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ESSENTIAL CALCULUS CH01 Functions & Limits. In this Chapter:. 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.
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In this Chapter: • 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
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
A function fis 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
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. Chapter 1, 1.1, P2
EXAMPLE 1 The graph of a function f is shown in Figure 6. • Find the values of f(1) and f(5) . • (b) What are the domain and range of f ? Chapter 1, 1.1, P2
EXAMPLE 3 Find the domain of each function. 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, P4
EXAMPLE 4 A function f is defined by 1-X if X≤1 X2 if X>1 f(x)= Evaluate f(0) ,f(1) , and f(2) and sketch the graph. Chapter 1, 1.1, P5
EXAMPLE 5 Sketch the graph of the absolute 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 0.39 if o<w≤1 0.63 if 1<w≤2 0.87 if 2<w≤3 1.11 if 3<w≤4 C(w)= 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
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. • f(x)=x5+x • g(x)=1-x4 • h(x)=2x=x2 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, 07
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
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
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, 08
53–54 ■ Graphs of f and g are shown. Decide whether each function is even, odd, or neither. Explain your reasoning. Chapter 1, 1.1, 10
A function P is called a polynomial if P(x)=anxn+an-1xn-1+‧‧‧+a2x2+a1x+a0 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. Chapter 1, 1.2, 13
A rational function fis a ratio of two polynomials: Where P and Q are polynomials. The domain consists of all values of x such that Q(x)≠0. Chapter 1, 1.2, 15
-1≤ son x≤1 -1≤ cos x≤1 Chapter 1, 1.2, 15
