# Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 1.

## Presentation on theme: "Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 1."— Presentation transcript:

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 1

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Chapter 8 Sequences, L’Hôpital’s Rule, and Improper Integrals

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 8.1 Sequences

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 4 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 5 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 6 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 7 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 8 What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence …and why Sequences arise frequently in mathematics and applied fields.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 9 Defining a Sequence

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 10 Example Defining a Sequence Explicitly

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 11 Example Defining a Sequence Recursively

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 12 Arithmetic Sequence

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 13 Example Defining Arithmetic Sequences

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 14 Geometric Sequence

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 15 Example Defining Geometric Sequences

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 16 Example Constructing a Sequence

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 17 Example Graphing a Sequence Using Parametric Mode

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 18 Example Graphing a Sequence Using Sequence Graphing Mode

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 19 Example Graphing a Sequence Using Sequence Graphing Mode

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 20 Limit

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 21 Properties of Limits

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 22 Example Finding the Limit of a Sequence

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 23 The Sandwich Theorem for Sequences

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 24 Absolute Value Theorem

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 8.2 L’Hôpital’s Rule

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 26 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 27 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 28 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 29 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 30 What you’ll learn about Indeterminate Form 0/0 Indeterminate Forms ∞/∞, ∞·0, ∞-∞ Indeterminate Form 1 ∞, 0 0, ∞ 0 …and why Limits can be used to describe the behavior of functions and l’Hôpital’s Rule is an important technique for finding limits.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 31 Indeterminate Form 0/0

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 32 L’Hôpital’s Rule (First Form)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 33 Example Indeterminate Form 0/0

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 34 L’Hôpital’s Rule (Stronger Form)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 35 Example Using L’Hôpital’s Rule with One-Sided Limits

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 36 Example Working with Indeterminate Form ∞/∞

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 37 Example Working with Indeterminate Form ∞·0

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 38 Indeterminate Forms 1 ∞, 0 0,∞ 0

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 39 Example Working with Indeterminate Form 1 ∞

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 40 Example Working with Indeterminate Form 1 ∞

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 41 Example Working with Indeterminate Form 0 0

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 42 Example Working with Indeterminate Form ∞ 0

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 43 Quick Quiz for Sections 8.1 and 8.2

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 44 Quick Quiz for Sections 8.1 and 8.2

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 45 Quick Quiz for Sections 8.1 and 8.2

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 46 Quick Quiz for Sections 8.1 and 8.2

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 47 Quick Quiz for Sections 8.1 and 8.2

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 48 Quick Quiz for Sections 8.1 and 8.2

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 8.3 Relative Rates of Growth

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 50 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 51 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 52 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 53 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 54 What you’ll learn about Comparing Rates of Growth Using L’Hôpital’s Rule to Compare Growth Rates Sequential versus Binary Search …and why Understanding growth rates as x→∞ is an important feature in understanding the behavior of functions.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 55 Faster, Slower, Same-rate Growth as x→∞

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 56 Example Comparing e x and x 3 as x→∞

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 57 Example Comparing ln x with x as x→∞

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 58 Example Comparing x with x+sin x as x→∞

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 59 Transitivity of Growing Rates

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 60 Example Growing at the Same Rate as x→∞

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 61 Example Finding the Order of a Binary Search

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 8.4 Improper Integrals

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 63 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 64 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 65 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 66 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 67 What you’ll learn about Infinite Limits of Integration Integrands with Infinite Discontinuities Test for Convergence and Divergence …and why The techniques of this section allow us to extend integration techniques to cases where the interval of integration [a,b] is not finite or where integrands are not continuous.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 68 Improper Integrals with Infinite Integration Limits

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 69 Example Evaluating an Improper Integral on [1,∞)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 70 Example Using L’Hôpital’s Rule with Improper Integrals

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 71 Example Evaluating an Integral on (-∞,∞)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 72 Improper Integrals with Infinite Discontinuities

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 73 Example Infinite Discontinuity at an Interior Point

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 74 Comparison Test

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 75 Example Finding the Volume of an Infinite Solid

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 76 Example Finding the Volume of an Infinite Solid

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 77 Quick Quiz Sections 8.3 and 8.4

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 78 Quick Quiz Sections 8.3 and 8.4

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 79 Quick Quiz Sections 8.3 and 8.4

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 80 Quick Quiz Sections 8.3 and 8.4

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 81 Quick Quiz Sections 8.3 and 8.4

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 82 Quick Quiz Sections 8.3 and 8.4

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 83 Chapter Test

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 84 Chapter Test

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 85 Chapter Test

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 86 Chapter Test

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 87 Chapter Test Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 88 Chapter Test Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 89 Chapter Test Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 90 Chapter Test Solution