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# SmithMinton_Calculus_ET_5e_C00_S01.pptx

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# SmithMinton_Calculus_ET_5e_C00_S01.pptx

basic calculus

basic calculus

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### SmithMinton_Calculus_ET_5e_C00_S01.pptx

1. 1. © 2017 by McGraw-Hill Education. Permission required for reproduction or display.
2. 2. CHAPTER 0 Preliminaries Slide 2 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 0.2 GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 0.3 INVERSE FUNCTIONS 0.4 TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNCTIONS 0.5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 0.6 TRANSFORMATIONS OF FUNCTIONS
3. 3. 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS The Real Number System and Inequalities Slide 3 The set of integers consists of the whole numbers and their additive inverses: 0, ±1,±2,±3, . . . . A rational number is any number of the form p/q , where p and q are integers and q ≠ 0. For example, 2/3 and −7/3 are rational numbers. Notice that every integer n is also a rational number, since we can write it as the quotient of two integers: n = n/1.
4. 4. 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS The Real Number System and Inequalities Slide 4 Irrational numbers are all those real numbers that cannot be written in the form p/q , where p and q are integers. Recall that rational numbers have decimal expansions that either terminate or repeat. By contrast, irrational numbers have decimal expansions that do not repeat or terminate.
5. 5. 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS The Real Number System and Inequalities Slide 5 We picture the real numbers arranged along the number line (the real line). The set of real numbers is denoted by the symbol .
6. 6. 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS The Real Number System and Inequalities Slide 6 For real numbers a and b, where a < b, we define the closed interval [a, b] to be the set of numbers between a and b, including a and b (the endpoints). That is, . Similarly, the open interval (a, b) is the set of numbers between a and b, but not including the endpoints a and b, that is, .
7. 7. THEOREM 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.1 Slide 7
8. 8. EXAMPLE 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.2 Solving a Two-Sided Inequality Slide 8
9. 9. EXAMPLE Solution 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.2 Solving a Two-Sided Inequality Slide 9
10. 10. EXAMPLE 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.3 Solving an Inequality Involving a Fraction Slide 10
11. 11. EXAMPLE Solution 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.3 Solving an Inequality Involving a Fraction Slide 11
12. 12. DEFINITION 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.1 Slide 12
13. 13. 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS The Real Number System and Inequalities Slide 13 Notice that for any real numbers a and b, |a · b| = |a| · |b|, although |a + b| ≠ |a| + |b|, in general. However, it is always true that |a + b| ≤ |a| + |b|. This is referred to as the triangle inequality. The interpretation of |a − b| as the distance between a and b is particularly useful for solving inequalities involving absolute values.
14. 14. EXAMPLE 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.7 Solving Inequalities Slide 14 Solve the inequality |x − 2|< 5.
15. 15. EXAMPLE Solution 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.7 Solving Inequalities Slide 15
16. 16. THEOREM 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.1 Slide 16 The distance between the points (x1, y1) and (x2, y2) in the Cartesian plane is given by
17. 17. EXAMPLE 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.8 Using the Distance Formula Slide 17 Find the distance between the points (1, 2) and (3, 4).
18. 18. EXAMPLE Solution 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.8 Using the Distance Formula Slide 18
19. 19. DEFINITION 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.2 Slide 19 For x1 ≠ x2, the slope of the straight line through the points (x1, y1) and (x2, y2) is the number When x1 = x2 and y1 ≠ y2, the line through (x1, y1) and (x2, y2) is vertical and the slope is undefined. Notice that a line is horizontal if and only if its slope is zero.
20. 20. EXAMPLE 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.10 Using Slope to Determine if Points Are Colinear Slide 20 Use slope to determine whether the points (1, 2), (3, 10) and (4, 14) are colinear.
21. 21. EXAMPLE Solution 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.10 Using Slope to Determine if Points Are Colinear Slide 21 Since the slopes are the same, the points must be colinear.
22. 22. 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS POINT-SLOPE FORM OF A LINE Slide 22
23. 23. EXAMPLE 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.12 Finding the Equation of a Line Given Two Points Slide 23 Find an equation of the line through the points (3, 1) and (4, −1), and graph the line.
24. 24. EXAMPLE Solution 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.12 Finding the Equation of a Line Given Two Points Slide 24
25. 25. THEOREM 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.2 Slide 25 Two (nonvertical) lines are parallel if they have the same slope. Further, any two vertical lines are parallel. Two (nonvertical) lines of slope m1 and m2 are perpendicular whenever the product of their slopes is −1 (i.e., m1 · m2 = −1). Also, any vertical line and any horizontal line are perpendicular.
26. 26. EXAMPLE 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.14 Finding the Equation of a Perpendicular Line Slide 26 Find an equation of the line perpendicular to y = −2x + 4 and intersecting the line at the point (1, 2).
27. 27. EXAMPLE Solution 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.14 Finding the Equation of a Perpendicular Line Slide 27 The slope of y = −2x + 4 is −2. The slope of the perpendicular line is then −1/(−2) = ½.
28. 28. DEFINITION 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.3 Slide 28 A function f is a rule that assigns exactly one element y in a set B to each element x in a set A. In this case, we write y = f (x). We call the set A the domain of f. The set of all values f(x) in B is called the range of f , written {y|y = f (x), for some x in set A}.
29. 29. DEFINITION 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.3 Slide 29 Unless explicitly stated otherwise, whenever a function f is given by a particular expression, the domain of f is the largest set of real numbers for which the expression is defined. We refer to x as the independent variable and to y as the dependent variable.
30. 30. 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS Functions Slide 30 By the graph of a function f, we mean the graph of the equation y = f (x). That is, the graph consists of all points (x, y), where x is in the domain of f and where y = f (x). Notice that not every curve is the graph of a function, since for a function, only one y-value can correspond to a given value of x.
31. 31. 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS Functions Slide 31 We can graphically determine whether a curve is the graph of a function by using the vertical line test: If any vertical line intersects the graph in more than one point, the curve is not the graph of a function since, in this case, there are two y-values for a given value of x.
32. 32. EXAMPLE Solution 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.16 Using the Vertical Line Test Slide 32 Determine which of the curves correspond to functions.
33. 33. EXAMPLE Solution 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.16 Using the Vertical Line Test Slide 33
34. 34. DEFINITION 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.4 Slide 34 A polynomial is any function that can be written in the form where a0, a1, a2, . . . , an are real numbers (the coefficients of the polynomial) with an ≠ 0 and n ≥ 0 is an integer (the degree of the polynomial).
35. 35. DEFINITION 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.5 Slide 35 Any function that can be written in the form where p and q are polynomials, is called a rational function.
36. 36. EXAMPLE 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.18 A Sample Rational Function Slide 36 Find the domain of the function
37. 37. EXAMPLE Solution 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.18 A Sample Rational Function Slide 37
38. 38. 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS Functions Slide 38 The square root function is defined in the usual way. When we write we mean that y is that number for which y2 = x and y ≥ 0. In particular, . Be careful not to write erroneous statements such as . In particular, be careful to write Similarly, for any integer n ≥ 2, whenever yn = x, where for n even, x ≥ 0 and y ≥ 0.
39. 39. EXAMPLE 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.19 Finding the Domain of a Function Involving a Square Root or a Cube Root Slide 39
40. 40. EXAMPLE Solution 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.19 Finding the Domain of a Function Involving a Square Root or a Cube Root Slide 40
41. 41. 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS Functions Slide 41 A solution of the equation f (x) = 0 is called a zero of the function f or a root of the equation f (x) = 0. Notice that a zero of the function f corresponds to an x- intercept of the graph of y = f (x).
42. 42. EXAMPLE 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.20 Finding Zeros by Factoring Slide 42 Find all x- and y-intercepts of f (x) = x2 − 4x + 3.
43. 43. EXAMPLE Solution 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.20 Finding Zeros by Factoring Slide 43 y-intercept x-intercepts For the quadratic equation ax2 + bx + c = 0 (for a ≠ 0), the solution(s) are also given by the familiar quadratic formula:
44. 44. THEOREM 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.3 Slide 44
45. 45. REMARK 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.3 Slide 45 Polynomials may also have complex zeros. For instance, f (x) = x2 + 1 has only the complex zeros x = ±i, where i is the imaginary number defined by . We confine our attention in this text to real zeros.
46. 46. THEOREM 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.4 Slide 46 (Factor Theorem)
47. 47. EXAMPLE 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.22 Finding the Zeros of a Cubic Polynomial Slide 47
48. 48. EXAMPLE Solution 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 1.22 Finding the Zeros of a Cubic Polynomial Slide 48 By calculating f (1), we can see that one zero of this function is x = 1. Factor: