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ESSENTIAL CALCULUS CH01 Functions & Limits

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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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ESSENTIAL CALCULUS CH01 Functions & Limits
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  1. ESSENTIAL CALCULUSCH01 Functions & Limits

  2. 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

  3. 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

  4. 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

  5. Chapter 1, 1.1, P2

  6. Chapter 1, 1.1, P2

  7. 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

  8. Chapter 1, 1.1, P2

  9. Chapter 1, 1.1, P2

  10. Chapter 1, 1.1, P2

  11. 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

  12. EXAMPLE 3 Find the domain of each function. Chapter 1, 1.1, P4

  13. 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

  14. Chapter 1, 1.1, P5

  15. 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

  16. Chapter 1, 1.1, P5

  17. EXAMPLE 5 Sketch the graph of the absolute value function f(x)=│X│. Chapter 1, 1.1, P6

  18. 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

  19. Chapter 1, 1.1, P6

  20. 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

  21. Chapter 1, 1.1, P6

  22. Chapter 1, 1.1, P6

  23. 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

  24. 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

  25. Chapter 1, 1.1, 07

  26. 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

  27. 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

  28. Chapter 1, 1.1, 08

  29. 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

  30. Chapter 1, 1.1, 08

  31. 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

  32. 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

  33. 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

  34. Chapter 1, 1.2, 14

  35. Chapter 1, 1.2, 14

  36. Chapter 1, 1.2, 14

  37. Chapter 1, 1.2, 14

  38. Chapter 1, 1.2, 14

  39. Chapter 1, 1.2, 14

  40. Chapter 1, 1.2, 14

  41. Chapter 1, 1.2, 14

  42. Chapter 1, 1.2, 14

  43. Chapter 1, 1.2, 15

  44. 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

  45. Chapter 1, 1.2, 15

  46. Chapter 1, 1.2, 15

  47. -1≤ son x≤1 -1≤ cos x≤1 Chapter 1, 1.2, 15

  48. Chapter 1, 1.2, 16

  49. Chapter 1, 1.2, 16