Chapter 1

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 1

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

Chapter 1

Prerequisites for Calculus


1.1

Lines


Quick Review

( )( )

1. Find the value of that corresponds to 3 in 2 4 3 .


In Exercises 3 and 4, find the value of that corresponds

to the values of and

y x y x

x y y x

m

x y

= =- + -

= = - +

.

33. 5, 2,

4

24. 1, 3,

3

yx y m

x

yx y m

x

-= = =

-

-=- =- =

-


Quick Review

In exercises 5 and 6, determine whether the ordered pair is a

solution to the equation.

5. 3 4 5 6. 2 5

1a) 2, b) 3, 1 a) 1,7 b) 2,1

4

In exercises 7 and 8, find the distance between the

x y y x

points.

17. 1,0 and 0,1 8. 2,1 and 1,

3


Quick Review

In Exercises 9 and 10, solve for in terms of .

9. 4 3 7 10. 2 5 3

y x

x y x y- = - + =-


Quick Review Solutions

( )

( )



In Exercises 3 and 4, find the value of that corresponds

to the value

2

1

s o

y

y

x y

x y y

x

x

x

m

=

= =-

-

=

-

- +

-

+

= =

f and .

33. 5, 2,

4

24. 1, 3,

1

5

3 4

x y

yx y m

x

yx y

x

m

mm

=--

= = =-

-=- - = ==

-



In exercises 5 and 6, determine whether the ordered pair is a

solution to the equation.

5. 3 4 5 6.

a) y

2 5

1a) 2, b) 3, 1 a) 1,7 b)

es

2,14

b) no a) yes

x y y x

In exercises 7 and 8, find the distance between the points.

17. 1,0 and

b) no

distance = 2

0,1 8. 2,1 and 1, 3

distance = 5/3



In Exercises 9 and 10, solve for in terms of .

9. 4 3 7 10. 2

4 7 2 3

3 3 5 5

5 3

y x

x y x y

y x y x

- = - +

= - = -

=-


Increments Slope of a Line Parallel and Perpendicular Lines Equations of Lines Applications

What you’ll learn about…

…and why.

Linear equations are used extensively in business and economic applications.


Increments

1 1 2 2

2 1 2 1

If a particle moves from the point ( , ) to the point ( , ), the

in its coordinates are

and

x y x y

x x x y y y increments

The symbols and are read delta and delta .

The letter is a Greek capital for difference.

Neither nor denotes multiplication;

is not delta times nor is delta times .

x y x y

d

x y

x x y y

D

D

D

D D


Example Increments

The coordinate increments from (8, 3) to (-6, 1) are:

6 8 14, 1 3 2x y


Slope of a Line

1 1 2 2

1

2 1

Let ( , ) and ( , ) be points on a nonvertical line, . The

of is

rise

run

1 2

2

P x y P x y L

L

y ym =

x x

-=

-

slope

A line that goes uphill as x increases has a positive slope. A line that goes downhill as x increases has a negative slope.


Slope of a Line

A horizontal line has slope zero since all of its points have the same

-coordinate, making 0.

For vertical lines, 0 and the ratio is undefined.

We express this by saying that vertical li

y y

yx

x

D =

DD =

D

nes have no slope.


Parallel and Perpendicular Lines

1 2

Parallel lines form equal angles with the -axis. Hence, nonvertical

parallel lines have the same slope.

=

If two nonvertical lines and are perpendicular, their slopes

and satisfy

1 2

1 2

x

m m

L L

m m 1 2

1 22 1

1, so each slope is the negative reciprocal

1 1of the other: ,

m m

m mm m

= -

=- =-


Equations of Lines

( )

( )

The vertical line through the point , has equation

since every -coordinate on the line has the same value .

Similarly, the horizontal line through , has equation .

a b x=a

x a

a b y b=


Example Equations of Lines

Write the equations of the vertical and horizontal lines through

the point ( 3,8).-

3 and 8x y =- =


Point Slope Equation

1

1

The equation ) is the of the line

through the point ( , ) with slope .1

1

y=m(x x y

x y m

point - slope equation- +


Example: Point Slope Equation

( ) ( )Write the point-slope equation for the line through 7, -2 and -5, 8 .

1 1

8 ( 2) 10 10 5The line's slope is = = =

5 (7) 12 12 6

We can use this slope with either of the two given points in the point-slope

equation. For ( , ) = (7, 2) we obtain

5= ( )

6

=

7 2

m

x y

y x

y

- -= - -

- - -

-

- - +-

5 35 2

6 65 23

= 6 6

x

y x

- + -

- +


Equations of Lines

The -coordinate of the point where a non-vertical line

intersects the -axis is the -intercept of the line.

Similarly, the -coordinate of the point where a non-horizontal

line intersects the -axis

y

y y

x

x

( )

is the -intercept of the line.

A line with slope and -intercept passes through 0, so( 0) , or

x

m y b by m x b y m x b= - + = +


Slope-Intercept Equation

The equation is the of the line

with slope and -intercept

y=m x + b

m y b.

slope - intercept equation


General Linear Equation

The equation ( and not both 0)

is a in and .

Ax By = C A B

x y

+

general linear equation

Although the general linear form helps in the quick identification of lines, the slope-intercept form is the one to enter into a calculator for graphing.


Example Analyzing and Graphing a General Linear Equation

Find the slope and y-intercept of the line 2 3 15. Graph the line. x y = -

Solve the equation for to put the equation in slope-intercept form:

3 = 2 15

2 15 =

3 32

= 53

y

y x

y x

y x

- - +

-+

- -

-

[-10, 10] by [-10, 10]


Example Determining a Function

The following table gives values for the linear function ( ) .

Determine and .

f x mx b

m b

= +

x f(x)

-1 -1

1 5

3 11

( ) ( ) ( )The graph of is a line. We know from the table that the following points are on

the line: 1, 1 , 1,5 , 3,11

11 5 6Using the last two points the slope is: = = = 3

3 1 2So = 3 Because f

f

m m

f(x) x b.

- -

--

+ (1) = 5, we have

( ) 3( )

5 = 3

= 2 Thus, 3, 2 and (

1 1

) 3 2

f b

b

b m b f x x

= +

+

= = = +


Example Reimbursed Expenses

A company reimburses its sales representatives $150 per day for lodging and

meals plus $0.34 per mile driven.

Write a linear equation giving the daily cost to the company in terms of ,

the number o

C x

f miles driven.

How much does it cost the company if a sales representative drives 137 miles on

a given day?

Because we know that the relationship is linear, we know that it conforms to the

equation ( ) .34 150.

If a sales representative drives 137 miles, then 137. Thus,

( ) .34( ) 150

( ) 46.58

137 137

13 17

C x x

x

C

C

= +

=

= +

= + 50

( ) 196.58

It will cost the company $196.58 for a sales representative to drive 137 miles a .

1

y

37

da

C =


1.2

Functions and Graphs


Quick Review

( )

( ) ( ) ( )( ) ( )

2 2

22

In Exercises 1 - 6, solve for .

1. 3 1 5 3 2. 2 0

3. 3 4 4. 2 5

5. 16 6. 9 0

In Exercises 7 and 8, describe how the graph of can be

transformed to the graph of .

7. , 2 3

8. , 5

x

x x x x

x x

x x

f

g

f x x g x x

f x x g x x

- £ + - >

- £ - ³

< - ³

= = + -

= = - +2


Quick Review

( )( ) ( ) ( ) ( )

( )

( ) ( ) ( ) ( )

2

In Exercises 9 - 12, find all real solutions to the equations.

9. 5

a 4 b 6

110.

a 5 b 0

f x x

f x f x

f xx

f x f x

= -

= =-

=

=- =


Quick Review

( )( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )

3

11. 7

a 4 b 1

12. 1

a 2 b 3

f x x

f x f x

f x x

f x f x

= +

= =

= -

=- =



( ) ( ) ( )

( )2 2

[ 2, ) ,0 2,

[ 1,7] ( , 3] [7,

In Exercises 1 - 6, solve for .

1. 3 1 5 3 2. 2 0

3. 3 4 4. 2 5

5. 16 6. 9 0

In Exercises 7 and 8, describe how the graph of can be

transformed to

)

4,4 [ 3

t

,3]

he

x

x x x x

x x

x x

f

- ¥ - ¥ È ¥

- - ¥ - È ¥

- £ + - >

- £ - ³

< - ³- -

( ) ( ) ( )( ) ( )

22 2 units left and 3 units d

graph of .

7. , 2 3

8. ,

ownward

5 units ri5 2 ght and 2 units upward

g

f x x g x x

f x x g x x

= = + -

= = - +



( )( ) ( ) ( ) ( )( ) ( )

( )

( ) ( ) ( ) ( )

( ) ( )

2

In Exercises 9 - 12, find all real solutions to the equations.

a 3,3 b no real solutio

9. 5

a 4 b 6

110.

a 5 b 0

n

1a b no real solution

5

f x x

f x f x

f xx

f x f x

= -

= =-

=

-

-

-

=

=



( )( ) ( ) ( ) ( )( ) ( )

( )( ) ( ) ( ) ( )( ) ( )

3

11. 7

a 4 b 1

12

a 9 b 6

a 7 b 28

. 1

a 2 b 3

f x x

f x f x

f x x

f x f x

= +

= =

= -

-

=- =

-


Functions Domains and Ranges Viewing and Interpreting Graphs Even Functions and Odd functions - Symmetry Functions Defined in Pieces Absolute Value Function Composite Functions

…and why

Functions and graphs form the basis for understanding mathematics applications.



Functions

A rule that assigns to each element in one set a unique

element in another set is called a function. A function is like

a machine that assigns a unique output to every allowable

input. The inputs make up the domain of the function; the

outputs make up the range.


Function

A function from a set D to a set R is a rule thatassigns a unique element in R to each element inD.

In this definition, D is the domain of the function and R is a set containing the range.


Function

( )

( )( )

The symbolic way to say " is a function of " is which is read as equals of .The notation gives a way to denote specific values of a function. The value of at can be written as , read

y x y f xy f x

f xf a f a

=

as " of ."f a


Example Functions

Evaluate the function ( ) 2 3 when = 6. f x x x= +

( )

( ) 2( ) 3

( ) 12 3

15

6 6

6

6

f

f

f

= +

= +

=


Domains and Ranges

( )When we define a function with a formula and the domain is

not stated explicitly or restricted by context, the domain is assumed to be

the largest set of -values for which the formula gives real

y f x

x

=

( )

-values -

the so-called natural domain. If we want to restrict the domain, we must say so.

The domain of 2 is restricted by context because the radius, ,

must always be positive.

y

C r r rp=


Domains and Ranges

The domain of 5 is assumed to be the entire set of real numbers.

If we want to restrict the domain of 5 to be only positive values,

we must write 5 , 0.

y x

y x

y x x

=

=

= >


Domains and Ranges

The domains and ranges of many real-valued functions of a real variable are intervals or combinations of intervals. The intervals may be open, closed or half-open, finite or infinite.

The endpoints of an interval make up the interval’s boundary and are called boundary points.

The remaining points make up the interval’s interior and are called interior points.


Domains and Ranges

Closed intervals contain their boundary points. Open intervals contain no boundary points


Domains and Ranges


Graph

( )( )

The points , in the plane whose coordinates are the

input-output pairs of a function make up the

function's .

x y

y f x

graph

=


Example Finding Domains and Ranges

2

Identify the domain and range and use a grapher

to graph the function .y x=

[-10, 10] by [-5, 15]

2y x=

( )

[ )

Domain: The function gives a real value of for every value of

so the domain is , .

Range: Every value of the domain, , gives a real, positive value of

so the range is 0, .

y x

x y

- ¥ ¥

¥


Viewing and Interpreting Graphs

Recognize that the graph is reasonable.

See all the important characteristics of the graph.

Interpret those characteristics.

Recognize grapher failure.

Graphing with a graphing calculator requires that you develop graph viewing skills.


Viewing and Interpreting Graphs

Being able to recognize that a graph is reasonable comes with experience. You need to know the basic functions, their graphs, and how changes in their equations affect the graphs.

Grapher failure occurs when the graph produced by a grapher is less than precise – or even incorrect – usually due to the limitations of the screen resolution of the grapher.


Example Viewing and Interpreting Graphs

2

Identify the domain and range and use a grapher to

graph the function 4y x= -

( ] [ )

Domain: The function gives a real value of for each value of 2

so the domain is , 2 2, .

Range: Every value of the domain, ,

gives a real, positive value of

so the range is [0, ).

y x

x

y

³

- ¥ - È ¥

¥

[-10, 10] by [-10, 10]

2 4y x= -


Even Functions and Odd Functions-Symmetry

The graphs of even and odd functions have important symmetry properties.

( )( ) ( )

A function ( )is a

if ( )

if

for every in the function's domain.

y f x

x f x f x

x f x f x

x

=

- =

- =-

even function of

odd function of


Even Functions and Odd Functions-Symmetry

The graph of an even function is symmetric about the y-axis. A point (x,y) lies on the graph if and only if the point (-x,y) lies on the graph.

The graph of an odd function is symmetric about the origin. A point (x,y) lies on the graph if and only if the point (-x,-y) lies on the graph.


Example Even Functions and Odd Functions-Symmetry

3Determine whether is even, odd or neither.y x x= -

( ) ( ) ( ) ( ) ( )

3

3 3 3

is odd because

x

y x x

f x xx x f xx x- -

= -

= - = - + = - - =--

3y x x= -


Example Even Functions and Odd Functions-Symmetry

Determine whether 2 5 is even, odd or neither.y x= +

( ) ( ) ( )2 5 is neither because

2 5 2 5 ( )x x

y x

f x f x f x

= +

= + =- + ¹ ¹ -- -

2 5y x= +


Functions Defined in Pieces

While some functions are defined by single formulas, others are defined by applying different formulas to different parts of their domain.

These are called piecewise functions.


Example Graphing a Piecewise Defined Function

2

Use a grapher to graph the following piecewise function :

2 1 0( )

3 0

x xf x

x x

2 1; 0y x x= - £

2 3; 0y x x= + >

[-10, 10] by [-10, 10]


Absolute Value Functions

The absolute value function is defined piecewise by the formula

, 0

, 0

y x

x xx

x x

=

ì - <ïï= íï ³ïî

The function is even, and its graph is symmetric about the y-axis


Composite Functions

Suppose that some of the outputs of a function can be used as inputs of

a function . We can then link and to form a new function whose inputs

are inputs of and whose outputs are the numbers

g

f g f

x g ( )( )( )( ) ( )

.

We say that the function read of of is

. The usual standard notation for the composite is ,

which is read " of ."

f g x

f g x f g x

f g

f g

the composite

of and og f


Example Composite Functions

( )Given ( ) 2 3 and 5 , find .f x x g x x f g= - = o

( ) ( )( )( )( )

( )

2 3

1

5

5

0 3

g x

x

f g x

f

x

x

f=

=

= -

= -

o


1.3

Exponential Functions


Quick Review

223

3 5

10

In Exercises 1 3, evaluate the expression. Round your answers

to 3 decimal places.

1. 5 2. 3

3. 3

In Exercises 4 6, solve the equation. Round your answers


4. 17 5. 24

6. 1

x x

x

1.5-

-

-

= =

= .4567


Quick Review

In Exercises 7 and 8, find the value of investing dollars for

years with the interest rate compounded annually.

7. $500, 4.75%, 5 years

8. 1000, 6.3%, 3 years

In Exercises 9 and 10, simplify the

P

n r

P r n

P r n

= = =

= = =

( )( )

2 2 13 2 3 2 4 2

3 4 34 3

exponential expression.

9. 10.x y a b a c

c bx y

-- - -æ ö æ ö÷ ÷ç ç÷ ÷ç ç÷ ÷ç ç÷ ÷è ø è ø



223

3

2.924 4.

In Exercises 1 3, evaluate the expression. Round your answers


1. 5 2. 3

3. 3

In Exercises 4 6, solve the equation. Round your answers


4.

729

0.192

x

1.5-

-

-

5

10

2.5713 1.88817 5. 24

6. 1.4567

2

1.0383

x

x

= =

= ±



In Exercises 7 and 8, find the value of investing dollars for

years with the interest rate compounded annually.

7. $500, 4.75%, 5 years

8. 1000, 6.3%, 3 y

$630.58

$1201.16ears

In Exercises 9 and 1

P

n r

P r n

P r n

= = =

= = =

( )( )

2 2 13 2 3 2 2

1

4 2

3 8 5 4 34 3 6

1

0, simplify the exponential expression.

9. 10.x ay a b a c

c bx y x y bc

-- - -æ ö æ ö÷ ÷ç ç÷ ÷ç ç÷ ÷ç ç÷ ÷è ø è ø


Exponential Growth Exponential Decay Applications The Number e

…and why

Exponential functions model many growth patterns.



Exponential Function

Let be a positive real number other than 1. The function

( )

is the .

x

a

f x a

a

=

exponential function with base

The domain of ( ) is ( , ) and the range is (0, ).

Compound interest investment and population growth are examples

of exponential growth.

xf x a= - ¥ ¥ ¥


Exponential Growth

If 1 the graph of looks like the graph

of 2 in Figure 1.22ax

a f

y=


Exponential Growth

If 0 1 the graph of looks like the graph

of 2 in Figure 1.22b.x

a f

y -=


Rules for Exponents

( )

( ) ( )

If 0 and 0, the following hold for all real numbers and .

1. 4.

2. 5.

3.

xx y x y x x

xx xx y

y x

y xx y xy

a b x y

a a a a b ab

a a aa

ba b

a a a

+

-

> >

× = × =

æö÷ç= =÷ç ÷çè ø

= =


Half-life

Exponential functions can also model phenomena that produce decrease over time, such as happens with radioactive decay. The half-life of a radioactive substance is the amount of time it takes for half of the substance to change from its original radioactive state to a non-radioactive state by emitting energy in the form of radiation.


Exponential Growth and Exponential Decay

The function , 0, is a model for

if 1, and a model for if 0 1.

xy k a k

a a

exponential growth

exponential decay

= × >

> < <


Example Exponential Functions

( )Use a grapher to find the zero's of 4 3.xf x = -

( ) 4 3xf x = -

[-5, 5], [-10,10]


The Number e

Many natural, physical and economic phenomena are best modeled

by an exponential function whose base is the famous number , which is

2.718281828 to nine decimal places.

We can define to be the numbe

e

e ( ) 1r that the function 1

approaches as approaches infinity.

x

f xx

x

æ ö÷ç= + ÷ç ÷çè ø


The Number e

The exponential functions and are frequently used as models

of exponential growth or decay.

Interest compounded continuously uses the model , where is the

initial investment, is t

x x

r t

y e y e

y P e P

r

-= =

= ×

he interest rate as a decimal and is the time in years.t


Example The Number e

( ) 0.03

The approximate number of fruit flies in an experimental population after

hours is given by 20 , 0.

a. Find the initial number of fruit flies in the population.

b. How large is the populat

tt Q t e t= ³

ion of fruit flies after 72 hours?

c. Use a grapher to graph the function .Q

( )

( ) ( ) ( )

( ) ( )

0.03 0

0.03 2.72 16

0

a. To find the initial population, evaluate at 0.

20 20 20 1 20 flies.

b. After 72 hours, the population size is

20 2

0

0 173 flies.

c.

72

Q t t

Q e e

Q e e

=

= = = =

= = »

[0,100] by [0,120] in 10’s

( ) 0.0320 , 0tQ t e t= ³


Quick Quiz Sections 1.1 – 1.3

( )

( )

( )

( )( )

You may use a graphing calculator to solve the following problems.

1. Which of the following gives an equation for the line through 3, 1

and parallel to the line: 2 1?

1 7A

2 21 5

B2 2

C 2 5

D 2

y x

y x

y x

y x

y x

-

=- +

= +

= -

=- +

=-

( )

7

E 2 1y x

-

=- +



( )

( )

( )

( )( )

You may use a graphing calculator to solve the following problems.

1. Which of the following gives an equation for the line through 3, 1

and parallel to the line: 2 1?

1 7A

2 21 5

B2 2

D 2

C 2 5

y x

y x

y x

y x

y x

=- +

-

=- +

= +

= -

=-

( )

7

E 2 1y x

-

=- +



( ) ( )( )( )

( )( )( )( )( )

22. If 1 and 2 1, which of the

following gives 2 ?

A 2

B 5

C 9

D 10

E 15

f x x g x x

f g

= + = -

o



( ) ( )( )( )

( )( )( )( )( )

22. If 1 and 2 1, which of the

following gives 2 ?

A 2

B 5

C 9

15

D 10

E

f x x g x x

f g

= + = -

o



( )( )( )( )

3. The half-life of a certain radioactive substance is 8 hours. There

are 5 grams present initially. Which of the following gives the

best approximation when there will be 1 gram remaining?

A 2

B 10

C 15

D 16

( )E 19



( )( )( )( )

3. The half-life of a certain radioactive substance is 8 hours. There

are 5 grams present initially. Which of the following gives the

best approximation when there will be 1 gram remaining?

A 2

B 10

C 15

D 16

( )E 19


1.4

Parametric Equations


Relations Circles Ellipses Lines and Other Curves


…and why

Parametric equations can be used to obtain graphs of relations and functions.


Quick Review

( ) ( )( )

( )

In Exercises 1 3, write an equation for the line.

1. the line through the points 1, 8 and 4, 3

2. the horizontal line through the point 3, 4

3. the vertical line through the point 2, 3

In Exercises 4 - 6

-

-

-

2 2 2

2

, find the - and -intercepts of the graph of the relation.

4. 1 5. 19 16 16 9

6. 2 1

x y

x y x y

y x

+ = - =

= +


Quick Review

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

2 2

2 2

In Exercises 7 and 8, determine whether the given points lie on

the graph of the relation.

7. 2 3

1a 1, 1 b 1, 1 c , 2

2

8. 9 18 4 27

a 1, 3 b 1, 3 c 1, 3

x y y

x x y

+ =

æ ö÷ç- - - ÷ç ÷çè ø

- + =

- -


Quick Review

( ) ( )

( )

( )

( )

2

2

2

9. Solve for .

a 2 3 5 b 3 2 1

10. For what values of is each equation true?

a

b

c 4 2

t

x t y t

a

a a

a a

a a

+ =- - =-

=

=±

=



( ) ( )

( )( )

In Exercises 1 3, write an equation for the line.

1. the line through the points 1, 8 and 4, 3

2. the horizontal line through the point 3, 4

3. the vertical line through the point 2,

5 2

3

I

9

3 34

2

y x

y

x

=- +

-

=-

=

-

-

2 2

2 2

2

n Exercises 4 - 6, find the - and -intercepts of the graph of th

3, 3; 4

e relation.

4. 19 16

5. 116 9

, 4

4, 4; no -intercepts

1 11; ,.

21

26 2

x y

xx y

x y

x y

y

x y

y x

=- =-

=-

=- =+ -

+ =

- =

=



( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

2 2

2 2

In Exercises 7 and 8, determine whether the given points lie on

the graph of the relation.

Yes No Yes

Yes Y

7. 2 3

1a 1, 1 b 1, 1 c , 2

2

8. 9 18 4 27

a 1, 3 b 1, 3 c 1, 3 N es o

x y y

x x y

+ =

æ ö÷ç- - - ÷ç ÷çè ø

- + =

- -



( ) ( )

( )

( )

( )

2

2

2

9. Solve for .

a 2 3 5 b 3 2 1

10. For what values of is each equation t

2 5 3 1

3 2

0

All Reals

All Re

rue?

a

a s

b

c 4 2 l

x yt t

a

t

x t y t

a

a a

a a

a a

+ =- - =-

=

-

=

=

- += =

³

±


Relations

A relation is a set of ordered pairs (x, y) of real numbers.

The graph of a relation is the set of points in a plane that

correspond to the ordered pairs of the relation.

If x and y are functions of a third variable t, called a parameter, then we can use the parametric mode of a grapher to obtain a graph of the relation.


Parametric Curve, Parametric Equations

( ) ( )

( ) ( ) ( )( )

If and are given as functions

,

over an interval of -values, then the set of points , ,

defined by these equations is a . The equations are

of th

x y

x f t y g t

t x y f t g t

= =

=

parametric curve

parametric equations e curve.


Relations

( ) ( )( ) ( ) ( )( )

The variable is a parameter for the curve and its domain is the

parameter interval. If is a closed interval, , the point

, is the initial point of the curve and the point ,

is the te

t I

I a t b

f a g a f b g b

£ £

rminal point of the curve. When we give parametric equations

and a parameter interval for a curve, we say that we have parametrized

the curve. A grapher can draw a parametrized curve only over a closed

interval, so the portion it draws has endpoints even when the curve being

graphed does not.


Example Relations

2Describe the graph of the relation determined by , 1 .x t y t= = -

21 1Set , 1 , and use the parametric mode

of the grapher to draw the graph.

x t y t= = -


Circles

In applications, t often denotes time, an angle or the distance a particle has traveled along its path from a starting point.

Parametric graphing can be used to simulate the motion of a particle.


Example CirclesDescribe the graph of the relation determined by

3cos , 3sin , 0 2 .

Find the initial points, if any, and indicate the direction in which the curve

is traced.

Find a Cartesian equation for a curve

x t y t t p= = £ £

that contains the parametrized curve.

( ) ( )2 2 2 2 2 2

2 2

9cos 9sin 9 cos sin 9 1 9

Thus, 9

This represents the equation of a circle with radius 3 and center at the origin.

As increases from 0 to 2 the curve is traced in a counter-clock

x y t t t t

x y

t p

+ = + = + = =

+ =

wise direction

beginning at the point (3,0).

2 2 9x y+ =


Ellipses

2 2

2 2

2 2

2 2

Parametrizations of ellipses are similar to parametrizations of circles.

Recall that the standard form of an ellipse centered at (0, 0) is

1.

For cos and sin , we have 1

which

x y

a b

x yx a t y a t

a b

+ =

= = + =

is the equation of an ellipse with center at the origin.


Lines and Other Curves

Lines, line segments and many other curves can be defined parametrically.


Example Lines and Other Curves

If the parametrization of a curve is , 2, 0 2 graph the curve.

Find the initial and terminal points and indicate the direction in which the curve

is traced. Find a Cartesian equation for a cu

x t y t t= = + £ £

rve that contains the parametrized

curve. What portion of the Cartesian equation is traced by the parametrized curve?

, 2, 0 2x t y t t= = + £ £

Initial point (0, 2) Terminal point (2, 4)

Curve is traced from left to right

( )( ) ( )

If and 2, then by substitution 2.

The graph of the Cartesian equation is a line through 0,2 with 1.

The segment of that line from 0,2 to 2, 4 is traced by the parametrized curve.

x t y t y x

m

= = + = +

=


1.5

Functions and Logarithms


Quick Review

( ) ( )

( )( ) ( )( )( )( ) ( )( )

23In Exercises 1 4, let 1, 1, and

evaluate the expression.

1. 1 2. 7

3. 4.

In Exercises 5 and 6, choose parametric equations and a parameter

interval to represent the function o

f x x g x x

f g g f

f g x g f x

- = - = +

-o o

o o

n the interval specified.

15. , 2 6. , 3

1y x y x x

x= ³ = <-

-


Quick Review

( ) ( )

( ) ( )

In Exercises 7 10, find the points of intersection of the two curves.

Round your answers to 2 decimal places.

7. 2 3, 5

8. 3 5, 3

9. a 2 , 3 b 2 , 1

10. a , 4 b , 1

x x

x x

y x y

y x y

y y y y

y e y y e y- -

-

= - =

=- + =-

= = = =-

= = = =-



( ) ( )

( )( ) ( )( )

( )( ) ( )( ) ( )2 23 3

23In Exercises 1 4, let 1, 1, and

evaluate the expression.

1. 1 2. 7

3. 4.

In Exercises 5 and 6, choose parametric equations and a parameter

interval to represent th

1 5

1 1

e

f x x g x x

f g g f

f g x g f xx x

- =

-

-

-

+

= +

o o

o o

( )possible answers ar

function on the interva

e give

l specified.

15.

n

1, ,

, 2 6.

,

1

2 ,1

3

3

,y

x t y t

x

x

y x x

t y t tt

x= ³ =

= = ³

-

= =

<-

<--



( )

( ) ( ) ( )

( )

In Exercises 7 - 10, find the points of intersection of the two curves.

Round your answers to 2 decimal places.

7. 2 3, 5

8. 3 5,

4,5

8, 3

3

1.58,3 No

3

9. a 2 , 3 b 2 , 1

10. a ,

ne

4

x x

x

y x y

y x y

y y y y

y e y-

æ ö÷ç - ÷ç ÷çè

= - =

=- + =-

= = = =-

= =

ø

-( ) ( )1.39,4 Noneb , 1xy e y-= =-


One-to-One Functions Inverses Finding Inverses Logarithmic Functions Properties of Logarithms Applications

…and why

Logarithmic functions are used in many applications including finding time in investment problems.



One-to-One Functions

A function is a rule that assigns a single value in its range to each point in its domain.

Some functions assign the same output to more than one input.

Other functions never output a given value more than once. If each output value of a function is associated with exactly

one input value, the function is one-to-one.



( ) ( ) ( )A function is on a domain if

whenever .

f x D f a f b

a b

¹

¹

one - to - one



( )The horizontal line test states that the graph of a one-to-one function

can intersect any horizontal line at most once.

If it intersects such a line more than once it assumes the sam

y f x=

e -value

more than once and is not a one-to-one function.

y


Inverses

Since each output of a one-to-one function comes from just one input, a one-to-one function can be reversed to send outputs back to the inputs from which they came.

The function defined by reversing a one-to-one function f is the inverse of f.

Composing a function with its inverse in either order sends each output back to the input from which it came.


Inverses

( )( )

( )( ) ( )( )

1

1 1

The symbol for the inverse of is , read " inverse."

1The -1 in is not an exponent; does not mean

If , then and are inverses of one another;

otherwise they are not.

f f f

f f xf x

f g x g f x f g

-

- -

=o o


Identity Function

The result of composing a function and its inverse in either order

is the identity function.


Example Inverses

( ) ( ) 2Determine via composition if and , 0

are inverses.

f x x g x x x= = ³

( )( ) ( )

( )( ) ( ) ( )

22

2

x

f g x f x x x

g x g x x

x

f

= = = =

= = =

o

o


Writing f -1as a Function of x.

( )

( )1

Solve the equation for in terms of .

Interchange and . The resulting formula

will be .

y f x x y

x y

y f x-

=

=


Finding Inverses


Example Finding Inverses

Given that 4 12 is one-to-one, find its inverse.

Graph the function and its inverse.

y x= -

Solve the equation for in terms of .

13

4Interchange and .

13

4

x y

x y

x y

y x

= +

= +

( ) 4 12f x x= -

( )1 13

4f x x- = +

[-10,10] by [-15, 8]

Notice the symmetry about the line y x=


Base a Logarithmic Function

( )

( )

( )

The base logarithm function log is the inverse of

the base exponential function 0, 1 .

The domain of log is 0, , the range of .

The range of log is , , the domain of .

a

x

xa

xa

a y x

a y a a a

x a

x a

=

= ¹

¥

- ¥ ¥


Logarithmic Functions

Logarithms with base e and base 10 are so important in applications that calculators have special keys for them.

They also have their own special notations and names.

log ln is called the .

log log is often called the .10

y x xe

y x x

natural logarithm function

common logarithm function

= =

= =


Inverse Properties for ax and loga x

log

ln

Base : , 1, 0

Base : , ln , 0

a x

x x

a a x a x

e e x e x x

= >

= =


Properties of Logarithms

For any real numbers 0 and 0,

log log log

log log log

log log

x y

Product Rule : xy x ya a a

xQuotient Rule : x ya a ay

yPower Rule : x y xa a

= +

= -

=


Example Properties of Logarithms

Solve the following for .

2 12x

x

=

Take logarithms of both sides

PowerRule

2 12

ln 2 ln12

ln 2 ln12

ln12 2.3025853.32193

ln 2 .693147

x

x

x

x

=

=

=

= = »


Example Properties of Logarithms

Solve the following for .

5 60x

x

e + =

Subtract 5

Take logarithm of both sides

5 60

55

ln ln55

ln55 4.007333

x

x

x

e

e

e

x

+ =

=

=

= »


Change of Base Formula

lnlog

lna

xx

a=

This formula allows us to evaluate log for any

base 0, 1, and to obtain its graph using the

natural logarithm function on our grapher.

a x

a a ¹


Example Population Growth

0.015The population of a city is given by 105,300

where 0 represents 1990. According to this model,

when will the population reach 150,000?

tP P e

t

=

=

0.015

0.015

0.015

0.015

Solve for t

Take logarithm of both sides

Inverse property

105,300 , 150,000

105,300

150,000

105,300

150,000ln ln

105,300

0.353822= 0.01

150,000

5

0.353822= 23.5881

0.015

t

t

t

t

P e P

e

e

e

t

t

= =

=

=

æ ö÷ç =÷ç ÷çè ø

» 33 years 0 is 1990, so

the population will reach 150,000 in the year 2013.

t =


1.6

Trigonometric Functions


Quick Review

In Exercises 1 4, convert from radians to degrees or

degrees to radians.

1. 2. 2.53

3. 40 4. 45

In Exercises 5 7, solve the equation graphically in the

given interval.

5. sin 0.6, 0 2

6. cos 0.4, 0 2

x x

x x

p

p

-

-

-

-

= £ <

=- £ <

° °

37. tan 1,

2 2x x

p

p p= - £ <


Quick Review

( )

( )

2

3

8. Show that 2 3 is an even function. Explain why

its graph is symmetric about the -axis.

9. Show that 3 is an odd function. Explain why

its graph is symmetric about the origin.

10. Give on

f x x

y

f x x x

= -

= -

( ) 4e way to restrict the domain of the function 2

to make the resulting function one-to-one.

f x x= -



In Exercises 1 - 4, convert from radians to degrees or

degrees to radians.

1. 2. 2.53

3. 40 4. 45

In Exercises 5 - 7, solve the equation graphically in the

given interval.

5. sin 0

60

.6,

143

9

0 2

.24

2

x x

p

p

p

p

-

-

-

-

= £ <

° °

° °

4

0.6435, 2.4981

1.98236. cos 0.4, 0 2

37. t

, 4.3009

0.7854, 3an 1, .92

2702

x x

x x

x

x

x

p

p p

=- £ <

= - »£ <

»

»



( )

( ) ( ) ( )

( )

2

2 2

8. Show that 2 3 is an even function. explain why

its graph is symmetric about the -

2 3 2 3

The graph is symmetric about the -axis

a

because if a point

, is on the graph, the

is.

n

x

so

f x x x f x

y

a b

f x x

y

- = - - = - =

= -

( ) is the point , .a b-



( )

( ) ( ) ( ) ( )

( )

3

3 3

9. Show that 3 is an odd function. Explain why

its graph is symmetric about the o

3 3

The graph is symmetric about the origin because if a point

, is on the graph, t

rigi

he

n.

n

f x x x x

f x x

x f x

a

x

b

- = - - - =-

-

+

=

=-

( )

( ) 410. Give one way to restrict the domain of the function 2

to make the resulti

so is the poin

ng functi

t , .

0

on one-to-one.

f x x

a b

x

- -

-

³

=



Radian Measure Graphs of Trigonometric Functions Periodicity Even and Odd Trigonometric Functions Transformations of Trigonometric Graphs Inverse Trigonometric Functions

…and why

Trigonometric functions can be used to model periodic behavior and applications such as musical notes.


Radian Measure

The radian measure of the angle ACB at the center of the unit circle equals the length of the arc that ACB cuts from the unit circle.


Radian Measure

An angle of measure θ is placed in standard position at the center of circle of radius r,


Trigonometric Functions of θ

The six basic trigonometric functions of are

defined as follows:

sine: sin cosecant: csc

cosine: cos secant: sec

tangent: tan cotangent: cot

y r

r y

x r

r xy x

x y

q

q q

q q

q q

= =

= =

= =


Graphs of Trigonometric Functions When we graph trigonometric functions in the coordinate plane, we usually

denote the independent variable (radians) by x instead of θ .


Angle Convention

Angle Convention: Use Radians

From now on in this book, it is assumed that all angles are measured in radians

unless degrees or some other unit is stated explicitly. When we talk about the angle

we 3

pmean radians ( which is 60°), not degrees.

3 3

When you do calculus, keep your calculator in radian mode.

p p


Periodic Function, Period

( )( ) ( )

A function is if there is a positive number such

that for every value of . The smallest value

of p is the of .

The functions cos , sin , sec and csc are periodic wi

f x p

f x p f x x

f

x x x x

periodic

period

+ =

th

period 2 . The functions tan and cot are periodic with

period .

x xp

p


Even and Odd Trigonometric Functions

The graphs of cos x and sec x are even functions because their graphs are symmetric about the y-axis.

The graphs of sin x, csc x, tan x and cot x are odd functions.

cosy x= siny x=


Example Even and Odd Trigonometric Functions

Show that csc is an odd function.x

( )( )1 1

csc cscsin sin

x xx x

- = = -- -


Transformations of Trigonometric Graphs

The rules for shifting, stretching, shrinking and reflecting the graph of a function apply to the trigonometric functions.

( )( )y a f b x c d= + +

Vertical stretch or shrink

Reflection about x-axis

Horizontal stretch or shrink

Reflection about the y-axisHorizontal shift

Vertical shift


Example Transformations of Trigonometric Graphs

( )Determine the period, domain, range and draw the graph of

2sin 4y x p=- +

[-5, 5] by [-4,4]

( )

We can rewrite the function as 2sin 44

2The period of sin is . In our example 4,

2so the period is = . The domain is .

4 2The graph is a basic sin curve w

y x

y a bx bb

x

p

p

p p

æ öæ ö÷ç ÷ç=- + ÷÷ç ç ÷÷ç ÷ç è øè ø

= =

- ¥ ¥,

ith an amplitude of 2. Thus, the range is [ 2, 2].

The graph of the function is shown together with the graphof the sin function.x

-


Inverse Trigonometric Functions

None of the six basic trigonometric functions graphed in Figure 1.42 is one-to-one. These functions do not have inverses. However, in each case, the domain can be restricted to produce a new function that does have an inverse.

The domains and ranges of the inverse trigonometric functions become part of their definitions.



1

1

1

1

1

1

cos 1 1 0

sin 1 1 2 2

tan 2 2

sec 1 0 ,2

csc 1 , 02 2

cot 0

y x x y

y x x y

y x x y

y x x y y

y x x y y

y x x y

Function Domain Range

pp p

p p

pp

p p

p

-

-

-

-

-

-

= - £ £ £ £

= - £ £ - £ £

= - ¥ < <¥ - < <

= ³ £ £ ¹

= ³ - £ £ ¹

= - ¥ < <¥ < <



The graphs of the six inverse trigonometric functions are shown here.


Example Inverse Trigonometric Functions

1 1Find the measure of sin in degrees and in radians.

2- æ ö÷ç- ÷ç ÷çè ø

1

1

1Put the calculator in degree mode and enter sin .

2

The calculator returns 30

1Put the calculator in radian mode and enter sin .

2

The calculator returns .52359877556 radians.

-

-

æ ö÷ç- ÷ç ÷çè ø

-

æ ö÷ç- ÷ç ÷çè ø

-

°.

This is the same as radians.6

p-



2

You should solve the following problems without using a graphing caluclator.

1. Which of the following is the domain of ( ) log 3 ?

(A) ,

(B) ,3

(C) 3,

(D) [ 3, )

(E) ( ,3]

f x x



2

You should solve the following problems without using a graphing caluclator.

1. Which of the following is the domain of ( ) log 3 ?

(

(C) 3,

A) ,

(B) ,3

(D) [ 3, )

(E) ( ,3]

f x x



2. Which of the following is the range of ( ) 5cos 3?

(A) ,

(B) 2,4

(C) 8,2

(D) 2,8

2 8(E) ,

5 5

f x x



2. Which of the following is the range of ( ) 5cos 3?

(A) ,

(B) 2,4

(C) 8,2

2 8(E) ,

5

(D) 2,8

5

f x x



3

3. Which of the following gives the solution of tan -1 in ?2

(A) 4

(B) 4

(C) 33

(D) 4

5(E)

4

x x



3

3. Which of the following gives the solution of tan -1 in ?2

(A) 4

(B) 4

(C) 33

5(E)

D)

4

( 4

x x


Chapter Test

( )( )

In Exercises 1 and 2, write an equation for the specified line.

1. through 4, 12 and parallel to 4 3 12

2. the line where has the following values:

3. Determine whether the graph of the function

x y

y f x f

- + =

=

1

5

4

3

is

symmetric about the -axis, the origin or neither.

14. Determine whether the function is even, odd or neither.

2

y x

y

xy

x x

=

+=

-

x - 2 2 4

f(x) 4 2 1


Chapter Test

( ) ( )( )

( )( )

In Exercises 5 and 6, find the a domain and b range, and

c graph the function.

5. 2sin 3 1

6. ln 3 1

y x

y x

p= + -

= - +


Chapter Test

7. Write a piecewise formula for the function.

21

1


Chapter Test

( )

( )

8. 5cos , 2sin , 0 2 is a parametrization

for a curve.

a Graph the curve.

Identify the initial and terminal points, if any.

Indicate the direction in which the curve is traced.

b Find a Cartesian equati

x t y t t p= = £ £

on for a curve that contains

the parametrized curve.

What portion of the graph of the Cartesian equation

is traced by the parametrized curve?


Chapter Test

( ) ( )( )

( ) ( )( ) ( )( )

( )

1 1 1

1

9. Give one parametrization for the line segment with

endpoints 2, 5 and 4,3 .

10. Given 2 3 ,

a find and show that

b graph and in the same viewing window

f x x

f f f x f f x

f f

- - -

-

-

= -

=○ ○


Chapter Test Solutions

( )

( )

In Exercises 1 and 2, write an equation for the specified line.

1. through 4, 12 and parallel to 4 3 12

2. the line where has the following values:

3. Determine whether the gra

4 2

p

0

3 3

13

2

x y

y f

y x

y

x f

x

=- -

=- +

- + =

=

1

5

4

3

Origin

h of the function is

symmetric about the -axis, the origin or neither.

14. Determine whether the function is even, odd

2

or neither O. dd

y x

y

xy

x x

=

+=

-

x - 2 2 4

f(x) 4 2 1



( ) ( )( )

( ) ( )( ) ( ) ( ) ( ) ( )

In Exercises 5 and 6, find the a domain and b range, and

c graph the f

a A

un

ll

ctio

Rea

n.

5. 2s

ls b

in 3 1 6

[ 3, 1] a 3, b All Rea

. ln 3

s

1

l

y x y xp= + - =

- ¥

- +

[-π, π] by [-5, 5]

[- 2, 10] by [- 2, 5]



7. Write a piecewise formula for the function.

( )1 , 0 1

2 , 1 2

x xf x

x x

ì - £ <ïï=íï - £ £ïî

21

1



( )

8. 5cos , 2sin , 0 2 is a parametrization

for a curve.

a Graph the curve.

Identify the initial and terminal points, if any.

Indicate the direction in which the curve is traced.

x t y t t p= = £ £

Initial Point (5, 0)Terminal Point (5, 0)



( )2 2

8. b Find a Cartesian equation for a curve that contains

the parametrized curve.

What portion of the graph of the Cartesian equation

is traced by the parametrized curve?

15 2

Al

x yæö æ ö÷ ÷ç ç+ =÷ ÷ç ç÷ ÷ç çè ø è ø

l



( ) ( )

( )

( ) ( )( ) ( )( )

( ) ( )

( )( )

1 1 1

1

1

9. Give one parametrization for the line segment with

endpoints 2, 5 and 4,3 .

10. Given 2 3 ,

a find and show that

(A possible answer)

2 6 , 5 2 , 0 1

2a

32

3

x t y t t

xf x

xf f x

f x x

f f f x f f x

f

-

-

-

- -

=- + = - £ £

-=

æ ö- ÷ç= ÷ççè

-

= -

=

ø

○ ○

( )

( )( ) ( )( )1 1

22 3 2 2

3

2 2 3 32 3

3 3

xx x

x xf f x f x x- -

æ ö- ÷ç= - = - - =÷ç÷ ÷çè ø

- -= - = = =



( ) 110. b graph and in the same viewing windowf f -

( ) 2 3f x x= -

( )1 2

3

xf x- -

=

[-5, 5] by [-5, 5]

Documents

Chapter 1