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Chapter 1 Slide for Market-leading APPLIED CALCULUS FOR THE MANAGERIAL, LIFE, AND SOCIAL SCIENCES. Applies math to your world in fun and interesting ways. It delivers just the right balance of teaching, technology, and enlightening real-life examples. And when it comes to study time, the Ninth Edition offers an exciting array of supplements that maximize your efforts and improve your results.
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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
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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
∞
∞
∞ ∞
∞
∞ ∞
∞
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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
− = =
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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
= =
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Algebraic Expressions
• Polynomials
• Rational Expressions
• Other Algebraic Fractions
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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− −
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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.
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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.
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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
+=
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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−
= =− + +
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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+ − = −
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Cartesian Coordinate System
y-axis
x-axis
(x, y)
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Cartesian Coordinate System
Ex. Plot (4, 2)
(4, 2)
Ex. Plot (-2, 1)
Ex. Plot (2, -3)
(2, -3)(-2, 1)
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
The Distance Formula
( )2 2,x y
( )1 1,x y
( ) ( )2 21 2 1 2d x x y y= − + −
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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
− + − =
− + =
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Straight Lines
• Slope
• Point-Slope Form
• Slope-Intercept Form
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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∆ − − −
= = = =−∆ −
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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
−
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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− = −
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
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 =
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Vertical Lines
x = 3
Can be expressed in the form x = a
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.
Horizontal Lines
y = 2
Can be expressed in the form y = b
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.