introductory mathematical analysis for business, economics, and the life and social sciences 2011...

INTRODUCTORY MATHEMATICAL INTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences

2011 Pearson Education, Inc.

Chapter 1 Chapter 1 Applications and More AlgebraApplications and More Algebra


• To model situations described by linear or quadratic equations.

• To solve linear inequalities in one variable and to introduce interval notation.

• To model real-life situations in terms of inequalities.

• To solve equations and inequalities involving absolute values.

• To write sums in summation notation and evaluate such sums.

Chapter 1: Applications and More Algebra

Chapter ObjectivesChapter Objectives



Chapter OutlineChapter Outline

Applications of Equations

Linear Inequalities

Applications of Inequalities

Absolute Value

Summation Notation

1.6) Sequence

1.1)

1.2)

1.3)

1.4)

1.5)


• Modeling: Translating relationships in the problems to mathematical symbols.


1.1 Applications of Equations1.1 Applications of Equations

A chemist must prepare 350 ml of a chemical

solution made up of two parts alcohol and three

parts acid. How much of each should be used?

Example 1 - Mixture


Solution:Let n = number of milliliters in each part.

Each part has 70 ml. Amount of alcohol = 2n = 2(70) = 140 mlAmount of acid = 3n = 3(70) = 210 ml

705

350

3505

35032

n

n

nn


1.1 Applications of Equations

Example 1 - Mixture


• Fixed cost is the sum of all costs that are independent of the level of production.

• Variable cost is the sum of all costs that are dependent on the level of output.

• Total cost = variable cost + fixed cost

• Total revenue = (price per unit) x (number of units sold)

• Profit = total revenue − total cost




The Anderson Company produces a product for which the variable cost per unit is $6 and the fixed cost is $80,000. Each unit has a selling price of $10. Determine the number of units that must be sold for the company to earn a profit of $60,000.



Example 3 – Profit


Solution:

Let q = number of sold units.

variable cost = 6q

total cost = 6q + 80,000

total revenue = 10q

Since profit = total revenue − total cost

35,000 units must be sold to earn a profit of $60,000.

q

q

qq

000,35

4000,140

000,80610000,60



Example 3 – Profit


A total of $10,000 was invested in two business ventures, A and B. At the end of the first year, A and B yielded returns of 6%and 5.75 %, respectively, on the original investments. How was the original amount allocated if the total amount earned was $588.75?



Example 5 – Investment


Solution:

Let x = amount ($) invested at 6%.

$5500 was invested at 6%

$10,000−$5500 = $4500 was invested at 5.75%.

5500

75.130025.0

75.5880575.057506.0

75.588000,100575.006.0

x

x

xx

xx



Example 5 – Investment




Example 7 – Apartment Rent

A real-estate firm owns the Parklane Garden Apartments, which consist of 96 apartments. At $550 per month, every apartment can be rented. However, for each $25 per month increase, there will be three vacancies with no possibility of filling them. The firm wants to receive $54,600 per month from rent. What rent should be charged for each apartment?


Solution 1:

Let r = rent ($) to be charged per apartment.

Total rent = (rent per apartment) x (number of apartments rented)





Solution 1 (Con’t):

Rent should be $650 or $700.

256756

500,224050

32

000,365,13440504050

0000,365,140503

34050000,365,1

25

34050600,54

25

165032400600,54

25

550396600,54

2

2

r

rr

rr

rr

rr

rr





Solution 2:

Let n = number of $25 increases.

Total rent = (rent per apartment) x (number of apartments rented)





Solution 2 (Con’t):

The rent charged should be either

550 + 25(6) = $700 or

550 + 25(4) = $650.

4 or 6

046

02410

0180075075

75750800,52600,54

39625550600,54

2

2

2

n

nn

nn

nn

nn

nn






1.2 Linear Inequalities1.2 Linear Inequalities

• Supposing a and b are two points on the real-number line, the relative positions of two points are as follows:


• We use dots to indicate points on a number line.

• Suppose that a < b and x is between a and b.

• Inequality is a statement that one number is less than another number.


1.2 Linear Inequalities


• Rules for Inequalities:

1. If a < b, then a + c < b + c and a − c < b − c.

2. If a < b and c > 0, then ac < bc and a/c < b/c.

3. If a < b and c < 0, then a(c) > b(c) and a/c > b/c.

4. If a < b and a = c, then c < b.

5. If 0 < a < b or a < b < 0, then 1/a > 1/b .

6. If 0 < a < b and n > 0, then an < bn.

If 0 < a < b, then .



nn ba




• Linear inequality can be written in the form

ax + b < 0where a and b are constants and a 0

• To solve an inequality involving a variable is to find all values of the variable for which the inequality is true.




Example 1 – Solving a Linear Inequality

Solve 2(x − 3) < 4.

Solution:Replace inequality by equivalent inequalities.

5 2

10

2

2

102

64662

462

432

x

x

x

x

x

x




Example 3 – Solving a Linear Inequality

Solve (3/2)(s − 2) + 1 > −2(s − 4).

7

20

207

16443

442232

422122

32

42122

3

s

s

ss

ss

s-s

ss

The solution is ( 20/7 ,∞).

Solution:



1.3 Applications of Inequalities1.3 Applications of Inequalities

Example 1 - Profit

• Solving word problems may involve inequalities.

For a company that manufactures aquarium heaters, the combined cost for labor and material is $21 per heater. Fixed costs (costs incurred in a given period, regardless of output) are $70,000. If the selling price of a heater is $35, how many must be sold for the company to earn a profit?


Solution:

profit = total revenue − total cost

5000

000,7014

0000,702135

0 cost total revenue total

q

q

qq

Let q = number of heaters sold.


1.3 Applications of Inequalities

Example 1 - Profit


After consulting with the comptroller, the president of the Ace Sports Equipment Company decides to take out a short-term loan to build up inventory. The company has current assets of $350,000 and current liabilities of $80,000. How much can the company borrow if the current ratio is to be no less than 2.5? (Note: The funds received are considered as current assets and the loan as a current liability.)



Example 3 – Current Ratio


Solution:

Let x = amount the company can borrow.

Current ratio = Current assets / Current liabilities

We want,

The company may borrow up to $100,000.

x

x

xx

x

x

000,100

5.1000,150

000,805.2000,350

5.2000,80

000,350



Example 3 – Current Ratio


• On real-number line, the distance of x from 0 is called the absolute value of x, denoted as |x|.

DEFINITIONThe absolute value of a real number x, written |x|, is defined by

0 if ,

0 if ,

xx

xxx


1.4 Absolute Value1.4 Absolute Value



1.4 Absolute Value

Example 1 – Solving Absolute-Value Equations

a. Solve |x − 3| = 2

b. Solve |7 − 3x| = 5

c. Solve |x − 4| = −3


Solution:a. x − 3 = 2 or x − 3 = −2 x = 5 x = 1

b. 7 − 3x = 5 or 7 − 3x = −5 x = 2/3 x = 4

c. The absolute value of a number is never negative. The solution set is .


1.4 Absolute Value



Absolute-Value Inequalities

• Summary of the solutions to absolute-value inequalities is given.


1.4 Absolute Value



1.4 Absolute Value


a. Solve |x + 5| ≥ 7

b. Solve |3x − 4| > 1

Solution:a.

We write it as , where is the union symbol.

b.

We can write it as .

2 12

75 or 75

xx

xx

,212,

3

5 1

143 or 143

xx

xx

,3

51,


Properties of the Absolute Value

• 5 basic properties of the absolute value:

• Property 5 is known as the triangle inequality.

baba

aaa

abba

b

a

b

a

baab

.5

.4

.3

.2

.1


1.4 Absolute Value



1.4 Absolute Value

Example 5 – Properties of Absolute Value

323251132 g.

222 f.

5

3

5

3

5

3 e.

3

7

3

7

3

7 ;

3

7

3

7

3

7 d.

77 c.

24224 b.

213737- a.

-

xxx

xx

Solution:



1.5 Summation Notation1.5 Summation Notation

DEFINITION

The sum of the numbers ai, with i successively taking on the values m through n is denoted as

nmmm

n

mii aaaaa

...21


Evaluate the given sums.

a. b.

Solution:

a.

b.


1.5 Summation Notation

Example 1 – Evaluating Sums

7

3

25n

n

6

1

2 1j

j

115

3328231813

275265255245235257

3

n

n

97

3726171052

1615141312111 2222226

1

2

j

j


• To sum up consecutive numbers, we have

where n = the last number

2

1

1

nni

n

i



6

)12(1

1

2

nnni

n

i


Evaluate the given sums.

a. b. c.

Solution:

a.

b.

c.

550,2510032

1011005 3535

100

1

100

1

100

1

kkk

kk

300,180,246

401201200999

200

1

2200

1

2

kk

kk



Example 3 – Applying the Properties of Summation Notation

2847144471

1

100

30

ij

100

1

35k

k

200

1

29k

k

100

30

4j


1.6 Sequence

• Arithmetic sequenceAn arithmetic sequence is a sequence (bk) defined

recursively by b1=a and, for each positive integer k, bk+1= d + bk

Example

1.5, 1.5+0.7 , 1.5+2*0.7, 1.5 +3*0.7, 1.5+ 4*0.7, 1.5+5*0.7

1.5, 2.2, 2.9, 3.6, 4.3, 5.0

• Geometric sequenceA geometric sequence is a sequence (ck) defined

recursively by

c1=a and, for each positive integer k, ck+1= ck*r

Example

2 2*3 , 2*3*3, 2*3*3*3, 2*3*3*3*3

2, 6, 18, 48, 144


Sums of sequences

• Sum of an arithmetic sequence - first n term

First term – a, common difference – d

• Sum of an geometric sequence

First term – a, common ratio – r

- Sum to first n term

- Sum of an infinite geometric sequence

for

)2)1((2

adnn

sn

1,1

)1(

rforr

ras

n

n

1

1

1,1

i

i

r

aarr


• Example 1

A rich woman would like to leave $100,000 a year, starting now, to be divided equally among all her direct descendants. She puts no time limit on his bequeathment and is able to invest for this long-term outlay of funds at 2% compounded annually. How much must she invest now to meet such a long-term commitment?


• Solution:

Let us write R=100,000, set the clock to 0 now, and measure times is years from now. With these conventions we are to account for payments for R payments of R at times 0,1,2…..k,.. By making a single investment now. Then the investment must equal to the sum

....)02.1(......)02.1()02.1( 21 kRRRR

First term=a=R=100,000

Common ratio=r=(1.02)-1. Since |r|<1, we can evaluate the required investment as

000,100,5

02.11

1

000,100

1

ra

introductory mathematical analysis for business, economics, and the life and social sciences 2011...

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