ESSENTIAL CALCULUSESSENTIAL CALCULUS

CH01 Functions & LimitsCH01 Functions & Limits

1.1 Functions and Their Representations

1.2 A Catalog of Essential Functions

1.3 The Limit of a Function

1.4 Calculating Limits

1.5 Continuity

1.6 Limits Involving Infinity

Review

In this Chapter:In this Chapter:

Some Terminologies ：domain ： set Arange ：independent varible ： A symbol representing any number in the domaindependent varible ： A symbol representing any number in the range

Chapter 1, 1.1, P2

AxBxf /)(

Chapter 1, 1.1, P2

A function f is a rule that assigns to each element x in a set A exactly one element, called f(x) , in a set B.

Chapter 1, 1.1, P2

Chapter 1, 1.1, P2

Chapter 1, 1.1, P2

If f is a function with domain A, then its graph is the set of ordered pairs

(Notice that these are input-output pairs.) In other words, the graph of f consists of all Points(x,y) in the coordinate plane such that y=f(x) and x is in the domain of f.

Axxfx /))(,(

Chapter 1, 1.1, P2

Chapter 1, 1.1, P2

Chapter 1, 1.1, P2

Chapter 1, 1.1, P2

EXAMPLE 1 The graph of a function f is shown in Figure 6.

(a)Find the values of f(1) and f(5) .

(b) What are the domain and range of f ?

Chapter 1, 1.1, P4

EXAMPLE 3 Find the domain of each function.

2)()( xxfaxx

xgb

2

1)()(

Chapter 1, 1.1, P4

THE VERTICAL LINE TEST A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once.

Chapter 1, 1.1, P5

Chapter 1, 1.1, P5

EXAMPLE 4 A function f is defined by

Evaluate f(0) ,f(1) , and f(2) and sketch the graph.

f(x)= 1-X if X≤1

X2 if X>1

Chapter 1, 1.1, P5

Chapter 1, 1.1, P6

EXAMPLE 5 Sketch the graph of theabsolute value function f(x)=│X│.

Chapter 1, 1.1, P6

EXAMPLE 6 In Example C at the beginning of this section we considered the cost C(w)of mailing a first-class letter with weight w. In effect, this is a piecewise defined function because, from the table of values, we have

C(w)=

0.39 if o<w≤1

0.63 if 1<w≤2

0.87 if 2<w≤3

1.11 if 3<w≤4

Chapter 1, 1.1, P6

Chapter 1, 1.1, P6

If a function f satisfies f(-x)=f(x) for every number x in its domain, then f is called an even function.

Chapter 1, 1.1, P6

Chapter 1, 1.1, P6

Chapter 1, 1.1, 07

If f satisfies f(-x)=-f(x) for every number x in its domain, then f is called an odd function.

Chapter 1, 1.1, 07

EXAMPLE 7 Determine whether each of the following functions is even, odd, or neither even nor odd.

(a) f(x)=x5+x(b) g(x)=1-x4

(c) h(x)=2x=x2

Chapter 1, 1.1, 07

Chapter 1, 1.1, 07

A function f is called increasing on an interval if

f (x1)< f (x2) whenever x1< x2 in I

It is called decreasing on I if

f (x1)> f (x2) whenever x1 < x2 in I

Chapter 1, 1.1, 08

1. The graph of a function f is given.(a) State the value of f(-1).(b) Estimate the value of f(2).(c) For what values of x is f(x)=2?(d) Estimate the values of x such that f(x)=0 .(e) State the domain and range of f .(f ) On what interval is f increasing?

Chapter 1, 1.1, 08

Chapter 1, 1.1, 08

2. The graphs of f and g are given.(a) State the values of f(-4)and g(3).(b) For what values of x is f(x)=g(x)?(c) Estimate the solution of the equation f(x)=-1.(d) On what interval is f decreasing?(e) State the domain and range of f.(f ) State the domain and range of g.

Chapter 1, 1.1, 08

Chapter 1, 1.1, 08

3–6 ■ Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function.

Chapter 1, 1.1, 10

53–54 ■ Graphs of f and g are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.

Chapter 1, 1.2, 13

A function P is called a polynomial if

where n is a nonnegative integer and the numbers a0,a1,a2,…..an are constants called the coefficients of the polynomial. The domain of any polynomial is R=(-∞,∞)If the leading coefficient an≠0, then the degree of the polynomial is n.

P(x)=anxn+an-1xn-1+‧‧‧+a2x2+a1x+a0

Chapter 1, 1.2, 14

Chapter 1, 1.2, 14

Chapter 1, 1.2, 14

Chapter 1, 1.2, 14

Chapter 1, 1.2, 14

Chapter 1, 1.2, 14

Chapter 1, 1.2, 14

Chapter 1, 1.2, 14

Chapter 1, 1.2, 14

Chapter 1, 1.2, 15

Chapter 1, 1.2, 15

A rational function f is a ratio of two polynomials:

)(

)()(

XQ

xPxf

Where P and Q are polynomials. The domain consists of all values of x such that Q(x)≠0.

Chapter 1, 1.2, 15

Chapter 1, 1.2, 15

Chapter 1, 1.2, 15

-1≤ son x≤1 -1≤ cos x≤1

Chapter 1, 1.2, 16

Chapter 1, 1.2, 16

Chapter 1, 1.2, 16

Chapter 1, 1.2, 16

Sin(x+2π)=sin x cos(x+2π)=cos x

Chapter 1, 1.2, 16

The exponential functions are the functions of the form f(x)=ax , where the base is a positive constant.

Chapter 1, 1.2, 16

The logarithmic functions f(x)=logax , where the base a is a positive constant,are the inverse functions of the exponential functions.

Chapter 1, 1.2, 17

■ Figure 15 illustrates these shifts by showing how the graph of y=(x+3)2+1 is obtained from the graph of the parabola y=x2: Shift 3 units to the left and 1 unit upward.

Y=(x+3)2+1

Chapter 1, 1.2, 17

VERTICAL AND HORIZONTAL SHIFTS Suppose c>0. To obtain the graph of

Y= f(x)+c, shift the graph of y=f(x) a distance c units c units upward

Y= f(x)- c, shift the graph of y=f(x) a distance c units c units downward

Y= f(x- c), shift the graph of y=f(x) a distance c units c units to the right

Y=f(x+ c), shift the graph of y=f(x) a distance c units c units to the left

Chapter 1, 1.2, 17

VERTICAL AND HORIZONTAL STRETCHING AND REFLECTINGSuppose c>1. To obtain the graph of

y=cf(x), stretch the graph of y=f(x) vertically by a factor of c

y=(1/c)f(x), compress the graph of y=f(x) vertically by a factor of c

Y=f(cx), compress the graph of y=f(x) horizontally by a factor of c

Y=f(x/c), stretch the graph of y=f(x) horizontally by a factor of c

Y=-f(x), reflect the graph of y=f(x) about the x-axis

Y=f(-x), reflect the graph of y=f(x) about the ｙ -axis

Chapter 1, 1.2, 17

Chapter 1, 1.2, 17

Chapter 1, 1.2, 18

EXAMPLE 2 Given the graph of y= , use transformations to graph y= -2 ,

y= , y=- , y=2 , and y=

xx

2X x x X

Chapter 1, 1.2, 18

Chapter 1, 1.2, 18

Chapter 1, 1.2, 18

Chapter 1, 1.2, 18

Chapter 1, 1.2, 18

Chapter 1, 1.2, 18

Chapter 1, 1.2, 18

EXAMPLE 3 Sketch the graph of the function y=1-sin x.

Chapter 1, 1.2, 18

Chapter 1, 1.2, 18

Chapter 1, 1.2, 18

(f+g)(x)=f(x)+g(x) (f-g)(x)=f(x)-g(x)

If the domain of f is A and the domain of g is B, then the domain of f + g is the intersection A ∩ B

Chapter 1, 1.2, 18

(fg)(x)=f(x)g(x) )(

)()(

xg

xfx

g

f

The domain of fg is A ∩B, but we can’t divide by 0 and so the domain of f/g is

0)(/ xgBAx

Chapter 1, 1.2, 19

DEFINITION Given two functions f and g , the composite function f 。 g (also called the composition of f and g ) is defined by (f 。 g)(x)=f(g(x))

Chapter 1, 1.2, 19

Chapter 1, 1.2, 20

EXAMPLE 5 If f(x)= and g(x)= , find each function and its domain.

X

x2

(a) f 。 g (b) g 。 f (c) f 。 f (d)g 。 g

Chapter 1, 1.2, 20

EXAMPLE 6 Given F(x)=cos2(x+9) , find functions f ,g ,and h such that F=f 。 g 。 H.

Chapter 1, 1.2, 22

17. The graph of y=f(x) is given. Match each equation with its graph and give reasons for your choices.

(a)y=f(x-4)(b)y=f(x)+3(c)y= f(x)(d)y=-f(x+4)(e)y=2f(x+6)

3

1

Chapter 1, 1.2, 22

18. The graph of f is given. Draw the graphs of the following functions. (a)y=f(x+4) (b) y=f(x)+4 (c) y=2f(x) (d) y=- f(x)+3

2

1

Chapter 1, 1.2, 22

19 The graph of f is given. Use it to graph the following functions. (a) y=f(2x) (b) y=f( x) (c) y=f(-x) (d)y=-f(-x)

2

1

Chapter 1, 1.2, 22

53 Use the given graphs of f and g to evaluate each expression, or explain why it is undefined.

(a) f(g(2)) (b) g(f(0)) (c) (f 。 g)(0)(b) (g 。 F)(6) (e) (g 。 g)(-2) (f) (f 。 f)(4)

Chapter 1, 1.3, 25

Chapter 1, 1.3, 25

Chapter 1, 1.3, 25

1 DEFINITION We write

and say “the limit of f(X), as x approaches , equals L ”

if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of ) but not equal to a.

limf(x)=LX→a

Chapter 1, 1.3, 25

limf(x)=LX→a

is f(x)→L as x→awhich is usually read “f(x) approaches L as x approaches a.”

Chapter 1, 1.3, 26

Chapter 1, 1.3, 26

Chapter 1, 1.3, 26

Chapter 1, 1.3, 26

Chapter 1, 1.3, 26

Chapter 1, 1.3, 28

Chapter 1, 1.3, 28

Chapter 1, 1.3, 29

limf(x)=LX→a-

2. DEFINITION We write

and say the left-hand limit of f(x) as X approaches a [or the limit of f(x) as Xapproaches a from the left] is equal to L if we can make the values of f(X) arbitrarily close to L by taking x to L be sufficiently close to a and x less than a.

Chapter 1, 1.3, 30

Chapter 1, 1.3, 30

Chapter 1, 1.3, 30

3 limf(x)=L if and only if limf(x)=L and limf(x)=L X→a X→a- X→a+

Chapter 1, 1.3, 30

EXAMPLE 7 The graph of a function g is shown is Figure 10. Use it to state the values(if they exist) of the following ：

(a)lim g(x) (b) lim g(x) (c)lim g(x)

(d) lim g(x) (e) lim g(x) (f)lim g(x)

X→2─ X→2+ X→2

X→5─ X→5+ x→5

Chapter 1, 1.3, 30

Chapter 1, 1.3, 31

Chapter 1, 1.3, 31

FINITION Let f be a function defined on some open interval that containsthe number a , except possibly at a itself. Then we say that the limit ofas approaches is , and we write

lim g(x)=LX→a

if for every number ε>0 there is a corresponding number δ>0 such that

if 0<│x-a│<δ then │f(x)-L│<ε

Chapter 1, 1.3, 32

Chapter 1, 1.31, 32

Chapter 1, 1.3, 32

Chapter 1, 1.3, 32

Chapter 1, 1.3, 32

Chapter 1, 1.3, 33

Chapter 1, 1.3, 33

3. Use the given graph of f to state the value of each quantity, if it exists. If it does not exist, explain why.

(a)Lim f(X) (b) lim f(X) (C)lim f(X)

(d) Lim f(X) (e)F(5)X→1─ X→1+ X→1

X→5

Chapter 1, 1.3, 33

4. For the function f whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why.(a_Lim f(X) (b) lim f(X) (C)lim f(X)

(d) Lim f(X) (e)F(5)X→0 X→3- X→3+

X→3

Chapter 1, 1.3, 33

5. For the function g whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why.

(a)lim g(t) (b) lim g(t) (c) lim g(t)

(d)lim g(t) (e) lim g(t) (f) lim g(t)(g)g(2) (h)lim g(t)

X→0- X→0+ X→0

X→2- X→2+ X→2

X→4

Chapter 1, 1.4, 35

LIMIT LAWS Suppose that c is a constant and the limits

lim f(X) and lim g(x)

Exist Then

1. lim﹝f(x)+g(x)﹞=lim f(x)+lim g(x)

2. lim﹝f(x)-g(x)﹞=limf(x)-lim g(x)

3. lim ﹝cf(x)﹞=c lim f(x)

4. lim ﹝f(x)g(x)﹞=lim f(x)‧lim g(x)

5. lim = if lim g(x)≠0)(

)(

xg

xf

aX

aX

xg

xf

)(lim

)(lim

X→a X→a X→a

X→a X→a X→a

X→a X→a

X→a X→a X→a

X→a X→a

X→a X→a

Chapter 1, 1.4, 36

Sum Law

Difference Law

Constant Multiple Law

Product Law

Quotient Law

Chapter 1, 1.4, 36

1. The limit of a sum is the sum of the limits.2. The limit of a difference is the difference of the limits.3. The limit of a constant times a function is the constant times the limit of the function.4. The limit of a product is the product of the limits.5. The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0).

Chapter 1, 1.4, 36

6. lim[f(x)]n=[limf(x)]n where n is a positive integerX→a X→a

Chapter 1, 1.4, 36

7. lim c=c 8. lim x=a X→a X→a

Chapter 1, 1.4, 36

9. lim xn=an where n is a positive integer X→a

Chapter 1, 1.4, 36

10. lim = where n is a positive integer

(If n is even, we assume that a>0.)

n x X→a

n a

Chapter 1, 1.4, 36

11.Lim = where n is a positive integer

[If n is even, we assume that lim f(X)>0.]

n xf )( n xf )(lim X→a X→a

X→a

Chapter 1, 1.4, 37

DIRECT SUBSTITUTION PROPERTY If f is a polynomial or a rational functionand is in the domain of f, then

lim f(X)>f(a) X→a

Chapter 1, 1.4, 38

If f(x)=g(x) when x ≠ a, then lim f(x)=lim g(x),

provided the limits exist. X→a X→a

Chapter 1, 1.4, 39

FIGURE 2

The graphs of the functions f (from Example 2) and g (from Example 3)

Chapter 1, 1.4, 39

2 THEOREM lim f(x)=L if and only if

lim f(x)=L=lim f(x) X→a

X→a- X→a+

Chapter 1, 1.4, 40

Chapter 1, 1.4, 40

Chapter 1, 1.4, 40

Chapter 1, 1.4, 41

3. THEOREM If f(x)≤g(x) when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a, then

lim f(x) ≤lim g(x) X→a X→a

Chapter 1, 1.4, 41

4. THE SQUEEZE THEOREM If f(x) ≤g(x) ≤h(x) when x is near a (except possibly at a) and

limf(x)=lim h(X) =L

Then lim g(X)=L

X→a X→a

X→a

Chapter 1, 1.4, 41

Chapter 1, 1.4, 41

Chapter 1, 1.4, 43

2. The graphs of f and g are given. Use them to evaluate each limit, if it exists. If the limit does not exist, explain why.

(a)lim[f(x)+g(x)] (b) lim [f(x)+g(x)]

(c)lim [f(x)g(x)] (d) lim

(e)Lim[x3f(x)] (f) lim)(

)(

Xg

Xf

)(3 Xf

X→2 X→1

X→0 X→ -1

X→2 X→1

Chapter 1, 1.5, 46

■ As illustrated in Figure 1, if f is continuous,then the points (x, f(x)) on the graph of f approach the point (a, f(a)) on the graph. So there is no gap in the curev.

Chapter 1, 1.5, 46

Chapter 1, 1.5, 46

1.DEFINITION A function f is continuous at a

number a if

lim f(X)=f(a) X→a

Chapter 1, 1.5, 46

Notice that Definition I implicitly requires three things if f is continuous at a:

1. f(a)is defined (that is, a is in the domain of f )

2. lim f(x) exists

3. lim f(x) = f(a) X→a X→a

Chapter 1, 1.5, 46

If f is defined near a(in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not continuous at a.

Chapter 1, 1.5, 47

2. DEFINITION A function f is continuous from the right t a number a if

lim f(x)=f(a)

And f is continuous from the left at a if

lim f(x)=f(a)

X→a+

X→a-

Chapter 1, 1.5, 48

3. DEFINITION A function f is continuous on an interval if it is continuous at every number in the interval. (If f is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint to mean continuousfrom the right or continuous from the left.)

Chapter 1, 1.5, 48

4. THEOREM If f and g are continuous at a and c is a constant, then the following functions are also continuous at a :

1. f+g 2 f-g 3 cf

4. fg 5. if g(a)≠0g

f

Chapter 1, 1.5, 49

5. THEOREM(a)Any polynomial is continuous everywhere;

that is, it is continuous on R=(-∞,∞).(b) Any rational function is continuous

wherever it is defined; that is, it is continuous on its domain.

Chapter 1, 1.5, 50

6. THEOREM The following types of functions are continuous at every number in their domains: polynomials, rational functions, root functions, trigonometric functions

Chapter 1, 1.5, 51

7. THEOREM If f is continuous at b and lim g(x)=b, then lim f(g(X))=f(b). in the words

lim f(g(X))=f(lim g(X))

X→a X→a

X→a X→a

Chapter 1, 1.5, 51

8. THEOREM If g is continuous at a and f is continuous at g(a), then the composite function f 。g given by(f 。 g)(x)=f(g(x)) is continuous at a.

Chapter 1, 1.5, 52

9. INTERMEDIATE VALUE THEOREM Suppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b) , where f(a)≠f(b). Then there exists a number c in(a,b) such that f(c)=N.

Chapter 1, 1.5, 52

Chapter 1, 1.5, 52

Chapter 1, 1.5, 53

Chapter 1, 1.5, 54

3 (a) From the graph of f, state the numbers at which f is discontinuous and explain why. (b) For each of the numbers stated in part (a), determine whether f is continuous from the right, or from the left, or neither.

Chapter 1, 1.5, 54

4. From the graph of g , state the intervals on which g is continuous.

Chapter 1, 1.6, 56

1 DEFINITION The notation lim f(x)=∞

means that the values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a (on either side of a) but not equal to a.

X→a

Chapter 1, 1.6, 57

Chapter 1, 1.6, 57

Chapter 1, 1.6, 57

Chapter 1, 1.6, 57

Chapter 1, 1.6, 57

Chapter 1, 1.6, 57

Chapter 1, 1.6, 57

2. DEFINITION The line x = a is called a vertical asymptote of the curve y=f(x) if at least one of the following statements is true:

lim f(x)=∞ lim f(x)=∞ lim f(x)=∞

lim f(x)=-∞ lim f(x)=-∞ lim f(x)=-∞

X→a X→a- X→a+

X→a X→a- X→a+

Chapter 1, 1.6, 58

Chapter 1, 1.6, 59

3. DEFINITION Let f be a function defined on some interval(a, ∞) . Then lim f(x)=L

means that the values of f(x) can be made as close to L as we like by taking x sufficiently large.

X→a

Chapter 1, 1.6, 59

Chapter 1, 1.6, 59

Chapter 1, 1.6, 59

Chapter 1, 1.6, 60

Chapter 1, 1.6, 60

Chapter 1, 1.6, 60

Chapter 1, 1.6, 60

4. DEFINITION The line y=L is called a horizontal asymptote of the curve y=f(x)if either

lim f(x)=L or lim f(x)=L X→∞ X→∞

Chapter 1, 1.6, 60

EXAMPLE 3 Find the infinite limits, limits at infinity, and asymptotes for the function f whose graph is shown in Figure 11.

Chapter 1, 1.6, 61

5. If n is a positive integer, then

lim =0 lin =0 xn

1

xn

1

X→∞ X→∞

Chapter 1, 1.6, 64

6. DEFINITION Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then lim f(x)=∞

means that for every positive number M there is a positive number δsuch that if 0<│x-a│<δ then f(x)>M

X→ a

Chapter 1, 1.6, 65

7. DEFINITION Let f be a function defined on some interval(a, ∞) . Then

lim f(x)=L

means that for every ε>0 there is a corresponding number N such that if x>N then │f(x)-L│<ε

X→∞

Chapter 1, 1.6, 65

Chapter 1, 1.6, 65

Chapter 1, 1.6, 66

8. DEFINITION Let f be a function defined on some interval(a, ∞) . Then lim f(x)=∞means that for every positive number M there is a corresponding positive number N such that if x>N then f(x)>M

X→∞

Chapter 1, 1.6, 66

1.For the function f whose graph is given, state the following.

(a) lim f(x) (b) lim f(x)

(c) lim f(x) (d) lim f(x)

(e) lim f(x)

(f) The equations of the asymptotes

X→ 2 X→ -1-

X→ -1+ X→∞

X→ -∞

Chapter 1, 1.6, 66

Chapter 1, 1.6, 67

2. For the function g whose graph is given, state the following.

(a) lim g(x) (b) lim g(x)

(c) lim g(x) (d) lim g(x)(e) lim g(x) (f) The equations of the asymptotes

X→∞ X→ -∞

X→3 X→ 0

X→ -2+

Chapter 1, Review, 70

1. Let f be the function whose graph is given.(a) Estimate the value of f(2).(b) Estimate the values of x such that f(x)=3.(c) State the domain of f.(d) State the range of f.(e) On what interval is increasing?(f ) Is f even, odd, or neither even nor odd? Explain.

Chapter 1, Review, 70

Chapter 1, Review, 71

2. Determine whether each curve is the graph of a function of x. If it is, state the domain and range of the function.

Chapter 1, Review, 71

8. The graph of f is given. Draw the graphs of the following functions.

(a)y=f(x-8) (b)y=-f(x)

(c)y=2-f(x) (d)y= f(x)-1 21

Chapter 1, Review, 71

21. The graph of f is given.

(a)Fine each limit, or explain why it doex not exist.

(i) lim f(x) (ii) lim f(x)

(iii) lim f(x) (iv) lim f(x)

(v) lim f(x) (vi) lim f(x)

(vii) lim f(x) (viii) lim f(x)(b)State the equations of the horizontal asymptotes.

(c)State the equations of the vertical asymptotes.

(d)At what number is f discontinuous? Explain.

X→ 2+ X→ -3+

X→ -3 X→ 4

X→0 X→2-

X→∞ X→ -∞

Chapter 1, Review, 71