Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Chapter 2 Limits and Continuity
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 3 Example Limits
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 4 Example Limits [-6,6] by [-10,10]
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 5 One-Sided and Two-Sided Limits
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 7 Example One-Sided and Two-Sided Limits o Find the following limits from the given graph.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 8 Sandwich Theorem
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 9 Sandwich Theorem
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.2 Limits Involving Infinity
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions [-12,12] by [-8,8][-6,6] by [-4,4]
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide What you’ll learn about Finite Limits as x→±∞ Sandwich Theorem Revisited Infinite Limits as x→a End Behavior Models Seeing Limits as x→±∞ …and why Limits can be used to describe the behavior of functions for numbers large in absolute value.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Finite limits as x→±∞ The symbol for infinity (∞) does not represent a real number. We use ∞ to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For example, when we say “the limit of f as x approaches infinity” we mean the limit of f as x moves increasingly far to the right on the number line. When we say “the limit of f as x approaches negative infinity (- ∞)” we mean the limit of f as x moves increasingly far to the left on the number line.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Horizontal Asymptote
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide [-6,6] by [-5,5] Example Horizontal Asymptote
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Sandwich Theorem Revisited
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Properties of Limits as x→±∞
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Properties of Limits as x→±∞ Product Rule: Constant Multiple Rule:
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Properties of Limits as x→±∞
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Infinite Limits as x→a
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Vertical Asymptote
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Vertical Asymptote [-6,6] by [-6,6]
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide End Behavior Models
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example End Behavior Models
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide End Behavior Models
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide End Behavior Models
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example “Seeing” Limits as x→±∞
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 2.1 and 2.2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 2.1 and 2.2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 2.1 and 2.2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 2.1 and 2.2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 2.1 and 2.2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 2.1 and 2.2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.3 Continuity
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide What you’ll learn about Continuity at a Point Continuous Functions Algebraic Combinations Composites Intermediate Value Theorem for Continuous Functions …and why Continuous functions are used to describe how a body moves through space and how the speed of a chemical reaction changes with time.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Continuity at a Point
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Continuity at a Point o
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Continuity at a Point
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Continuity at a Point If a function f is not continuous at a point c, we say that f is discontinuous at c and c is a point of discontinuity of f. Note that c need not be in the domain of f.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Continuity at a Point The typical discontinuity types are: a)Removable(2.21b and 2.21c) b)Jump(2.21d) c)Infinite(2.21e) d)Oscillating (2.21f)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Continuity at a Point
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Continuity at a Point [-5,5] by [-5,10]
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Continuous Functions A function is continuous on an interval if and only if it is continuous at every point of the interval. A continuous function is one that is continuous at every point of its domain. A continuous function need not be continuous on every interval.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Continuous Functions [-5,5] by [-5,10]
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Properties of Continuous Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Composite of Continuous Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Intermediate Value Theorem for Continuous Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Intermediate Value Theorem for Continuous Functions The Intermediate Value Theorem for Continuous Functions is the reason why the graph of a function continuous on an interval cannot have any breaks. The graph will be connected, a single, unbroken curve. It will not have jumps or separate branches.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.4 Rates of Change and Tangent Lines
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide What you’ll learn about Average Rates of Change Tangent to a Curve Slope of a Curve Normal to a Curve Speed Revisited …and why The tangent line determines the direction of a body’s motion at every point along its path.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Average Rates of Change The average rate of change of a quantity over a period of time is the amount of change divided by the time it takes. In general, the average rate of change of a function over an interval is the amount of change divided by the length of the interval. Also, the average rate of change can be thought of as the slope of a secant line to a curve.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Average Rates of Change
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Tangent to a Curve In calculus, we often want to define the rate at which the value of a function y = f(x) is changing with respect to x at any particular value x = a to be the slope of the tangent to the curve y = f(x) at x = a. The problem with this is that we only have one point and our usual definition of slope requires two points.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Tangent to a Curve The process becomes: 1.Start with what can be calculated, namely, the slope of a secant through P and a point Q nearby on the curve. 2.Find the limiting value of the secant slope (if it exists) as Q approaches P along the curve. 3.Define the slope of the curve at P to be this number and define the tangent to the curve at P to be the line through P with this slope.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Tangent to a Curve
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Tangent to a Curve
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Slope of a Curve To find the tangent to a curve y = f(x) at a point P(a,f(a)) calculate the slope of the secant line through P and a point Q(a+h, f(a+h)). Next, investigate the limit of the slope as h→0. If the limit exists, it is the slope of the curve at P and we define the tangent at P to be the line through P with this slope.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Slope of a Curve
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Slope of a Curve at a Point
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Slope of a Curve
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Normal to a Curve The normal line to a curve at a point is the line perpendicular to the tangent at the point. The slope of the normal line is the negative reciprocal of the slope of the tangent line.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Normal to a Curve
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Speed Revisited
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 2.3 and 2.4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 2.3 and 2.4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 2.3 and 2.4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 2.3 and 2.4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 2.3 and 2.4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 2.3 and 2.4
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Chapter Test Solutions