Calculus Early Transcendental Functions PDF

© 2017 by McGraw-Hill Education. Permission required for reproduction or display.  CHAPTER Preliminaries 0 0.1 POLYNOMIALS AND RATIONAL FUNCTIONS 0.2 GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 0.3 INVERSE FUNCTI...

© 2017 by McGraw-Hill Education. Permission required for reproduction or display.  CHAPTER Preliminaries 0 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 Slide 2  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS The Real Number System and Inequalities 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. Slide 3  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS The Real Number System and Inequalities 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. Slide 4  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS The Real Number System and Inequalities We picture the real numbers arranged along the number line (the real line). The set of real numbers is denoted by the symbol. Slide 5  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS The Real Number System and Inequalities 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,. Slide 6  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS THEOREM 1.1 Slide 7  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS EXAMPLE 1.2 Solving a Two-Sided Inequality Slide 8  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS EXAMPLE 1.2 Solving a Two-Sided Inequality Solution Slide 9  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS EXAMPLE 1.3 Solving an Inequality Involving a Fraction Slide 10  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS EXAMPLE 1.3 Solving an Inequality Involving a Fraction Solution Slide 11  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS DEFINITION 1.1 Slide 12  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS The Real Number System and Inequalities 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. Slide 13  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS EXAMPLE 1.7 Solving Inequalities Solve the inequality |x − 2|< 5. Slide 14  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS EXAMPLE 1.7 Solving Inequalities Solution Slide 15  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS THEOREM 1.1 The distance between the points (x1, y1) and (x2, y2) in the Cartesian plane is given by Slide 16  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS EXAMPLE 1.8 Using the Distance Formula Find the distance between the points (1, 2) and (3, 4). Slide 17  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS EXAMPLE 1.8 Using the Distance Formula Solution Slide 18  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS DEFINITION 1.2 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. Slide 19  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS EXAMPLE 1.10 Using Slope to Determine if Points Are Colinear Use slope to determine whether the points (1, 2), (3, 10) and (4, 14) are colinear. Slide 20  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS EXAMPLE 1.10 Using Slope to Determine if Points Are Colinear Solution Since the slopes are the same, the points must be colinear. Slide 21  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS POINT-SLOPE FORM OF A LINE Slide 22  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS EXAMPLE 1.12 Finding the Equation of a Line Given Two Points Find an equation of the line through the points (3, 1) and (4, −1), and graph the line. Slide 23  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS EXAMPLE 1.12 Finding the Equation of a Line Given Two Points Solution Slide 24  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS THEOREM 1.2 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. Slide 25  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS EXAMPLE 1.14 Finding the Equation of a Perpendicular Line Find an equation of the line perpendicular to y = −2x + 4 and intersecting the line at the point (1, 2). Slide 26  POLYNOMIALS AND RATIONAL 0.1 FUNCTIONS EXAMPLE 1.14 Finding the Equation of a Perpendicular Line Solution The slope of y = −2x + 4 is −2. The slope of the perpendicular line is then −1/(−2) = ½. Slide 27