1

Powerpoint slides copied from or based upon:

Connally,

Hughes-Hallett,

Gleason, Et Al.

Copyright 2007 John Wiley & Sons, Inc.

Functions Modeling Change

A Preparation for Calculus

Third Edition

INPUT AND OUTPUT

Chapter 2Section 1

2

You will remember the following problem from Chapter 1, Section 1:

3Page 62

The number of gallons of paint needed to paint a house depends on the size of the house. A gallon of paint typically covers 250 square feet.

Thus, the number of gallons of paint, n, is a function of the area to be painted, A ft2. We write n = f(A).

Which is the output and which is the input for:

n = f(A) ?4Page 62

A reminder from Chapter 1:

Output = f(Input) 

Or:

Dependent = f(Independent)

5Page 4

Which is the output and which is the input for:

n = f(A) ?

6Page 62

Which is the output and which is the input for:

n = f(A) ?

n=f(A) => output, A => input

7Page 62

n=f(A) => output, A => input

For example, f(20,000) represents ?

8Page 62

n=f(A) => output, A => input

f(20,000) represents the # of gallons of paint to cover a house of 20,000 sq ft.

(ft2)

9Page 62

Using the fact that 1 gallon of paint covers 250 ft2, evaluate the expression f(20,000).

10Page 62 Example 1

( )n f A

11Page 62

2

2

( )

250

n f A

Aftn

ft

gal

12Page 62

2

2

2

2

( )

250

( )

250

n f A

Aftn

ft

gal

Aftf A

ft

gal

13Page 62

2

2

2

2

2

2

( )

250

( )

250

20,000(20,000) ?

250

n f A

Aftn

ft

gal

Aftf A

ftgal

ftf

ft

gal

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2

2

2

2

2

2

( )

250

( )

250

20,000(20,000) 80

250

n f A

Aftn

ft

gal

Aftf A

ft

gal

ftf gallons

ft

gal

15Page 62

Area of a circle of radius r: A = q(r) = πr2.

 Use the formula to evaluate q(10) and q(20).

What do your results tell you about circles?

16Page 62 Example 2

Area of a circle of radius r: A = q(r) = πr2.

 Use the formula to evaluate q(10) and q(20).

2(10) (10)q

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Area of a circle of radius r: A = q(r) = πr2.

 Use the formula to evaluate q(10) and q(20).

2(10) (10)

(10) (100) 100 314.159

q

q

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Area of a circle of radius r: A = q(r) = πr2.

 Use the formula to evaluate q(10) and q(20).

2(20) (20)q

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Area of a circle of radius r: A = q(r) = πr2.

 Use the formula to evaluate q(10) and q(20).

2(20) (20)

(20) (400) 400 1256.637

q

q

20Page 62

Area of a circle of radius r: A = q(r) = πr2.

 What do your results tell you about circles?

21Page N/A

Area of a circle of radius r: A = q(r) = πr2.

 What do your results tell you about circles?

If we increase the radius by 2x (factor of 2), we increase the Area by 4x (factor of 4).

Or, we double r we quadruple A. 22Page N/A

Let:

Evaluate: g(3), g(-1), g(a)

2 1( )

5

xg x

x

23Page 62 Example 3

g(3):

23 1 10(3) 1.25

5 3 8g

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g(-1):

2( 1) 1 2( 1) .50

5 ( 1) 4g

25Page 62

g(a):

2 1( )

5

ag a

a

26Page 62

Let h(x) = x2 − 3x + 5. Evaluate and simplify the following expressions. (a)  h(2)(b)  h(a − 2)(c) h(a) − 2(d)  h(a) − h(2)

27Page 63 Example 4

h(2):

2( ) 3 5h x x x

28Page 63

h(2):

2

2

( ) 3 5

(2) 2 3(2) 5

4 6 5

3

h x x x

h

29Page 63

h(a-2):

2( ) 3 5h x x x

30Page 63

h(a-2):

2

2

2

2

( ) 3 5

( 2) ( 2) 3( 2) 5

4 4 3 6 5

7 15

h x x x

h a a a

a a a

a a

31Page 63

h(a)-2:

2( ) 3 5h x x x

32Page 63

h(a)-2:

2

2

2

( ) 3 5

( ) 2 3 5 2

3 3

h x x x

h a a a

a a

33Page 63

h(a)-h(2):

2( ) 3 5h x x x

34Page 63

h(a)-h(2):

2

2 2

2

2

2

( ) 3 5

( ) (2) ( 3 5) (2 3(2) 5)

( 3 5) (4 6 5)

( 3 5) 3

3 2

h x x x

h a h a a

a a

a a

a a

35Page 63

Finding Input Values: Solving Equations

Given an input, we evaluate the function to find the output. (Input Output)

Sometimes the situation is reversed; we know the output and we want to find the corresponding input. (Output Input)

36Page 63

Back to the "Cricket" function, but now

if T = 76, R = ?

140

4T R

37Page 63 Example 5

140

41

76 4041

76 4041

364

144

T R

R

R

R

R

38Page 63

Area of a circle of radius r (cm.): A = q(r) = πr2.

 What is the radius of a circle whose area is 100 cm2?

39Page64 Example 7

2

2

2

2

2

( )

100

100

31.83098862

31.83098862

5.641895835 Are we done?

A q r r

r

r

r

r

r

40Page 64

5.641895835

5.641895835

r

r

Since a circle CAN'T have a negative radius, we conclude:

41Page 64

Finding Output and Input Values from Tables and Graphs

42Page 64

Table 2.1 shows the revenue, R = f(t), received or expected, by the National Football League,1 NFL, from network TV as a function of the year, t, since 1975.

(a) Evaluate and interpret f(25).(b) Solve and interpret f(t) = 1159.

43Page 64 Example 8

R = f(t)

(a) Evaluate and interpret f(25). (b) Solve and interpret f(t) = 1159.

Year, t (since 1975)

0 5 10 15 20 25 30

Revenue, R(million $)

201

364

651

1075

1159

2200

2200

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R = f(t)

(a) Evaluate and interpret f(25).

f(25) = 2200.

Therefore, in 2000 (1975+25), revenue was $2,200 million.

Year, t (since 1975)

0 5 10 15 20 25 30

Revenue, R(million $)

201

364

651

1075

1159

2200

2200

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R = f(t)

(b) Solve and interpret f(t) = 1159.

Year, t (since 1975)

0 5 10 15 20 25 30

Revenue, R(million $)

201

364

651

1075

1159

2200

2200

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R = f(t)

(b) Solve and interpret f(t) = 1159.

When were Revenues $1159 million?Year, t (since 1975)

0 5 10 15 20 25 30

Revenue, R(million $)

201

364

651

1075

1159

2200

2200

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R = f(t)

When were Revenues $1159 million?

t=20. Therefore, 1995.Year, t (since 1975)

0 5 10 15 20 25 30

Revenue, R(million $)

201

364

651

1075

1159

2200

2200

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End of Section 2.1

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