Copyright © by Houghton Mifflin Company, All rights reserved. Calculus Concepts 2/e LaTorre, Kenelly, Fetta, Harris, and Carpenter Chapter 1 Ingredients

Copyright © by Houghton Mifflin Company, All rights reserved.Copyright © by Houghton Mifflin Company, All rights reserved.

Calculus Concepts 2/eCalculus Concepts 2/eLaTorre, Kenelly, Fetta, Harris, and CarpenterLaTorre, Kenelly, Fetta, Harris, and Carpenter

Chapter 1Chapter 1Ingredients of Change: Ingredients of Change:

Functions and Linear ModelsFunctions and Linear Models


Chapter 1 Key ConceptsChapter 1 Key Concepts• FunctionsFunctions

• Discrete and Continuous Functions Discrete and Continuous Functions

• Special FunctionsSpecial Functions

• Limits of FunctionsLimits of Functions

• Linear ModelsLinear Models


FunctionsFunctions• For every input there is exactly one output.For every input there is exactly one output.

x y

1

3

4

7

11

2

3

6

6

3

FunctionFunction Non-FunctionNon-Function

x y

1

3

3

5

8

2

6

7

8

2


Functions: ExampleFunctions: Example

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

f (x)

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

f (x)



Functions: Exercise 1.1 #21Functions: Exercise 1.1 #21• You want to finance a car for 60 months at 10% You want to finance a car for 60 months at 10%

interest with no down payment. The graph shows interest with no down payment. The graph shows the car value as a function of the monthly payment. the car value as a function of the monthly payment. Estimate the car value when the monthly payment is Estimate the car value when the monthly payment is $200 and estimate the monthly payment for a $200 and estimate the monthly payment for a $16,000 car.$16,000 car.

100 200 300 400 500

4,000

8,000

12,000

16,000

24,000

MonthlyPayment(dollars)

Car Value (dollars)

20,000$200 payment $200 payment $9400 car$9400 car

$16,000 car $16,000 car $340 payment$340 payment


Discrete and Continuous FunctionsDiscrete and Continuous Functions

• Discrete - graph is a scatter plotDiscrete - graph is a scatter plot• Continuous - graph doesn’t have any breaksContinuous - graph doesn’t have any breaks• Continuous with Discrete Interpretation - continuous Continuous with Discrete Interpretation - continuous

graph with meaning only at certain pointsgraph with meaning only at certain points• Continuous without Restriction - continuous graph Continuous without Restriction - continuous graph

with meaning at all pointswith meaning at all points


Discrete and Continuous: ExampleDiscrete and Continuous: Example

DiscreteDiscrete Continuous Continuous (without restriction)(without restriction)


Discrete and Continuous: ExampleDiscrete and Continuous: Example

Continuous with Continuous with Discrete Discrete

InterpretationInterpretation


Discrete/Continuous: Exercise 1.2 #5Discrete/Continuous: Exercise 1.2 #5• On the basis of data recorded in the spring of On the basis of data recorded in the spring of

each year between 1988 and 1997, the each year between 1988 and 1997, the number of students of osteopathic medicine in number of students of osteopathic medicine in the U.S. may be modeled as the U.S. may be modeled as O(t)=0.027tO(t)=0.027t22 - 4.854t + 218.929 thousand - 4.854t + 218.929 thousand students in year 1900 + t. students in year 1900 + t.

• Is this function discrete, continuous with Is this function discrete, continuous with discrete interpretation or continuous without discrete interpretation or continuous without restriction?restriction?

• The function is continuous without restriction.The function is continuous without restriction.


Special FunctionsSpecial Functions• Composite FunctionsComposite Functions

• Inverse Functions Inverse Functions

• Piecewise Continuous FunctionsPiecewise Continuous Functions


Composite Functions: ExampleComposite Functions: Example

t = time into flight (minutes)

F(t) = feet above sea level

0

1

2

3

4

4,500

7,500

13,000

19,000

26,000

72

17

-34

-55

-62

4,500

7,500

13,000

19,000

26,000

F = feet above sea

level

A(F) = air temperature (Fahrenheit)

FMin.Min. FeetFeet AFeetFeet DegreesDegrees


Composite Functions: ExampleComposite Functions: Example

72

17

-34

-55

-62

A(F(t)) = air temperature (Fahrenheit)

t = time into flight (minutes)

0

1

2

3

4

A(F(t)) =A(F(t)) =

FMinutesMinutes AFeetFeet DegreesDegrees


Composite Functions: Exercise 1.3 #9Composite Functions: Exercise 1.3 #9• Draw and label the input/output diagram for Draw and label the input/output diagram for

the composite function given P(c) is the profit the composite function given P(c) is the profit from the sale of c computer chips and C(t) is from the sale of c computer chips and C(t) is the number of computer chips produced after the number of computer chips produced after t hours. t hours.

CHours (t)Hours (t) PChips C(t) Chips C(t) Profit P(C(t))Profit P(C(t))

Profit = P(C(t))Profit = P(C(t))


Inverse Functions: ExampleInverse Functions: Example

Year

Average Home Sales

Price ($)

1970

1980

1985

1990

1995

23,400

64,600

84,300

122,900

133,900

Year

Average Home Sales

Price ($)

1970

1980

1985

1990

1995

23,400

64,600

84,300

122,900

133,900

fYearYear PricePrice f-1PricePrice YearYear


Inverse Functions: Exercise 1.3 #25Inverse Functions: Exercise 1.3 #25

Age

% with Flex Work Schedule

16-19

20-24

25-34

35-44

45-54

20.7

22.7

28.4

29.1

27.3

• Determine if the tables are inverse functions.Determine if the tables are inverse functions.

Age

% with Flex Work Schedule

16-19

20-24

25-34

35-44

45-54

20.7

22.7

28.4

29.1

27.3



Piecewise Continuous: ExamplePiecewise Continuous: Example

93t90when7.1098t7.7

90t85when667.3903t514.23)t(P

93t90when7.1098t7.7

90t85when667.3903t514.23)t(P


• Yearly water ski sales (in millions of dollars) in Yearly water ski sales (in millions of dollars) in the continental U.S. between 1985 and 1992 the continental U.S. between 1985 and 1992 can be modeled by can be modeled by

Piecewise : Exercise 1.3 #43Piecewise : Exercise 1.3 #43

92x89when9.1414x8.14

88x85when4.905x1.12)x(S

92x89when9.1414x8.14

88x85when4.905x1.12)x(S

• Find S(85), S(88), S(89), and S(92).Find S(85), S(88), S(89), and S(92).• S(85) = 123.1S(85) = 123.1• S(88) = 159.4S(88) = 159.4• S(89) = 97.79S(89) = 97.79• S(92) = 53.39S(92) = 53.39


Limits of FunctionsLimits of Functions• The limit of f(x) as x approaches c is writtenThe limit of f(x) as x approaches c is written

• In order for the limit to exist, the graph of the In order for the limit to exist, the graph of the function must approach a finite value, L, as x function must approach a finite value, L, as x approaches c from the left and from the right.approaches c from the left and from the right.

)x(flimcx

)x(flimcx

)x(flimcx

)x(flimcx

)x(flimcx

)x(flimcx

- right-hand limit- right-hand limit

- left-hand limit- left-hand limit


Limits of Functions: ExampleLimits of Functions: Example

5.0)t(Plim10t

5.0)t(Plim10t

1)t(Plim10t

1)t(Plim10t

DNE)t(Plim10t

DNE)t(Plim10t


Limits of Functions: Exercise 1.4 #4Limits of Functions: Exercise 1.4 #4

1)t(mlim1t

1)t(mlim1t

3)t(mlim1t

3)t(mlim1t

).t(mlimFind1t

).t(mlimFind1t

DNE)t(mlim1t

DNE)t(mlim1t


Limits of Functions: ExampleLimits of Functions: Example

8)t(rlim4t

8)t(rlim4t

t 4- r(t)

3.8

3.9

3.99

3.999

7.8

7.9

7.99

7.999

8)t(rlim4t

8)t(rlim4t

t 4+ r(t)

4.2

4.1

4.01

4.001

8.2

8.1

8.01

8.001

).t(rlimfind,4t

16t)t(rGiven

4t

2

).t(rlimfind,4t

16t)t(rGiven

4t

2

8)t(rlim4t

8)t(rlim4t


Limits of Functions: Exercise 1.4 #27Limits of Functions: Exercise 1.4 #27

)t(flim

3

1t

)t(flim

3

1t

)t(flim

3

1t

)t(flim

3

1t

DNE)t(flim3

1t

DNE)t(flim3

1t

t-1/3+ f(t)

-0.3

-0.33

-0.333

-0.3333

-18.27

-201.6

-2035

-20368

t-1/3- f(t)

-0.4

-0.34

-0.334

-0.3334

12.32

104.0

1020

10187

).t(flimfind,1t3

t6t)t(fGiven

3

1t

3

).t(flimfind,1t3

t6t)t(fGiven

3

1t

3


Linear ModelsLinear Models• Constant rate of changeConstant rate of change

• Equation of form f(x) = ax + bEquation of form f(x) = ax + b

• Graph is a lineGraph is a line

• a is the constant rate of change (slope)a is the constant rate of change (slope)

• b is the vertical intercept of the graph of fb is the vertical intercept of the graph of f


Linear Models: ExampleLinear Models: Example

t (years) P (%)

5

6

7

8

50

47

44

41

The table shows the percent P of companies The table shows the percent P of companies in business after t years of operation.in business after t years of operation.

Rate of change: Rate of change: -3 percentage points / year-3 percentage points / year

Model: Model: P = -3t + 65P = -3t + 65

Note: This model assumes Note: This model assumes 5 5 t t 8. 8.


Linear Models: Exercise 1.5 #1Linear Models: Exercise 1.5 #1Estimate slope:Estimate slope:-0.5-0.5

Find rate of change of profit:Find rate of change of profit:$0.5 million dollar decrease $0.5 million dollar decrease per yearper year

What is the significance of What is the significance of vertical intercept?vertical intercept?Profit at year 0 is about 2.5 Profit at year 0 is about 2.5 million dollars.million dollars.

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Copyright © by Houghton Mifflin Company, All rights reserved. Calculus Concepts 2/e LaTorre, Kenelly, Fetta, Harris, and Carpenter Chapter 1 Ingredients