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

LINEAR FUNCTIONS AND CHANGE

FUNCTIONS & FUNCTION NOTATION

Chapter 1

Section 1

2

A function is a rule which takes certain numbers as inputs and assigns to each input number exactly one output number. The output is a function of the input.

The inputs and outputs are also called variables.

3Page 2

4Page N/A

“Oecanthus Fultoni”

5Page N/A

“The Snowy Tree Cricket”

6Page N/A

“Nature’s Thermometer"

7Page N/A

By counting the number of times a snowy tree cricket chirps in 15 seconds...

8Page 2 (Example 1)

By counting the number of times a snowy tree cricket chirps in 15 seconds & adding 40...

9Page 2

By counting the number of times a snowy tree cricket chirps in 15 seconds & adding 40...

We can estimate the temperature (in degrees Fahrenheit)!!!

10Page 2

For instance, if we count 20 chirps in 15 seconds, then a good estimate of the temperature is?

11Page 2

For instance, if we count 20 chirps in 15 seconds, then a good estimate of the temperature is?

20 + 40 = 60°F!!!!

12Page 2

0 1Estimated temp (in F)= Chirp rate (in chirps/min.)+40

4

13Page 3

0 1Estimated temp (in F)= Chirp rate (in chirps/min.)+40

4

14Page 3

T R1

404

T R

140

4T R

If R = 80:

1(80) 40 ?

4T

15Page 3

140

4T R

If R = 80:

1(80) 40 60

4T

16Page 3

By doing more substitutions into the formula, we can create:

17Page 3

R, chirp rate (chirps/minute)

T, predicted temperature (°F)

20 45

40 50

60 55

80 60

100 65

120 70

140 75

160 80

18Page 3

From this table, we can create:

19Page 3

20Page 3

When we use a function to describe an actual situation, the function is referred to as a mathematical model.

is a mathematical model of the relationshipbetween the temperature and the cricket's chirp rate.

140

4T R

21Page 3

R, chirp rate (chirps/minute)

T, predicted temperature (°F)

20 45

40 50

60 55

80 60

100 65

120 70

140 75

160 80

What is the chirp rate when the temperature is 40 degrees?

22Page 4

140

4T R

23Page 4

140

4

140 ( ) 40

4

T R

R

24Page 4

140

4

140 ( ) 40

4

140 40 ( )

4

T R

R

R

25

140

4

140 ( ) 40

4

140 40 ( )

4

10 ( )

4

T R

R

R

R

26Page 4

140

4

140 ( ) 40

4

140 40 ( )

4

10 ( )

4

10 4 4 ( )

4

T R

R

R

R

R

27Page 4

140

4

140 ( ) 40

4

140 40 ( )

4

10 ( )

4

10 4 4 ( )

4

0

T R

R

R

R

R

R28Page 4

140

4T R

What if the temperature is 30 degrees? What is R?

29Page 4

140

4

130 ( ) 40

4

130 40 ( )

4

110 ( )

4

110 4 4 ( )

4

40 !!!!!

T R

R

R

R

R

R30Page 4

Let's verify:

140

4

130 ( 40) 40

4

30 10 40

30 30

T R

31Page 4

What is the moral here?

32Page 4

Whether the model's predictions are accurate for chirp rates down to zero and temperatures as low as 40°F is a question that mathematics alone cannot answer; an understanding of the biology of crickets is needed.

However, we can safely say that the model does not apply for temperatures below 40°F, because the chirp rate would then be negative. For the range of chirp rates and temperatures in Table 1.1, the model is remarkably accurate.

33Page 4

R, chirp rate (chirps/minute)

T, predicted temperature (°F)

20 45

40 50

60 55

80 60

100 65

120 70

140 75

160 80

34Page 4

140

4T R

Is T a function of R, or vice-versa?

35Page 4

140

4T R

T is a function of R.

36Page 4

140

4T R

Will making the cricket chirp faster (or slower) result in a temperature change upward (or downward)?!?

37Page 4

140

4T R

Will making the cricket chirp faster (or slower) result in a temperature change upward (or downward)?!?

No

38Page 4

140

4T R

Saying that the temperature (T) depends on the chirp rate (R) means:

Knowing the chirp rate (R) is sufficient to tell us the temperature (T).

39Page 4

140

4T R

Saying that the temperature (T) depends on the chirp rate (R) means:

Knowing the chirp rate (R) is sufficient to tell us the temperature (T). Again, a change in the chirp rate (R) doesn't cause a change in the temperature (T).

40Page 4

A function is a rule which takes certain numbers as inputs and assigns to each input number exactly one output number. The output is a function of the input.

The inputs and outputs are also called variables.

41Page 2

Function Notation

Q is a function of quantity, t

Or:

Q is a function of t

We abbreviate:

Q = “f of t” or Q = f(t).

42Page 4

Q = f(t)

This means:

applying the rule f to the input value, t, gives the output value, f(t).

Here:

Q = dependent variable (unknown, depends on t)t = independent variable (known)

43Page 4

Q = f(t).

In other words:

Output = f(Input)

Or:

Dependent = f(Independent)

44Page 4

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).

(a) Find a formula for f.(b) Explain in words what the statement f(10,000) = 40 tells us about painting houses.

45Page 4 (Example 2)

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).

(a) Find a formula for f.

If n = 1, A = ? ft2

46Page 4

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).

(a) Find a formula for f.

If n = 1, A = 250 ft2

47Page 4

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).

(a) Find a formula for f.

If n = 2, A = ? ft2

48Page 4

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).

(a) Find a formula for f.

If n = 2, A = 500 ft2

49Page 4

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).

(a) Find a formula for f.

If n = 3, A = ? ft2

50Page 4

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).

(a) Find a formula for f.

If n = 3, A = 750 ft2

51Page 4

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).

(a) Find a formula for f.

In general, we have:

A=(250)(n)52Page 4

In general, we have:

A=(250)(n)

Now solve for n:

53Page 4

In general, we have:

A=(250)(n)

Now solve for n:

n=A/250

54Page 4

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).

(a) Find a formula for f : n=A/250.

55Page 4

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).

(a) Find a formula for f : n=A/250. So f(A)=A/250

56Page 4

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).

(a) Find a formula for f : n=A/250. So f(A)=A/250(b) Explain in words what the statement f(10,000) = 40 tells us about painting houses.

57Page 4

(a) Find a formula for f : n=A/250. So f(A)=A/250

(b) Explain in words what the statement f(10,000) = 40 tells us about painting houses.

Since f(A)=A/250, what does f(10,000) mean?

58Page 4

(a) Find a formula for f : n=A/250. So f(A)=A/250

(b) Explain in words what the statement f(10,000) = 40 tells us about painting houses.

Since f(A)=A/250, what does f(10,000) mean?

A=10,000

59Page 4

(a) Find a formula for f : n=A/250. So f(A)=A/250

(b) Explain in words what the statement f(10,000) = 40 tells us about painting houses.

Since f(A)=A/250, what does f(10,000) mean?

A=10,000

What can we deduce next?

60Page 4

(a) Find a formula for f : n=A/250. So f(A)=A/250

(b) Explain in words what the statement f(10,000) = 40 tells us about painting houses.

Since f(A)=A/250, what does f(10,000) mean?

A=10,000

What can we deduce next?

Since f(A)=A/250, then f(10,000)=10,000/25061Page 4

(a) Find a formula for f : n=A/250. So f(A)=A/250

(b) Explain in words what the statement f(10,000) = 40 tells us about painting houses.

Since f(A)=A/250, then f(10,000)=10,000/250

Therefore,

f(10,000)=40

In English this means?62Page 4

(a) Find a formula for f : n=A/250. So f(A)=A/250

(b) Explain in words what the statement f(10,000) = 40 tells us about painting houses.

f(10,000)=40

In English this means:

An area of A=10,000 sq. ft. requiresn= 40 gallons of paint.

63Page 4

Functions Don't have to be Defined by Formulas:

People sometimes think that functions are always represented by formulas.

However, the next example shows a function which is not given by a formula.

64Page 5

The average monthly rainfall, R, at Chicago's O'Hare airport is given in Table 1.2, where time, t, is in months and t = 1 is January, t = 2 is February, and so on.

The rainfall is a function of the month, so we write R = f(t). However there is no equation that gives Rwhen t is known. Evaluate f(1) and f(11). Explain what your answers mean.

t 1 2 3 4 5 6 7 8 9 10 11 12

R 1.8 1.8 2.7 3.1 3.5 3.7 3.5 3.4 3.2 2.5 2.4 2.1

65Page 5 (Example 4)

Evaluate f(1) and f(11). Explain what your answers mean.

Remember: R = f(t).

t 1 2 3 4 5 6 7 8 9 10 11 12

R 1.8 1.8 2.7 3.1 3.5 3.7 3.5 3.4 3.2 2.5 2.4 2.1

f(1) = ?

66Page 5

Evaluate f(1) and f(11). Explain what your answers mean.

Remember: R = f(t).

t 1 2 3 4 5 6 7 8 9 10 11 12

R 1.8 1.8 2.7 3.1 3.5 3.7 3.5 3.4 3.2 2.5 2.4 2.1

f(1) = 1.8

67Page 5

Evaluate f(1) and f(11). Explain what your answers mean.

Remember: R = f(t).

t 1 2 3 4 5 6 7 8 9 10 11 12

R 1.8 1.8 2.7 3.1 3.5 3.7 3.5 3.4 3.2 2.5 2.4 2.1

f(11) = ?

68Page 5

Evaluate f(1) and f(11). Explain what your answers mean.

Remember: R = f(t).

t 1 2 3 4 5 6 7 8 9 10 11 12

R 1.8 1.8 2.7 3.1 3.5 3.7 3.5 3.4 3.2 2.5 2.4 2.1

f(11) = 2.4

69Page 5

t 1 2 3 4 5 6 7 8 9 10 11 12

R 1.8 1.8 2.7 3.1 3.5 3.7 3.5 3.4 3.2 2.5 2.4 2.1

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

70Page 5 (Not shown)

As was stated, functions don’t have to be defined by formulas.

You can do a linear regression analysis and get:

rainfall(R) = 2.6424242 + 0.0255245*month (t)

71Page 5 (Not shown)

As was stated, functions don’t have to be defined by formulas.

You can do a linear regression analysis and get:

rainfall(R) = 2.6424242 + 0.0255245*month (t)

BUT...

72Page 5 (Not shown)

As was stated, functions don’t have to be defined by formulas.

You can do a linear regression analysis and get:

rainfall(R) = 2.6424242 + 0.0255245*month (t)

BUT...

What if t = 12?73Page 5 (Not shown)

You can do a linear regression analysis and get:

rainfall(R) = 2.6424242 + 0.0255245*month (t)

BUT…

What if t=12?

rainfall(R) = 2.6424242 + 0.0255245*month (12)

74Page 5 (Not shown)

You can do a linear regression analysis and get:

rainfall(R) = 2.6424242 + 0.0255245*month (t)

BUT…

What if t=12?

rainfall(R) = 2.6424242 + 0.0255245*(12)R = 2.6424242 +.306294

R = 2.9487182

75Page 5 (Not sh0wn)

t 1 2 3 4 5 6 7 8 9 10 11 12

R 1.8 1.8 2.7 3.1 3.5 3.7 3.5 3.4 3.2 2.5 2.4 2.1

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

76Page 5 (Not shown)

t 1 2 3 4 5 6 7 8 9 10 11 12

R 1.8 1.8 2.7 3.1 3.5 3.7 3.5 3.4 3.2 2.5 2.4 2.1

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Let’s try to fit a quadratic curve:

77Page 5 (Not shown)

rainfall(R) = 3.3403409 +

0.0255245*month (t) -

0.0585664*(month (t)-6.5)^2

R = 3.3403409 + 0.0255245*t - 0.0585664*(t-6.5)2

78Page 5 (Not shown)

Again, for t=12:

R = 3.3403409 + 0.0255245*t - 0.0585664*(t-6.5)2

R = 3.3403409 + .306294 - 0.0585664*(5.5) 2

R = 3.3403409 + .306294 - 0.0585664*30.25R = 3.3403409 + .306294 - 0.0585664*30.25R = 3.3403409 + .306294 – 1.7716336R = 1.8750013

Closer, but still not 2.1!!!

79Page 5 (Not shown)

rainfall(R) = 3.3403409 +

0.0255245*month (t) -

0.0585664*(month (t)-6.5)^2

For t=12, R = 1.875001380Page 5 (Not shown)

When is a Relationship not a Function?

It is possible for two quantities to be related and yet for neither quantity to be a function of the other.

81Page 5

t 0 1 2 3 4 5 6 7 8 9 10 11

R 1000 750 567 500 567 750 1000 1250 1433 1500 1433 1250

F 150 143 125 100 75 57 50 57 75 100 125 143

A national park contains foxes that prey on rabbits. Table 1.3 gives the two populations, F and R, over a 12-month period, where t = 0 means January 1, t = 1 means February 1, and so on.

(a) Is F a function of t? Is R a function of t?(b) Is F a function of R? Is R a function of F?

82Page 5 (Example 5)

t 0 1 2 3 4 5 6 7 8 9 10 11

R 1000 750 567 500 567 750 1000 1250 1433 1500 1433 1250

F 150 143 125 100 75 57 50 57 75 100 125 143

(a) Is F a function of t? Is R a function of t?(b) Is F a function of R? Is R a function of F?

Please create 4 graphs with pen/paper:

1) Plot F on the y axis & t on the x axis.2) Plot R on the y axis & t on the x axis.3) Plot F on the y axis & R on the x axis.4) Plot R on the y axis & F on the x axis.

83Page 5 (Example 5)

t 0 1 2 3 4 5 6 7 8 9 10 11

R 1000 750 567 500 567 750 1000 1250 1433 1500 1433 1250

F 150 143 125 100 75 57 50 57 75 100 125 143

(a) Is F a function of t?

84Page 5

t 0 1 2 3 4 5 6 7 8 9 10 11

R 1000 750 567 500 567 750 1000 1250 1433 1500 1433 1250

F 150 143 125 100 75 57 50 57 75 100 125 143

(a) Is F a function of t?

Yes: for each value of t, there is exactly one value of F.

85Page 5

t 0 1 2 3 4 5 6 7 8 9 10 11

R 1000 750 567 500 567 750 1000 1250 1433 1500 1433 1250

F 150 143 125 100 75 57 50 57 75 100 125 143

(a) Is R a function of t?

86Page 5

t 0 1 2 3 4 5 6 7 8 9 10 11

R 1000 750 567 500 567 750 1000 1250 1433 1500 1433 1250

F 150 143 125 100 75 57 50 57 75 100 125 143

(a) Is R a function of t?

Yes: for each value of t, there is exactly one value of R.

87Page 5

t 0 1 2 3 4 5 6 7 8 9 10 11

R 1000 750 567 500 567 750 1000 1250 1433 1500 1433 1250

F 150 143 125 100 75 57 50 57 75 100 125 143

(b) Is F a function of R? Is R a function of F?

88Page 5

t 0 1 2 3 4 5 6 7 8 9 10 11

R 1000 750 567 500 567 750 1000 1250 1433 1500 1433 1250

F 150 143 125 100 75 57 50 57 75 100 125 143

(b) Is F a function of R?

89Page 5

t 0 1 2 3 4 5 6 7 8 9 10 11

R 1000 750 567 500 567 750 1000 1250 1433 1500 1433 1250

F 150 143 125 100 75 57 50 57 75 100 125 143

(a) Is F a function of R?

90Page 5

t 0 1 2 3 4 5 6 7 8 9 10 11

R 1000 750 567 500 567 750 1000 1250 1433 1500 1433 1250

F 150 143 125 100 75 57 50 57 75 100 125 143

(a) Is F a function of R?

No, F is not a function of R. Suppose R = 567. This happens both at t = 2 & at t = 4. Since there are R values which correspond to more than one F value, F is not a function of R.

91Page 5

t 0 1 2 3 4 5 6 7 8 9 10 11

R 1000 750 567 500 567 750 1000 1250 1433 1500 1433 1250

F 150 143 125 100 75 57 50 57 75 100 125 143

(a) Is R a function of F?

92Page 5

t 0 1 2 3 4 5 6 7 8 9 10 11

R 1000 750 567 500 567 750 1000 1250 1433 1500 1433 1250

F 150 143 125 100 75 57 50 57 75 100 125 143

(a) Is R a function of F?No, R is not a function of F. Suppose F = 57. This happens both at t = 5 & at t = 7. Since there are F values which correspond to more than one R value, R is not a function of F.

93Page 5

How to Tell if a Graph Represents a Function: Vertical Line Test

Graphically, this means:

Look at the graph of y against x. For a function, each x-value corresponds to exactly one y-value. This means that the graph intersects any vertical line at most once.

If a vertical line cuts the graph twice, the graph would contain two points with different y-values but the same x-value; this would violate the definition of a function. Thus, we have the following criterion:

94Page 6

How to Tell if a Graph Represents a Function: Vertical Line Test

Vertical Line Test: If there is a vertical line which intersects a graph in more than one point, then the graph does not represent a function.

95Page 6

This concludes Section 1.1

96