

f t h f t

h

Guided PracticeFind the speed of a falling rock (acted on by Earth’s gravityonly) at t = 1 sec.

Position function of the rock: 216f t tAverage speed of the rock over the interval between t = 1and t = 1 + h sec:

1 1f h f

h

2 216 1 16 1h

h

216 2h h

h

16 2h

Guided PracticeFind the speed of a falling rock (acted on by Earth’s gravityonly) at t = 1 sec.

Position function of the rock: 216f t t

The rock’s speed at the instant t = 1:

0

lim16 2h

h

ft/sec32

Average speed of the rock over the interval between t = 1and t = 1 + h sec: 16 2h

Guided PracticeAt t sec after lift-off, the height of a rocket is ft. Howfast is the rocket climbing after 10 sec?

23t

Let 23f t t

0

10 10limh

f h f

h

This is the position function for therocket… we seek the instantaneousrate of change of this function at t = 10…

2

0

300 60 3 300limh

h h

h

0lim 60 3h

h

60The rocket’s speed at t = 10 sec is 60 ft/sec

Speed:

Guided PracticeWhat is the rate of change of the volume of a sphere withrespect to the radius when the radius is r = 2 in.?

We need a function for thevolume with respect to the radius: 34

3V f r r

0

2 2limh

f h f

h

3 3

0

4 3 2 4 3 2limh

h

h

2 3

0

4 8 12 6 8lim

3 h

h h h

h

Rate of change of this function:

Guided PracticeWhat is the rate of change of the volume of a sphere withrespect to the radius when the radius is r = 2 in.?

We need a function for thevolume with respect to the radius: 34

3V f r r

2 3

0

4 8 12 6 8lim

3 h

h h h

h

20

4lim 12 6

3 hh h

16The volume is changingat a rate of 16 in perinch of radius

Rate of change of this function:

412

3 3

Guided PracticeAt what point is the tangent to the given function horizontal?

23 4f x x x First, find the slope of the tangent at x = a:

0

limh

f a h f am

h

2 2

0

3 4 3 4limh

a h a h a a

h

2 2 2

0

3 4 4 2 3 4limh

a h a ah h a a

h

Guided PracticeAt what point is the tangent to the given function horizontal?

23 4f x x x First, find the slope of the tangent at x = a:

2

0

4 2limh

h ah h

h

0lim 4 2h

a h

4 2a

The tangent at x = a is horizontal where the slope is zero:

4 2 0a 2a The tangent line is horizontal at the point:

2, 2f 2,7 Can we support this answer graphically???

Guided PracticeFind the equations of all lines tangent to thatpass through the point (1, 12).

29y x

First, sketch a graph of this situation.

2 2

0

9 9limh

a h a

h

Slope of the curve at x = a:

2 2 2

0

9 2 9limh

a ah h a

h

2

0

2limh

ah h

h

0lim 2h

a h

2a

Guided PracticeFind the equations of all lines tangent to thatpass through the point (1, 12).

29y x

1,12Our points:

2,9a a 2aSlope of the tangentthrough these points:

29 122

1

aa

a

Equation for slope:

29 12 2 1a a a 2 23 2 2a a a 2 2 3 0a a

1 3 0a a 1,3a

Guided PracticeFind the equations of all lines tangent to thatpass through the point (1, 12).

29y x

1a x At , the slope is 2 1 2

12 2 1y x Point-slope form (with point (1, 12)):

2 10y x

3a x At , the slope is 2 3 6

12 6 1y x Point-slope form (with point (1, 12)):

6 18y x

Th

e tw

o t

ang

ent

lines