Preliminaries 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 0.2 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 0.2 LINES AND FUNCTIONS 0.3 GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 0.4 TRIGONOMETRIC FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 2

GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 0.3 GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS Intercepts, Extrema, and Inflection Points Graphs are drawn to provide visual displays of the significant features of a function. What qualifies as significant will vary from problem to problem, but often, the x- and y-intercepts and points known as extrema are of interest. The function value f (M) is called a local maximum of the function f if f (M) ≥ f (x) for all x’s “nearby” x = M. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 3

GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 0.3 GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS Intercepts, Extrema, and Inflection Points Similarly, the function value f (m) is a local minimum of the function f if f (m) ≤ f (x) for all x’s “nearby” x = m. A local extremum is a function value that is either a local maximum or local minimum. Whenever possible, you should produce graphs that show all intercepts and extrema. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 4

GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 0.3 GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS Intercepts, Extrema, and Inflection Points Many curves (including all cubics) have what’s called an inflection point, where the curve changes its shape (from being bent upward, to being bent downward, or vice versa). © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 5

GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 0.3 GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 3.3 Sketching the Graph of a Rational Function © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 6

GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 0.3 GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 3.3 Sketching the Graph of a Rational Function We say that f (x) tends to ∞ (or –∞) as x approaches 2 and there is a vertical asymptote at x = 2. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 7

GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 0.3 GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 3.6 Finding Zeros Approximately © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 8

GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 0.3 GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 3.6 Finding Zeros Approximately x ≈ 1.452626878 and x ≈ −1.164035140 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 9

GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 0.3 GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 3.8 Solving an Equation by Calculator: An Erroneous Answer © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 10

GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 0.3 GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 3.8 Solving an Equation by Calculator: An Erroneous Answer Some computer algebra systems report the answer as x = 0, others as some number close to zero; typically 1 × 10-7. Neither answer is correct. The equation has no solution. Be an intelligent user of technology and don’t blindly accept everything your calculator tells you. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 11