Chapter 1 APPLIED CALCULUS FOR THE MANAGERIAL, LIFE, AND SOCIAL SCIENCES: A BRIEF APPROACH

Preliminaries

1• Precalculus Review

• The Cartesian Coordinate System

• Straight Lines

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

The Real Numbers

The real numbers can be ordered and represented in order on a number line

-3 -2 -1 0 1 2 3 4

-1.87

0

4.552


Inequality Graph Interval

3 7x< ≤

5x >

13

x ≤ −

( ]3,7

( )5,∞

1,

3 −∞ −

]

( ]

(5

3 7

13

−

) or ( means not included in the solution] or [ means included in the solution

Inequalities, graphs, and notation


IntervalsInterval Graph

( )

[ ]

( ]

[ )

(

)

[

]

a b

Example

(a, b)

[a, b]

(a, b]

[a, b)

(a, )

(- , b)

[a, )

(- , b]

(3, 5)

[4, 7]

(-1, 3]

[-2, 0)

(1, )

(- , 2)

[0, )

(- , -3]

( )

[ ]

( ]

[ )

(

)

[

]

a b

a b

a b

a

a

b

b

3 5

-2 0

4 7

-1 3

-3

2

1

0

∞

∞

∞ ∞

∞

∞ ∞

∞


Absolute Value

if 0 if 0

a aa

a a≥

= − <To evaluate:

3 8−

5 2 5 3− − − 5 2 (3 5)= − − − 2 5 5= −

( 5) 5= − − =5= −

Notice the opposite sign


Exponents

na 35 5 5 5 125= ⋅ ⋅ =...na a a a a= ⋅ ⋅ ⋅ ⋅

Definition

n factors

Examplen,m positive integers

0a

na−

( )0 1 0a a= ≠

( )10n

na aa

− = ≠

032 1=

44

1 12

162− = =

/m na

/m na−

/ nm n ma a=

/ 1m n

n ma

a− =

32 / 3 2125 125 25= =

3 / 2 34 9 279 4 16

− = =


Laws of Exponents

m n m na a a +=

Law Example

( )nm mna a=m

m nn

aa

a−=

( )n n nab a b=n n

n

a ab b

=

3 12 3 12 15x x x x+= =

( )65 5(6) 303 3 3= =14

14 12 212

yy y

y−= =

( )4 4 4 43 3 81r r r= =3 3

3 3

4 4 64x x x

= =


Algebraic Expressions

• Polynomials

• Rational Expressions

• Other Algebraic Fractions


Polynomials• Addition

( ) ( )3 2 33 2 7 15 5 13 12x x x x x− + + + − +3 2 3

3 2

3 2 7 15 5 13 12

8 2 6 27

x x x x x

x x x

− + + + − +

− − + Combine like terms

• Subtraction

( ) ( )3 2 3 26 1 3 2x x x x x x− + + − − +3 2 3 2

3

6 1 3 2

2 4 1

x x x x x x

x x

− + + − + −

− + + Combine like terms

Distribute


Polynomials• Multiplication

( )( )2 5 3 2x x− +

Combine like terms

Distribute2 (3 2) 5(3 2)x x x+ − +

Distribute26 4 15 10x x x+ − −26 11 10x x− −


Factoring Polynomials

3 26 36t t−

• Greatest Common Factor

• Grouping

( )26 6t t −

2 2 2mx mx x+ − −

( ) ( )1 2 1mx x x+ − +

The terms have 6t2 in common

( )( )2 1mx x− +

Factor mx Factor –2


Factoring Polynomials

• Sum/Difference of Two Cubes:

• Difference of Two Squares:

2 9m −

38 1x +( ) ( )22 1 4 2 1x x x+ − +

( ) ( )3 3m m− +

( ) ( )2 2x y x y x y− = − +

( )( )3 3 2 2x y x y x xy y± = ± +m

Ex.

Ex.


Factoring Polynomials• Trinomials

2 5 6x x− +

3 26 27 12x x x+ +

( ) ( )3 2x x− −

Ex.

Ex.

Trial and Error

( )23 2 9 4x x x+ +

Trial and Error( ) ( )3 2 1 4x x x+ +

Greatest Common Factor


Roots of Polynomials• Finding roots by factoring

( ) ( )2 1 3 0

2 1 0 or 3 0

1 or 3

2

x x

x x

x x

− + =

− = + =

= = −

22 5 3 0x x+ − =

(find where the polynomial = 0)

Ex.


Roots of Polynomials

• The Quadratic Formula:

If ( )2 0 0ax bx c a+ + = ≠

2 42

b b acx

a− ± −

=

• Finding roots by the Quadratic Formula

with a, b, and c real numbers then


ExampleUsing the Quadratic Formula:

( )( )( )

27 7 4 3 1 7 372 3 6

x− ± − − ±

= =

Ex. Find the roots of 23 7 1x x+ +

Here a = 3, b = 7, and c = 1

Plug in

Note values

7 37 7 37 or

6 6x

− + − −=Simplify


Rational ExpressionsP, Q, R, and S are polynomials

Addition

Operation

Multiplication

Subtraction

Division

P Q P QR R R

++ =

P Q P QR R R

−− =

P Q PQR S RS

⋅ =

P Q P S PSR S R Q RQ

÷ = ⋅ =

Notice the common denominator

Reciprocal and Multiply


Rational Expressions• Simplifying

2

225

7 10x

x x−

+ +

( ) ( )( ) ( )

5 52 5

x xx x

− +=

+ +

Cancel common factorsFactor

• Multiplying2 2

3 22 1 6 6

1x x x x

x x+ + −

⋅−

( ) ( ) ( )( ) ( )3

1 1 6 11 1

x x x xx xx

+ + −= ⋅

− +

FactorCancel common factors

2

Multiply Across

52

xx

−=

+

( )2

6 1x

x

+=


Rational Expressions• Adding/Subtracting

3 24x x

++

Combine like terms

( )( )

3 4 2( 4) 4x x

x x x x+

= ++ +

Must have LCD: x(x + 4)

( )3 12 2 5 12

( 4) 4x x xx x x x+ + +

= =+ +

Distribute and combine fractions


Other Algebraic Fractions• Complex Fractions

32

9 4

x

xx

−

−

Simplify to get to here

Distribute and reduce to get here

32

94

xx

x xx

− =

−

2

3 29 4

xx

−=

−

Multiply by the LCD: x

( )( )3 2 1

3 2 3 2 3 2x

x x x−

= =− + +


Other Algebraic Fractions

• Rationalizing a Denominator

73 y−

Simplify

( )( )( )

7 3

3 3

y

y y

+=

− +

21 79

yy

+=

−

Multiply by the conjugate

Notice: ( )( )a b a b a b+ − = −


Cartesian Coordinate System

y-axis

x-axis

(x, y)


Cartesian Coordinate System

Ex. Plot (4, 2)

(4, 2)

Ex. Plot (-2, 1)

Ex. Plot (2, -3)

(2, -3)(-2, 1)


The Distance Formula

( )2 2,x y

( )1 1,x y

( ) ( )2 21 2 1 2d x x y y= − + −


The Distance Formula

( )7,5

( )3, 2− −

( ) ( )

( ) ( )

2 21 2 1 2

2 27 ( 3) 5 ( 2)

100 49 149

d x x y y

d

d

= − + −

= − − + − −

= + =

Ex. Find the distance between (7, 5) and (-3, -2)

7

10


The Equation of a Circle

( ) ( )2 2 2x h y k r− + − =

A circle with center (h, k) and radius of length r can be expressed in the form:

Ex. Find an equation of the circle with center at (4, 0) and radius of length 3

( ) ( )( )

2 2 2

2 2

4 0 3

4 9

x y

x y

− + − =

− + =


Straight Lines

• Slope

• Point-Slope Form

• Slope-Intercept Form


Slope – the slope of a non-vertical line that passes through the points

is given by: ( )2 2,x y and ( )1 1,x y

2 1

2 1

y yym

x x x−∆

= =∆ −

Ex. Find the slope of the line that passes through the points (4,0) and (6, -3)

3 0 3 36 4 2 2

ym

x∆ − − −

= = = =−∆ −


Slope

Two lines are parallel if and only if their slopes are equal or both undefined

Two lines are perpendicular if and only if the product of their slopes is –1. That is, one slope is the opposite sign and reciprocal of the other slope (ex. ).3 4

and 4 3

−


Point-Slope Form

( )( )

1 1

1 4 3

1 4 124 11

y y m x x

y x

y xy x

− = −

− = −

− = −= −

An equation of a line that passes through the point with slope m is given by:

( )1 1,x y

Ex. Find an equation of the line that passes through (3,1) and has slope m = 4

( )1 1y y m x x− = −


Slope-Intercept Form

4 45

y mx b

y x

= +

= −

An equation of a line with slope m and y-intercept can be given by:( )0,b

Ex. Find an equation of the line that passes through (0,-4)

and has slope .

y mx b= +

45

m =


Vertical Lines

x = 3

Can be expressed in the form x = a


Horizontal Lines

y = 2

Can be expressed in the form y = b


Chapter 1 APPLIED CALCULUS FOR THE MANAGERIAL, LIFE, AND SOCIAL SCIENCES: A BRIEF APPROACH

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Transcript of Chapter 1 APPLIED CALCULUS FOR THE MANAGERIAL, LIFE, AND SOCIAL SCIENCES: A BRIEF APPROACH