M103LecNotesF09

8/14/2019 M103LecNotesF09

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CALCULUSIntroduction to Limits

Section 10.1

ccsun 2009

definition of the limit

limits from a graph

limits from an algebraic expression

indeterminate forms

1

Definition of the limit

Let f(x) be a function, and let c and L be real numbers.

limxc f(x) =L,

means that the value off(x) is close to the number L wheneverx is close to c.

Sometimes a limit does not exist. That is, there are functions

f and numbers c where f(x) doesnt get close to any number L

even through x is close to c.

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Limits from a graph

6 5 4 3 2 1 1 2 3 4 5 6

6

5

4

3

2

1

1

2

3

4

5

6

limx0 f(x) =

limx1 f(x) =

limx3f(x) =

limx4 f(x) =

3

Limits from a graph

6 5 4 3 2 1 1 2 3 4 5 6

6

5

4

3

2

1

1

2

3

4

5

6

limx0 f(x) =

limx1 f(x) =limx2f(x) =

limx4 f(x) =

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One-sided limits

6 5 4 3 2 1 1 2 3 4 5 6

6

5

4

3

2

1

1

2

3

4

5

6

limx4+ f(x) =

limx4 f(x) =

limx2+f(x) =

limx2 f(x) =

5

Limits that go to infinity

Example: f(x) = 1

x2 Here, lim

x0 f(x) =.

2 1 0 1 2

10

20

30

40

50

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Limits from an algebraic expression

Often, you can evaluate the limit of a function by just evalu-

ating the function at the desired point.

Here are some examples:

limx0 2x + 3 = 3

limx1 2x + 3 = 5

limx5

x + 3

2x 1 = 5 + 3

2 5 1=8

9

lim

x12x + 9 =

7

7

More difficult examples

Sometimes, you cannot simply evaluate the function to find thelimit.

Here are some examples:

A. limx0

1

x B. lim

x1x2 1x 1

C. lim

x12

1

2x 1

4x 2

x + 1

D. lim

x4

x2 16

x3

64In each case, a direct evaluation would require division by zero.

Still, some of these functions have a limit at the designated value

of x.

In all but one of these examples (which one? ? .), both the

numerator and the denominator have zero as their limit. In this

case, the limit problem is called a 0/0 indeterminate form.

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Indeterminate forms

If limxc f(x) = 0 and limxc g(x) = 0, then limxc

f(x)

g(x)

is indeterminate.

The term indeterminate is used because the limit may or may

not exist. And if the limit exists it can be any number.

9

Example of a 0/0 indeterminate form:

limx1

x2 1x 1

x .9 .99 .999 1.1 1.01 1.001x21x1 1.9 1.99 1.999 2.1 2.01 2.001

(.999)2 1.999

1

=.001999

.001

= 1.999

Why does x2 1

x 1 get close to 2 as x gets close to 1?Algebraic cancellation:

x2 1x 1 =

(x 1)(x + 1)x 1 =x + 1

So limx1

x2 1x 1 = ? .

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Example of a 0/0 indeterminate form:

limx2

x2 4x2 x 2

x 2.1 2.01 2.001

x24x2x2 1.32258 1.33223 1.33322

(2.01)2 42.012 (2.01) 2=

0.0401

0.0301= 1.33223

Why does x2 4x2 x 2 get close to

43 as x gets close to 2?

Algebraic cancellation:

x2 4x2 x 2

= =

So

limx2

x2 4x2 x 2=

?? .

11

Practice with 0/0 indeterminate forms

limx3

x2 9x + 3

limx3x2

9

x 3

limx5

x2 + 3x 10x2 + 5x

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Limits of rational functions at x horizontal asymptotes

Example: f(x) =1

x.

limx

1

x= 0 When x gets large (positively), 1/x gets close to 0.

limx

1

x= 0 When x gets large (negatively), 1/x gets close to 0.

What about?

limx 2x

limx6x

limx

5

x lim

x8

x

13

Limits of rational functions at x

Example: f(x) = x 2x + 3

.

Evaluate limxx 2x + 3

1412108642 2 4 6 8 10 12 14

14

12

10

8

6

4

2

2

4

6

8

10

12

14

x 2x + 3

= 1 2/x1 + 3/x

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More examples of horizontal asymptotes

Evaluate: Horizontal asymptotes

limx

2x 73x + 1

limx

x + 42x + 3

limx

3x 102x + 1

limx

x 15x + 11

15

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Example:

10 8 6 4 2 2 4 6 8 10

10

8

6

4

2

2

4

6

8

10

19

Example:

10 8 6 4 2 2 4 6 8 10

10

8

6

4

2

2

4

6

8

10

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Example: Is the function f(x) = x 2x + 3

continuous at

x= 0?

x= 3?

x= 6?1412108642 2 4 6 8 10 12 14

14

12

10

8

6

4

2

2

4

6

8

10

12

14

21

Example from business

A car rental agency charges $30 per day (or partial day) or $150per week, whichever is least. The graph the cost function C(x)

is shown below. Discuss continuity.

1 2 3 4 5 6 7 8 9 10Days

30

60

90

120

150

180

210

240

270

Cost

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The graph of T(x):

1 00 00 0 2 00 00 0 3 00 00 0 4 00 00 0Income

20000

40000

60000

80000

100000

120000

Tax

Is T(x) continuous?

Would it be good tax policy

to have an income tax func-

tion that is not continuous?

Between But Not Over Base Tax Rate Of theAmount Over

$0 $7,550 0 10% $0.00$7,550 $30,650 $755.00 15% $7,550

$30,650 $74,200 $4,220.00 25% $30,650$74,200 $154,800 $15,107.50 28% $74,200

$154,800 $336,550 $37,675.50 33% $154,800$336,550 $97,653.00 35% $336,550

23

A tax function that is not continuous:

0 100 200 300 400 500Income0

20

40

60

80

100

120

140

160

Tax

Near the end of the tax year, your income is exactly 300.What is your tax?

What is your net (after-tax) income?

You have an opportunity to take a small job and earn an extra

10 before the year ends. Should you take it? Why?

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The Derivative

Section 10.4

ccsun 2009

Average rate of change = Slope of secant line

The derivative as instantaneous rate of change

Instantaneous rate of change = Slope of tangent line

Computing the derivative using the definition

(The four-step process)

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Average rate of change: an example:

Revenue is given by R(x) =x(75

3x) for 0

x

20.

What is the change in revenue if sales change from 9 to 12?

What is the average rate of change in revenue if sales change

from 9 to 12?

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Revenue is given by R(x) =x(75 3x) for 0 x 20.

What is the average change in revenue if sales change from 9 to

11?


10?


9.1?

27

Average rate of change is slope: rise over run

7 8 9 10 11 12 13

414

420

426

432

438

444

450

456

462

468

474 Graphically, we are findingthe slope of the secant line

between the points (9, 432)

and (9 + h, R(9 + h)) on the

graph of R(x) =x(75 3x).The four values of h from

the previous slides are h =

3, 2, 1, 0.1.

R(9) = 432

R(9.1) = 434.07R(10) = 450

R(11) = 462

R(12) = 468

slope = riserunR(12)R(9)

129 = 36

3 = 12 R(11)R(9)

119 = 30

2 = 15R(10)R(9)

109 = 18

1 = 18 R(9.1)R(9)

9.19 = 2.07

.1 = 20.7

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Difference quotient: The slope of the secant line equalsthe difference quotient:

Difference quotient =f(x + h) f(x)

h .

x xh

fx

fxh

29

Instantaneous rate of change: The limit of the dif-ference quotient gives the instantaneous rate of change of the

function f(x) passing through the point (x, f(x)).

instantaneous rate of change at x = lim

h0

f(x + h) f(x)

h

The instantaneous rate of change depends on the function f(x)

and on the value of x.

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An example of the four-step process:

f(x) =x2 3x, find f(x)Step 1. Compute f(x + h)

Step 2. Compute f(x + h) f(x)

Step 3. Compute and simplify f(x + h) f(x)

h

Step 4. Find the limit as h approaches 0

f(x) = limh0

f(x + h) f(x)

h

.

33

An example of the four-step process:

R(x) = 60x

.02x2 Find R(x)

Step 1. Compute R(x + h)

Step 2. Compute R(x + h)R(x)

Step 3. Compute and simplify R(x + h)R(x)

h

Step 4. Find the limit as h approaches 0

R(x) = limh0

R(x + h)R(x)h

.

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Application of the derivative:

The revenue in dollars from the sale of x car seats for infants is

R(x) = 60x .02x2.Find the revenue and the instantaneous rate of change in rev-

enue at a sales level of 1000 car seats. Write a summary for theinstantaneous rate of change at x= 1000.

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Derivatives of Constants,Power Functions, and Sums

Section 10.5

ccsun 2009

Notation for the derivative

Derivative of a constant function

Derivatives of powers (power rule)

Derivatives of sums and differences

Marginal cost

Slope

37

Notation for the derivative

The derivative of a function f(x) may be represented by any of

the following:f(x), y, or dy/dx.

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The derivative of a constant function

Let f(x) = 5.

What is the slope at the point (2, 5)?

What is the slope at (3, 5)?

Does the slope depend on the particular value of x?

Theorem: If f(x) =c is a constant function, then f(x) = 0.

39

Power rule

Use the four-step process to compute the derivative of the func-

tion f(x) =x2:

The same pattern holds for other powers of x:

f(x) =x3 f(x) =x4 f(x) =x5

f(x) = 3x2 f(x) = 4x3 f(x) = 5x4

Theorem: If f(x) =xn, then f(x) =nxn1.

This is true for any number n, not just integers.

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Examples of the power rule

Find the derivatives of these functions:

f(x) =x3 f(x) =x10

f(x) =x3.2 f(x) =x2

f(x) =x4 f(x) =x1/2

41

Examples of the power rule (in disguise)


f(x) = 1x2

f(x) = 1x10

f(x) =

x f(x) = 1x

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Multiplication by a constant

An example: f(x) = 5x3.

We know that the derivative of u(x) = x3 is u(x) = 3x2. Thederivative of 5u(x) = 5x3 is 5u(x) = 5(3x2) = 15x2.

Theorem: Let f(x) =k u(x) where k is a constant. Then

f(x) =k u(x).

The proof follows from the fact that the constant k factors out

of the difference quotient:

f(x + h) f(x)h

=k u(x + h) k u(x)

h =k

u(x + h) u(x)h

.

43

Examples:

If f(x) = 10x4, then f(x) = 40x3.

If f(x) = 7x2, then f(x) = 14x.

If f(x) = 2

x3= 2x3, then f(x) = 6x4 = 6

x4.

If f(x) = 5

x= 5x1/2, then f(x) =5

2

x1/2.

If f(x) = 2x7, the f(x) =

If f(x) = 4x5

, then f(x) =

If f(x) = 30

x, then f(x) =

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Sum and Difference Properties

Theorem: If f(x) =u(x) + v(x), then f(x) =u(x) + v(x).

Examples

If f(x) =x2 + x3, then f(x) = 2x + 3x2.

If f(x) = 5x2 + x3, then f(x) = 10x + 3x2.

If f(x) = 3x +1

x, then f(x) = 3 1

x2.

If f(x) = x4 + 3x2, then f(x) =.

If f(x) = 2x +

x, then f(x) =.45

Use of the Derivative

For a particular value of x, the derivative f(x) gives the

instantaneous rate of change

slope of the line tangent to the graph of f(x)

marginal cost

If C(x) is a cost function, then

C(x) approximates the cost of producing on more item at aproduction level of x items.

C(x) is called the marginal cost.

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Marginal Cost

The total cost (in dollars) of producing x portable radios per day

is

C(x) = 1000 + 100x 0.5x2, for 0 x 100.

1. Find the marginal cost at a production level of x radios per

day.

2. Find the marginal cost at a production level of 80 radios.

47

Example continued

Cost function: C(x) = 1000 + 100x 0.5x2

3. Find the actual cost of producing the 81st radio and compare

this with the marginal cost in part 2.

4. Find the marginal cost at a production level of 20 radios and

compare this with the actual cost of producing the 21st radio.

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Slope of a Tangent Line

The derivative of a function f(x) at a point a gives the slope of

the line tangent to the graph of f(x) at the point (a, f(a)).

Example: The graph of the function f(x) = x3 3x2 + 2x+ 1and a tangent line are shown below.

0.5 0.5 1.0 1.5 2.0 2.5

1.0

0.5

0.5

1.0

1.5

2.0

What are the coordi-

nates of the point shown

on the graph?

What is the slope of the

tangent line?

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Marginal Analysis in Businessand Economics

Section 10.7

ccsun 2009

marginal cost, revenue, and profit functions

interpreting the derivative as marginal cost, revenue, and

profit

Marginal refers to an instantaneous rate of change, that is, aderivative.

53

Example of marginal cost

The total cost of producing x electric guitars is

C(x) = 1, 000 + 100x 0.25x2.1. Find the exact cost of producing the 51st guitar.

2. Compute the derivative C(x) and its value C(50) at x= 50.

3. Compare C(50) with the exact cost from part 1.

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Marginal Cost

The marginal cost function is just the derivative of the cost func-

tion. In business, we use marginal costC(x) to approximate the

exact cost to produce the (x + 1)st unit.

Why does the approximation work? Because

C(a) = limh0

C(x + h) C(x)h

and when h= 1, the difference quotient equals

C(x + 1) C(x),which is the exact cost to produce the (x + 1)st unit.

55

Graphical interpretation of exact cost and marginalcost

The total cost to produce x items is C(x) and the total costto produce x + 1 items isC(x + 1). Therefore the exact cost toproduce the (x + 1)st item is

C(x + 1) C(x).The marginal cost is an approximation of the exact cost:

C(x) .=C(x + 1) C(x).

x x1

Cx

Cx1

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Example:

Let

C(x) = 10, 000 + 90x .05x2

be the total weekly cost (in dollars) of manufacturing x fuel tanksfor cars.

Compute the marginal cost function and use it to approximate

the exact cost to manufacture the 101st fuel tank.

Marginal cost function: C(x) = 90 .10xMarginal cost at x= 100: C(100) = 90 .10(100) = 80

Summary: It costs approximately $80.00 to produce the 101stfuel tank.

57

Example continued

The marginal cost function:

C(x) = 90 .10x

Is the cost of a fuel tank increasing, decreasing or remaining the

same as x increases?

The marginal cost function C(x) gives the (approximate) costto produce the (x+ 1)st tank. So the first tank costs about

C(0) = 90 dollars. But the second tank cost about C(1) =89.90 dollars. Apparently the cost to produce the (x + 1)st tank

is $.10 less than the cost to produce the xth tank.

Compute and interpret C(200):

Compute and interpret C(500):

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Marginal Revenue & Profit

The technique of approximating the cost to produce a single

item by the marginal cost also applies to revenue and profit.

Revenue: If R(x) is the revenue from selling x units, then theadditional revenue earned by selling x + 1 units rather than x

units is

R(x+1)R(x). (additional revenue from selling (x+1)st unit)The marginal revenue, R(x) is approximately equal to the addi-tional revenue:

R(x) .=R(x + 1)R(x).

Profit: The same thing holds for the profit function, P(x):P(x) .=P(x + 1) P(x).

59

Application

The price-demand equation and the cost function for the pro-

duction and sales of television sets are

x= 6, 000 30p, C(x) = 150, 0 0 0 + 3x.where p is the price of a TV and x is the number produced and

sold. Profit and cost are in dollars.

a. Express the price p as a function of x:

b. Express revenue R(x) as a function of x:

c. Express profit P(x) as a function of x:

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Application continued

d. Find the marginal cost, marginal revenue, and marginal profit

functions:

e. Find and interpret R(3000):

f. Find and interpret P(1500) and P(1500):

61

Application continued

The cost and revenue functions are graphed below.

1000 2000 3000 4000 5000 6000

x

50000

100000

150000

200000

250000

300000

dollars

Identify which curve goes with

which function

Shade the profit area

Mark the break even points

Mark the point where profit is

maximized

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Derivatives of Products andQuotients

Section 11.3

ccsun 2009

Formulas for computing:

the derivative of a product of two functions, such as

S(t) = (9000t2 + t)(t2 + 50)

the derivative of a quotient of two functions, such as

S(t) =9000t2 + tt2 + 50

65

Derivative of a product: the product rule

If f

(x

) =F

(x

)S

(x

),

then f(x) =F(x)S(x) + S(x)F(x).

Example:

f(x) = (5x2)(x3 + 2):

Identify F(x) and S(x) (the first and second functions)

Find the derivatives F(x), S(x):

Find the derivative f(x):

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Practice problems


1. f(x) = (5x2 + x + 1)(3x 2)

2. f(x) = (2x1/2 + x + 1)(3x 2)

3. f(x) = (x2

+

1

x+ 1)(x 7)

67

Answers to practice problems

1.

5x2 + x + 1

3 + ( 3x 2)(10x + 1) = 45x2 14x + 1

2. (2x1/2 + x + 1 ) 3 + (x1/2 + 1)(3x 2) = 6x + 9x + 1 2x

3. (x2 + x1 + 1 ) + (x 7)(2x x2) = 3x2 14x + 1 + 7x2

Verify the algebraic simplifications.

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Answers to practice problems

(10x + 1)(3x 2) (5x2 + x + 1)3

(3x 2)2

=5

3x2 4x 1

(3x 2)2

(x1/2 + 1)(3x 2) (2x1/2 + x + 1)3(3x 2)2 =

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

(2x x2)(x 7) (x2 + x1 + 1)(x 7)2 =

x4 14x3 x2 2x + 7(x 7)2x2

Verify the algebraic simplifications.

71

Application: CD sales

The monthly sales of a new CD often increases and then levels

off. The graph below shows the monthly sales S(t) (in thou-

sands of CDs) as function of the number of months t since it

was released.

10 20 30 40t months

20

40

60

80

sales in thousands

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CD sales continued

The formula for the monthly sales is

S(x) = 90t2

t2 + 50.

Find the derivative S(x).

Find and interpret S(10) and S(10).

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The Power Rule

Section 11.4

ccsun 2009

the derivative of a function raised to a power,

such as f(x) = (x3 + 2)5

combining different rules to get the derivative of more com-

pliated functions, such as f(x) = (x3 + 2)5(x 1)

77

Different methods for computing the derivative

of a simple power function

f(x) = (3x + 1)2

Multiply out

f(x) = 9x2 + 6x + 1 f(x) = 18x + 6

Product Rulef(x) = (3x + 1)(3x + 1) f(x) = 3(3x + 1 ) + ( 3x + 1)3

= 18x + 6

Power Rule

f(x) = (3x + 1)2 f(x) = 2(3x + 1)1(3)= 18x + 6

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The Power Rule

If f(x) = [u(x)]n, where u(x) i s a d ifferentiable function and

n is any real number, then

f(x) =n[u(x)]n

1u(x).

Example: Find the derivative of f(x) = (x3 + 2)5.

f(x) = [u(x)]n, where u(x) =x3 + 2 and n= 5.

The Power Rule calls for u(x). But we know that:

u(x) = 3x2.

Apply the Power Rule:

f(x) =n[u(x)]n1 u(x) = 5(x3 + 2)4(3x2) = 15x2(x3 + 2)4.

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Work out the derivatives of these functions:

f(x) = (x3 2x + 1)3

f(x) = (5x2 x)4

f(x) = (x1 2x + 1)3

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Work out the derivatives of these functions:

f(x) = (x3 2x + 1)1/2

f(x) =

x3 2x + 1

f(x) = (x3

2x + 1)3

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Combining rules of differentiation

Some functions require more than one rule to find their deriva-

tive. For example:

f(x) = (3x 2)4(2x + 1).The function f(x) is the product of two other functions, F(x) =

(3x 2)4 and S(x) = 2x + 1. So we will need the product rule.But to differentiate the first function F(x), we also need the

power rule.

First apply the product rule:

f(x) =F(x) S(x) + S(x) F(x) = (3x 2)4 2 + ( 2x + 1)F(x).At this point, we still need to differentiateF(x) using the power

rule:

F(x) = 4(3x 2)3 3.Combining the results we get

f(x) = 2(3x 2)4 + 12(2x + 1)(3x 2)3.

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Derivatives that require two rules:practice problems

Differentiate these functions:

f(x) = (x2 + 1)5(5x 7)

g(x) = (6x 11)2x + 3

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Relative Rate of Change,Elasticity of Demand Part 1

Section 11.7

ccsun 2009

Relative rate of change (percentage increase or decrease)

Elasticity of demand at a certain price

Elasticity: inelastic, unitary, or elastic

Estimate the percentage decrease in demand if price in-

creases

85

Example: Relative rate of change

You intend to invest $1000 in one of two stocks, Biotech or

Comstat. Your broker estimates that Biotechs market value will

increase by $2 per share over the next year, while Comstats will

increase by only $1 per share.

Is this sufficient information for you to choose betwenn these

two stocks? What other information might you request from

the broker to help you decide?

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Example: Relative rate of change (continued)

You need to know the price of each stock.

Suppose Biotech is selling for $100 per share and Comstat is

selling of $25 per share. Find the percentage increase in the

market value of these two stocks.

Biotech increases $2 per share.

Comstat increases $1 per share.

87

Relative rate of change

The relative rate of change of a function f(x) is

f(x)f(x)

.

The relative rate of change compares the amount of the change,

f(x) to the value of f(x).

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Relative rate of change of demand

The most important example of the relative rate of change is

how demand changes as a function of price.

Example: A price-demand equation isx = f(p) = 5002p, wheref(p) is the demand for an product if the price of p.

What is the relative rate of change in demand f(p) as a function

of price p?

What is the relative rate of change in demand at a price of $50?

91

Elasticity of demand

We know the demand for a product decreases when the priceof the price increases. Elasticity of demand is the ratio of the

relative decrease in demand and the relative increase in price.

E(p) = percent change in demandpercent change in price

.

So if the elasticity of demand E(p) for a product is 1.5 and the

price of that product increases by 1%. then the demand will

decrease by 1.5%.

If price p and demand x are related by x = f(p), then the formula

for the elasticity of demand, E(p) at a price of p is

E(p) = pf(p)

f(p) .

Note that the price-demand equation must be in the form x =

f(p). That is, demand must be expressed as a function of price.

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Example: sunglasses

A sunglass retailer currently sells one type of sunglasses for $10

per pair. The price-demand function is x= f(p) = 7000 500p.

1. Find the elasticity of demand function, E(p).

93

Example: sunglasses (continued)

E(p) =

2. Find the elasticity of demand at a price of $10.

3. If the price is increased by 5%, what is the change in demand?

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Demand is: inelastic, unit, elastic

Elasticity Demand Interpretation

0< E(p)< 1 Inelastic

Percentage increase in price results

in smaller percentage decrease in

demand.

E(p) = 1 Unitary

Percentage change in price results in

same percentage change in demand.

E(p)> 1 Elastic

Percentage increase in price results

in larger percentage decrease in de-mand.

95

Elasticity of demand, complete example

Begin with a price-demand equation: 2p + 0.01x= 50.

1. Express the demand x as a function of the price p.

2. Find the elasticity of demand function E(p).

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Elasticity of demand, complete example (contin-ued)

E(p) =

3. Find the elasticity of demand at a price of $15. Is demand

inelastic, unitary, or elastic at that price?

4. If the price of $15 is increased 20%, what is the percentage

change in demand?

97

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99

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Elasticity of Demand Part 2

Section 11.7

ccsun 2009

An example from the book: Problem 17

Another complete example

101

Elasticity of Demand: Sec 11.7, Problem 17

E(p) = pf(p)f(p)

.

Begin with the price-demand equation:

p + 0.005x= 30.

(A) Express the demand x as a function of the price p.

(B) Find the elasticity of demand E(p).

(C) What is the elasticity of demand when p= $10?

If this price is increased by 10%, what is the approximate changein demand?

(D) What is the elasticity of demand when p= $25?

If this price is increased by 10%, what is the approximate change

in demand?

(E) What is the elasticity of demand when p= $15?


in demand?

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E(p) =



in demand?

105


E(p) =

(E) What is the elasticity of demand when p= $15?


in demand?

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Elasticity of Demand and Revenue,Sec 11.7, Problem 17

5 10 15 20 25 30

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

Revenue as a function of price. R(p) =xp = 200(30p)p.

Which price maximizes revenue?

How does this relate to elasticity?

107

Elasticity of Demand and Revenue,Sec 11.7, Problem 17

Elasticity of demand is 1 at the price where marginal revenue is

zero:

E(p) = 1:

R(p) = 0:

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109

Elasticity of Demand: New Example

E(p) = pf(p)

f(p) .


2p + 0.10x= 50.



(C) What is the elasticity of demand when p= $15?If this price is increased by 20%, what is the approximate change

in demand?



in demand?

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Elasticity of Demand: New Example continued

E(p) = pf(p)

f(p) .


2p + 0.10x= 50.



111


E(p) = pf(p)f(p)

= .

(C) What is the elasticity of demand when p= $15?


in demand?

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E(p) = pf(p)

f(p) = .



in demand?

113

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115

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Maxima and Minima

Section 12.5

ccsun 2009

What is an absolute maximum (or minimum)?

Where do they occur?

117

A graphical example

The graph of a function f(x) is shown below.

2,4

1, 7

2,20

4, 32

2 1 1 2 3 4

20

10

10

20

30

The highest point on the graph is (4, 32). So the absolute max-imum of f(x) is 32 and it occurs where x= 4.

The lowest point on the graph is (2,20). So the absolute min-imum of f(x) is20 and it occurs where x= 2.The points (2,4) and (4, 32) are called the end pointsof thegraph. x= 2, 4 are the end pointsof the domain of f.The points (1, 7) and (2,20) are the critical points of thegraph. x= 1, 2 are the critical values of x.

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Definition of absolute maximum/minimum

Let f(x) be a function.

f(c) is an absolute maximumof the function f iff(c) f(x) forall x in the domain of f. In other words, the value of f at c is

greater than or equal to the values of f for every other x in the

domain

f(c) is an absolute minimum of the function f if f(c) f(x) forall x in the domain of f.

The example above suggests that the absolute maximum and

the absolute minimum must occur at an end point of the graph

or at a point where the slope of the tangent line is zero, that isf(x) = 0; these are critical points on the graph.

119

Identifying types of points on a graph

Identify the end points, points where the tangent is horizontal,

absolute maximums, and absolute minimums for the functionf(x) with the following graph:

0, 0

1, 467

3, 1971

4, 1792

5, 3275

1 1 2 3 4 5

1000

2000

3000

4000

What are the end points?

What are the points where the slope of the tangent line is zero?

What is the absolute maximum(s)?

What is the absolute minimum(s)?

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Extreme value theorem

A functionfthat is continuous on a closed interval [a, b] has both

an absolute maximum value and an absolute minimum value on

that interval.

If the function has a derivative for every point in the interval [a, b],

then the absolute maximum and the absolute minimum must

occur either at an end point of the interval or at a point where

the slope of the tangent line is zero, that is where f(x) = 0.

121

Steps in finding absolute maximums and mini-mums

The values of x in the domain of f where f(x) = 0 or wheref(x) does not exist are called the critical values.

The absolute maximum and minimum of f(x) o n [a, b] always

must occur at critical values or at the end points of the interval.

1. Check to make sure f is continuous on the interval [a, b].

2. Find the critical values of x in the interval [a, b].

3. Evaluate f(x) at the end points a and b and at the critical

values of x.

4. The absolute maximum and minimum of f(x) o n [a, b] are

the largest and smallest of the values found in step 3

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127

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Maximizing Profit, Part 1

Section 12.6

ccsun 2009

Find the maximum profit from a graph

Profit is maximized when marginal revenue equals marginal

cost: R(x) =C(x).

129

Maximizing revenue: an example

A company can sell x fake tattoos per year at p dollars per

tattoo. The price-demand equation for the tattoos is

p= 10 .001x.To maximize its revenue, what price should the company charge

for each tattoo?

Summary:

What is the maximum revenue?

Summary:

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The graph of revenue and cost: tattoos contin-ued

2000 4000 6000 8000 10 000

Number of tattoos

4000

8000

12 000

16 000

20 000

24 000

28 000

Revenue and Cost in Dollars

Locate the point on the graph where profit is maximized.

Locate the point on the graph where revenue is maximized.

133

Profit on the graph

Where does the profit appear on the graph?

Since profit is the difference between cost and revenue, the profit

for a particular value ofx is represented by the length of the ver-tical line between the cost and revenue functions. These verticallines have been added to the graph. For example, if the produc-

tion/sales level is x = 6000 tattoos, the profit is represented bythe thick line on the graph.

Which of the vertical lines is the longest?

What does this mean in terms of profit?

2000 4000 6000 8000 10 000

Number of tattoos

4000

8000

12000

16000

20000

24000

28000

Revenue and Cost in Dollars

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Summary

To find the maximum profit over an interval [a, b]:

1. Find the critical values in the interval by solving R(x) =C(x).

2. Evaluate the profit function P(x) at the end points a and b

and at the critical values found in step 1.

3. The maximum profit on [a, b] is the largest of the values

found in step 2.

Most often, the interval is [0,) and because of the fixed cost,the profit is negative at x = 0. In these cases, the maximum

profit does not occur at an end point. It will occur at a point

where marginal cost equals marginal revenue.

135

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Maximizing Profit, Part 2

Section 12.6

ccsun 2009

A complete example

Changes in the cost function

137

A complete example:

ArtCo produces and sells special drawing kits. The cost and

price-demand information is given below:

Fixed costs $4105Per item cost $30Price-demand equation p= 440 5x

(A) Find the production/sales level that maximizes profit.(B) Find the price that maximizes profit.(C) Find the production costs at the production levelthat maximizes profit.(D) Find the revenue at the sales level that maximizes profit.(E) Find the maximum profit.

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(D) Find the revenue at the sales level that maximizes profit.

Summary:

(E) Find the maximum profit.

Summary:

141

The revenue and cost functions are shown on the graph below

along with the line tangent to the revenue curve at the point

where profit is maximum.

The maximum profit occurs when the production/sales level is

x= 41 kits.

20 40 60 80Kits

2000

4000

6000

8000

10000

12000

Cost, Revenue in dollars

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What happens if the fixed costs increase from $4105 to $5000?

Compare the two graphs:

20 40 60 80Kits

2000

4000

6000

8000

10000

12000


20 40 60 80Kits

2000

4000

6000

8000

10000

12000


Fixed costs are $4105 Fixed costs are $5000

It is clear from the graph that the production/sales level that

maximizes profit stays at x = 41. That is, the derivative of the

cost function, C(x) = 30, does not depend on the fixed costs,only the per-item cost.

The profit at that level decreases by the increase in the fixed

costs of $895. The break-even points are closer together when

the fixed costs are higher.

143

What happens if the per-item cost increases from $30 to $50?

Again we compare the two graphs:

20 40 60 80Kits

2000

4000

6000

8000

10000

12000


20 40 60 80Kits

2000

4000

6000

8000

10000

12000


Per-item cost is $30 Per-item cost is $50

This time the production/sales level that maximizes profit is

x = 39. The derivative of the cost function is now 50 instead of

30. So the point on the revenue curve where the slope is steeper

(50 instead of 30) is to the left of the point where x= 41. That

is, the solution to the equation R(x) = 50 (x= 39) is less thanthe solution to R(x) = 30 (x= 41).

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Another example:PatTech produces and sells patio furniture. The cost and price-

demand information for their outdoor lava lamp is given below:

Fixed costs $6156Per item cost $20

Price-demand equation p= 420 4x

(A) Find the production/sales level that maximizes profit.(B) Find the price that maximizes profit.(C) Find the production costs at the production levelthat maximizes profit.(D) Find the revenue at the sales level that maximizes profit.(E) Find the maximum profit.(F) If fixed costs decrease by $400, does the production/saleslevel that maximizes profit decrease, increase, or stay the same?

(G) If the per-item cost decreases, does the production/saleslevel that maximizes profit decrease, increase, or stay the same?

145

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Answers:

(A) x= 50(B) p= 220(C) C(50) = 7156(D) R(50) = 11, 000(E) P(50) = 3844(F) stays the same(G) increases

20 40 60 80 100Lamps

2000

4000

6000

8000

10000

12000Cost, Revenue in dollars

147

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Systems of Linear Equationsin Two Unknowns

Section 4.1

ccsun 2009

Solve by graphing

Sole using substitution

Solve by elimination (addition method)

149

Warm-up problem

A restaurant serves two types of fish dinnerssmall for $5.99

and large of $8.99.

Yesterday, there were 134 total orders of fish with total receipts

of $1024.66. How many small and how many large fish dinners

were ordered?

Set up the two equations in two unknowns, where s is the num-

ber of small dinners ordered andl

is the number of large dinners.

We will solve it later.

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Systems of two equations in two unknowns

This is an example of a system of two linear equations in two

unknowns:

3x + 5y = 9x + 4y = 10.

A solution is an ordered pari (x0, y0) that satisfies each equation.

The solution set is the set of all solutions.

Is (3, 0) a solution?

Is (2,3) a solution?

151

Solve the system by graphing

3x + 5y = 9x + 4y = 10.

5 4 3 2 1 1 2 3 4 5

5

4

3

2

1

1

2

3

4

5 Three points are shown on

the graph.

Write the coordinates of

each point.

Which of the points is a

solution to the system of

equations?

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Solve this system using substitution

3x 2y = 7y = 2x 3

155

Independent consistent systems

A system of linear equations is consistent if it has one or

more solutions.

If a consistent system has exactly one solution then the sys-

tem is independent.

This system of equations has only one solution, (x, y) = (2, 5):

x + 2y = 12

x 3y = 13Is the system consistent?

Is the system independent?

What can you say about the graphs of the equations?

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Dependent consistent systems

If a consistent system has more than one solution, then the

system is dependent.

The system

p = 0.1x + 32p = 0.2x + 6

has infinitely many solutions: (x, p) = (0, 3), (10, 2), (20, 1), . . ..

So this system is dependent consistent.

What can you say about the graphs of these dependent equa-

tions?

157

Inconsistent systems

An inconsistent linear system is one that has no solutions. For

example, this system has no solutions:

p = 0.1x + 3p = 0.1x + 6

What can you say about the graphs of these inconsistent equa-tions?

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Test your understanding

For each of the system below, state whether it is consistent

or inconsistent. If the system is consistent, is it dependent or

independent?

x + y = 4

2x y = 2

6x 3y = 92x y = 3

2x y = 46x 3y = 18

159

Solution by elimination (addition)

2x 7y = 3 (1)5x + 3y = 7 (2)

You may do any of the following and the solutions will not

change:

1. Write the equations in any order.

2. Multiply any equation by a nonzero constant.

3. Multiply any equation by a nonzero constant and add it to

another equation.

Follow these steps to solve the system above:

1. Multiply Equation (1) by 5 and Equa-

tion (2) by 2.

2. Add the two equations and replace

Equation (2) with the sum.

3. Use the second equation to solve for y.

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Solve using elimination

2x 7y = 35x + 3y = 7

161

Solve using any method

The US Postal Service charges a base price for overnight de-

livery of packages that weigh one ounce or less and a surchargefor each additional ounce or partial ounce. The cost of mailing

a 6 ounce package is $1.68 and the cost of mailing a 13 ounce

package is $2.87. Find the base price and the surcharge. (Useb

for the base price and s for the surcharge.)

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Solve the warmup (fish) problem

A restaurant serves two types of fish dinnerssmall for $5.99

and large of $8.99. Yesterday, there were 134 total orders of fish

with total receipts of $1024.66. How many small and how many

large fish dinners were ordered?

163

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Systems of Linear Equationsand the Augmented Matrix

Section 4.2

ccsun 2009

In this section, we develop a matrix method for solving systems

of two linear equations in two unknowns. This method is a

generalization of the elimination method from Section 4.1 and

can be used by computers and calculators to solve systems of

equations with many equations and many unknowns.

165

Matrices

A matrix is a rectangular array of numbers written within brack-

ets.

Example with three rows and three columns:

1 6 82 4 1

7 3 5

Example with three rows and two columns: 1 34 3

1 9 How many rows (columns) does the matrix A have?

A=

1 3 0 01 3 4 31 0 7 9

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Entries of a matrix

The subscripts give the address

of each entry of the matrix. For

example the entry a23 is found in

row 2 and column 3.

a11 a12 a13 a14a21 a22 a23 a24

a31 a32 a33 a34

Circle a21 in each matrix, if it exists.

1 35 0

309

2 5 8 1

4 57 3

10 48 8

11 6

167

Size of a matrix

When a matrix has m rows and n columns, its size is written

m n, as in A is an m-by-n matrix.

So

1 3 65 7 0

is a 2 3 matrix.

What is the size of each matrix?

1 35 0

309

2 5 8 1

4 57 3

10 48 8

11 6

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Matrix method for solving a linear system

First, represent the linear system of equations as an augmented

matrix.

Linear system Augmented matrix

x + 3y = 52x y = 3

1 3 52 1 3

4x y = 6x y = 0

4 1 61 1 0

a11x1+ a12x2 = k1

a21x1+ a22x2 = k2

a11 a12 k1a21 a22 k2

Write the augmented matrix for this system:

4x y = 6x y = 0

169

Matrix method continued

Then, manipulate the augmented matrix (with row operations

coming in the next slide) to obtain one of the following forms.

Now you can just read off the solution. What is the solution to

each system?

1 0 50 1 3

1 2 50 0 0

1 2 50 0 3

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Row operations, which do not change the solu-tions to the system

Augmented matrix Symbol System of equationsInterchange rows i and j Ri Rj Change order of equa-

tions i and jMultiply row i by a nonzero

constant k

kRi Ri Multiply both sides ofequation i by a nonzero

constant k

Add k times row i to row j kRi+ Rj Rj Replace equationi by the

sum of equation i and k

times equation j

171

An example

Solve

x + 3y = 5

2x y = 3 1 3 52 1 3

Augmented matrix

2R1+ R2

R2

1 3 50

7

7 This makes a12 = 0

17R2 R2

1 3 50 1 1

This makes a22 = 1

3R2+ R1 R1

1 0 20 1 1

The matrix is in

reduced form

Solution: x= 2, y = 1 or (x, y) = (2, 1)

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New problem

Solve:

2x + 6y = 65x + 7y = 1

Write the symbols for the row operations. Start with the row

operation 12R1 R1. Read as replaces.

173

Intersecting lines

Use matrix methods to find the coordinates (x, y) of the pointof intersection for the lines with these equations:

x + 2.0y = 4.0

x + 0.5y = 2.5

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Another problem

Solve this system using matrix methods. Write the symbols for

the row operations.

10x1 2x2 = 65x1+ x2 = 3

175

Equilibrium points

Find the equilibrium point for these price/demand and price/supply

equations.

price/supply: 2p= x 66price/demand: 5p= 0.5x + 105

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Summary

There are three possible reduced forms for the augmented matrix

of a system of two equations in two unknowns:

Form 1: Unique soluion: x= m, y =n (consistent

and independent)

1 0 m0 1 n

Form 2: Infinitely many solutions: y can be any

real number and x = m ry (consistent and de-pendent)

1 r m0 0 0

Form 3: n = 0 No solution: (inconsistent) 1 r m0 0 n

177

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179

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Gauss-Jordan Elimination

Section 4.3

ccsun 2009

A linear system must have exactly one solution, no solution, or an

infinite number of solutions. Just as in the case of two equations

in two unknowns:

the term consistent and independent is use to describe a

system with only one solution.

consistent and dependentis used for a system with an infinite

number of solutions.

inconsistent is used to describe a system with no solution.

181

An illustrative example

To solve this system using Gauss-Jordan elimination, start with

the augmented matrix.x + y z=2

2x y+ z= 5x + 2y+ 2z= 1

1 1 1 22 1 1 51 2 2 1

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Matrix representations of independent, inconsis-tent, and dependent systems

The following matrices represent systems of three linear equa-

tions in three unknowns. Write the solution for each.

consistent and independent

1 0 0 30 1 0 40 0 1 5

inconsistent

1 2 3 40 0 0 6

0 0 0 0

consistent and dependent

1 2 3 40 0 0 00 0 0 0

1 0 3 40 1 9 4

0 0 0 0

183

Reduced row echelon form

A matrix is in reduced row echelon form or more simply, in re-duced form, if:

1. The rows consisting entirely of zeros are below the other

rows.

2. The leftmost nonzero entry in each row is 1.

3. All other entires in the column containing the leftmost 1 of

the row are zeros.

4. The leftmost 1 in any row is to the right of the leftmost 1in the row above it.

All but one of these matrices are in reduced form. Which one is

not and which of the four conditions is not satisfied? 1 0 0 40 1 0 2

0 0 1 6

1 2 0 30 5 1 2

0 0 0 0

1 2 3 40 1 0 8

0 0 0 0

1 0 0 70 1 0 7

0 0 0 9

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More on reduced form

Which matrices are in reduced form? If a matrix is not in reduced

form, use a row operation to get it into reduced form. 2 0 4 40 1 0 7

0 0 1 5

1 0 3 40 1 2 9

0 0 0 0

1 2 0 40 1 0 5

0 0 1 8

185

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187

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Matrices: Basic Operations

Section 4.4

ccsun 2009

Add and subtract two matrices

Multiply a number and a matrix

Multiply two matrices

Matrix models

189

Addition and subtraction of Matrices

To add or subtract two matrices, they must be the same size.

To add (or subtract) matrices of the same size, add (or subtract)

their corresponding entries.

For example, if

A=1 2

5 2

, and B =4 5

9 1

,

then

A + B =

, and AB =

.

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More examples:

4 3 10 1 5

+

2 1 04 2 1

=

6 1 14 1 52 1 1

2 1 04 2 13 2 7

=3 1

1 5

+

2 1 04 2 1

=

2 110 4

1 0

+

1 02 10 0

=

14

10

+

1

231

=

191

Scalar Multiiplication

To find the scalar product of a number k and a matrix A,

multiply each entry in A by the number k. The number k is

called a scalar.

For example, if k = 2 and

A=

6 1 14 1 5

2 1 1

, then 2A=

.

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Combining operations

2

4 3 11 5 1

2 1 04 2 1

=

2

3 1 0

2 1 50 0 2

+ 3

2 1 0

4 2 15 2 1

=

193

Multiplying a Row Matrix and a Column Matrix

Observe how a 1 4 row matrix is multiplied by a 4 1 columnmatrix:

1 2 3 4

5678

= 1 5 + 2 6 + 3 7 + 4 8 = 70

Now try this example:

1 0 5 2

74

13

=

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Revenue of a Clothing Store

The number of items sold at a clothing store and the price of

each item are given below. Use a matrix operation to determine

the total revenue each day.

Monday: 3 T-shirts at $10 each, 4 hats at $15 dollars each, and

1 pair of shorts at $20.

Tuesday: 4 T-shirts at $10 each, 2 hats at $15 each, and 3 pairs

of shorts at $20.

199

Revenue of a Clothing Store:Solution

Price matrix P and quantity matrix Q:

P =

10 15 20

Q=

3 44 2

1 3

Total revenues for the two days:

P Q=

10 15 20 3 44 2

1 3

= 110 130

Summary: The total revenue for Monday is $110.00 and for

Tuesday is $130.00.

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Labor and Wage Requirements

A boat-manufacturing company has plants in Massachusetts and

Virginia. The labor-hour and wage requirements for manufactur-

ing three types of boats are recorded in these tables and matrices:

cutting assembly packagedept. dept. dept.

1 man boat 0.6 0.6 0.22 man boat 1.0 0.9 0.33 man boat 1.5 1.2 0.4

M=

0.6 0.6 0.21.0 0.9 0.3

1.5 1.2 0.4

MA VAcutting $17 $ 14assembly $12 $10package $10 $9

N =

17 1412 1010 9

Which is the labor-hours matrix?

Interpret the entry in row 1, column 3 of the matrix M.

201

Labor and Wage Requirements, continued

M =

0.6 0.6 0.2

1 0 0 9 0 3

N =

17 14

12 10

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