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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
14Page 3
T R
If R = 80:
1(80) 40 ?
4 T
15Page 3
140
4 T R
If R = 80:
1(80) 40 60
4 T
16Page 3
By doing more substitutions into the formula, we can create:
17Page 3
R, chirp rate (chirps/minute)
T, predicted temperature (°F)
20 4540 5060 5580 60
100 65120 70140 75160 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 relationship between the temperature and the cricket's chirp rate.
21Page 3
R, chirp rate (chirps/minute)
T, predicted temperature (°F)
20 4540 5060 5580 60
100 65120 70140 75160 80
What is the chirp rate when the temperature is 40 degrees?
22Page 4
140
4 T R
23Page 4
140
41
40 ( ) 404
T R
R
24Page 4
140
41
40 ( ) 404
140 40 ( )
4
T R
R
R
25Page 4
140
41
40 ( ) 404
140 40 ( )
41
0 ( )4
T R
R
R
R
26Page 4
140
41
40 ( ) 404
140 40 ( )
41
0 ( )4
10 4 4 ( )
4
T R
R
R
R
R
27Page 4
140
41
40 ( ) 404
140 40 ( )
41
0 ( )4
10 4 4 ( )
40
T R
R
R
R
R
R28Page 4
What if the temperature is 30 degrees? What is R?
29Page 4
140
41
30 ( ) 404
130 40 ( )
41
10 ( )4
110 4 4 ( )
440 !!!!!
T R
R
R
R
R
R30Page 4
Let's verify:
140
41
30 ( 40) 404
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 4540 5060 5580 60
100 65120 70140 75160 80
34Page 4
140
4 T R
Is T a function of R, or vice-versa?
35Page 4
T is a function of R.
36Page 4
140
4 T R
Will making the cricket chirp faster (or slower) result in a temperature change upward (or downward)?!?
37Page 4
Will making the cricket chirp faster (or slower) result in a temperature change upward (or downward)?!?
No38Page 4
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
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 ft249Page 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 R when 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
2.165Page 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 12R 1.
81.8
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 12R 1.
81.8
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 12R 1.
81.8
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 12R 1.
81.8
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
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
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
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