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SECTION 1.4 Function Models

� Mathematical modeling

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Example 1.4.1

The highway department is planning to build a picnic area for motorists along a major highway. It is to be rectangular with an area of 5,000 square yards and is to be fenced off on the three sides not adjacent to the highway. Express the number of yards of fencing required as a function of the length of the

unfenced side.

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Example 1.4.2

A cylindrical can is to have capacity (volume) of 24π cubic inches. The cost of the material used for the top and bottom of the can is 3 cents per square inch, and the cost of the material used for the curved side is 2 cents per square inch. Express the cost of constructing the can as a function of its radius.

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Example 1.4.3

During a drought, residents of Marin County, California, were faced with a severe water shortage. To discourage excessive use of water, the county water district initiated drastic rate increases. The monthly rate for a family of four was $1.22 per 100 cubic feet of water for the first 1,200 cubic feet, $10 per 100 cubic feet for the next 1,200 cubic feet, and $50 per 100 cubic feet thereafter. Express the monthly water bill for a family of four as a function of the amount of water used.

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Example 1.4.4(1/2)

When environmental factors impose an upper bound on its size, population grows at a rate that is jointly proportional to its current size and the difference between its current size and the upper bound. Express the rate of population growth as a function of the size of the population.

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A manufacturer can produce blank premium-quality videotapes at a cost of $2 per cassette. The cassettes have been selling for $5 apiece, and at that price, consumers have been buying 4,000 cassettes a month. The manufacturer is planning to raise the price of the cassettes and estimates that for each $1 increase in the price, 400 fewer cassettes will be sold each month.

a.Express the manufacturer’s monthly profit as a function of the price at which the cassettes are sold.

b.Sketch the graph of the profit function. What price corresponds to maximum profit? What is the maximum profit?

Example 1.4.5

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Market Equilibrium

� Demand function D(x) for a commodity relates the number of units

x that are produced to the unit price p = D(x) at which all x units are

demanded (sold) in the marketplace.

� Supply function S(x) gives the corresponding price p = S(x) at

which producers are willing to supply x units to the marketplace

� The law of supply and demand says that in a competitive market

environment, supply tends to equal demand, and when this occurs,

the market is said to be in equilibrium.

� When the market is not in equilibrium, it has a shortage when

demand exceeds supply [D(x) > S(x)] and surplus when supply

exceeds demand [S(x) > D(x)].

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Example 1.4.6

Market research indicates that manufacturers will supply xunits of a particular commodity to the marketplace when the price is p=S(x) dollars per unit and that the same number of units will be demanded (bought) by consumers when the price is p=D(x) dollars per unit, where the supply and demand functions are give by

S(x)=x2+14 and D(x)=174-6x

a.At what level of production x and unit price is market equilibrium achieved?b.Sketch the supply and demand curves, p=S(x) and p= D(x), on the same graph and interpret.

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� Suppose x denotes the number of units manufactured and

sold, and let C(x) and R(x) be the corresponding total

cost and total revenue, respectively.

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Break-Even Analysis

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Example 1.4.7

A manufacturer can sell a certain product for $110 per unit. Total cost consists of a fixed overhead of $7,500 plus production costs of $60 per unit.

a.How many units must the manufacturer sell to break even?b.What is the manufacturer’s profit or loss if 100 units are sold?c.How many units must be sold for the manufacturer to realize a profit of $1,250?

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Example 1.4.8

A certain car rental agency charges $25 plus 60 cents per mile. A second agency charges $30 plus 50 cents per mile. Which agency offers the better deal?

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SECTION 1.5 Limits

� The Limit of a Function

If f(x) gets closer and closer to a number L as x gets

closer and closer to c from both sides, then L is the limit

of f(x) as x approaches c. The behavior is expressed by

writing

( ) Lxfcx

=→

lim

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Use a table to estimate the limit

Example 1.5.1

1

1lim

1 −

−

→ x

x

x

x →→→→ 1 ←←←← x

x 0.99 0.999 0.9999 1 1.00001 1.0001 1.001

f(x) 0.50126 0.50013 0.50001 0.499999 0.49999 0.49988

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� Algebraic Properties of Limits

Properties of Limits

( ) ( )[ ] ( ) ( )

( ) ( )[ ] ( ) ( )

( )[ ] ( )

( ) ( )[ ] ( )[ ] ( )[ ]( )

( )

( )

( )( )

( )[ ] ( )[ ] ( )[ ] exists lim if limlim

0lim if lim

limlim

lim limlim

constant any for limlim

limlimlim

limlimlim

p

cx

p

cx

p

cx

cx

cx

cx

cx

cxcxcx

cxcx

cxcxcx

cxcxcx

xfxfxf

xgxg

xf

xg

xf

xgxfxgxf

kxfkxkf

xgxfxgxf

xgxfxgxf

→→→

→

→

→

→

→→→

→→

→→→

→→→

=

≠=

=

=

−=−

+=+

( ) ( ) thenexists, lim and limIf xgxfcxcx →→

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Properties of Limits

� Limits of Two Linear Functions

FIGURE 1.47 Limits of two linear functions.

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Example 1.5.2

).843(lim Find 3

-1x+−

→xx

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2

83lim Find

2

1x −

−

→ x

x

Example 1.5.3

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� Limits of Polynomials and Rational Functions

If p(x) and q(x) are polynomials, then

and

( ) ( )cpxpcx

=→

lim

( )

( )

( )

( )( ) 0 if lim ≠=

→cq

cq

cp

xq

xp

cx

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2

1lim Find

2x −

+

→ x

xExample 1.5.4

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Example 1.5.5

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1lim Find

2

2

1x +−

−

→ xx

x

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.1

1lim Find

1x −

−

→ x

x

Example 1.5.6

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Limits Involving Infinity

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.21

lim Find2

2

x xx

x

+++∞→

Example 1.5.7

x →→→→ + ∞∞∞∞

x 100 1,000 10,000 100,000

f(x) 0.49749 0.49975 0.49997 0.49999

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.253

132lim Find

2

2

x +−

++

+∞→ xx

xx

Example 1.5.8

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Example 1.5.9

If a crop is planted in soil where the nitrogen level is N, then

the crop yield Y can be modeled by the Michaelis-Menten

function

Where A and B are positive constants. What happens to

crop yield as the nitrogen level is increased indefinitely?

0 )( ≥+

= NNB

ANNY

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� Limits at Infinity

We say that is an infinite limit if f(x) increase or

decreases without bound as x→c. We write

if f(x) increases without bound as x→c or

If f(x) decreases without bound as x→c.

( )xfcx→

lim

( ) +∞=→

xfcx

lim

( ) −∞=→

xfcx

lim

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Example 1.5.10

.3

12lim Find

3

x −

++−

+∞→ x

xx

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SECTION 1.6 ONE-SIDED LIMITS AND

CONTINUITY

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Example 1.6.1

For the function

evaluate the on-sided limits

≥+

<≤=

2 xif 12

2x0 if -1)(

2

x

xxf

)(lim and )(lim2x2x

xfxf+−

→→

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Example 1.6.2

Find as x approaches 4 from the left and from the right.4

2lim

−

−

x

x

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Example 1.6.3Determine whether exists, where )(lim

1xxf

→

≥−+−

<+=

1 xif 14

1 xif 1)(

2xx

xxf

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Continuity

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Continuity

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Example 1.6.5

Show that the rational is continuous at

x = 3. 2

1)(

−

+=

x

xxf

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Example 1.6.6

Discuss the continuity of each of the following functions:

a. b. C.

xxf

1)( =

1

1)(

2

+

−=

x

xxg

≥−

<+=

1 xif 2

1 xif 1)(

x

xxh

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For what value of the constant A is the following function

continuous for all real x?

≥+−

<+=

1 xif 43

1 xif 5)(

2 xx

Axxf

Example 1.6.7

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Example 1.6.8

Discuss the continuity of the function

On the open interval -2 < x < 3 and on the closed interval -2 ≤

x ≤ 3.

3

2)(

−

+=

x

xxf

� If f(x) is continuous on the interval a ≤ x ≤ b and L is the

number between f(a) and f(b), then f(c) = L for some c

between a and b

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The Intermediate Value Property

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Example 1.6.9

Show that the equation has a solution for 1 <x < 2.1

112

+=−−

xxx