Chapter 2 Limits and Continuity Section 2.3 Continuity.

Slides:

Advertisements

Similar presentations

AP Calculus Review First Semester Differentiation to the Edges of Integration Sections , 3.9, (7.7)

Advertisements

2.3 Continuity When you plot function values generated in a laboratory or collected in a field, you can connect the plotted points with an unbroken curve.

1.5 Continuity. Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without.

Sec 2.5: CONTINUITY. Study continuity at x = 4 Sec 2.5: CONTINUITY Study continuity at x = 2.

Continuity Section 2.3a.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 1.

1.5 Infinite Limits. Copyright © Houghton Mifflin Company. All rights reserved Figure 1.25.

Chapter Two Limits and Their Properties. Copyright © Houghton Mifflin Company. All rights reserved. 2 | 2 The Tangent Line Problem.

Continuity Section 2.3.

Continuity Grand Canyon, Arizona Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.

2.3 Continuity. What you’ll learn about Continuity at a Point Continuous Functions Algebraic Combinations Composites Intermediate Value Theorem for Continuous.

SAT Prep. Continuity and One-Sided Limits Determine continuity at a point and an open interval Determine one-sided limits and continuity on a closed interval.

Continuity Chapter 2: Limits and Continuity.

A function, f, is continuous at a number, a, if 1) f(a) is defined 2) exists 3)

2.3 Continuity.

Informal Description f(x) is continuous at x=c if and only if there are no holes, jumps, skips or gaps in the graph of f(x) at c.

Review Limits When you see the words… This is what you think of doing…  f is continuous at x = a  Test each of the following 1.

Chapter 2 Review Calculus. Quick Review 1.) f(2) = 0 2.) f(2) = 11/12 3.) f(2) = 0 4.) f(2) = 1/3.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.3 Continuity.

1.4 Continuity  f is continuous at a if 1. is defined. 2. exists. 3.

Chapter 2  2012 Pearson Education, Inc. 2.3 Section 2.3 Continuity Limits and Continuity.

5.3 Definite Integrals and Antiderivatives. What you’ll learn about Properties of Definite Integrals Average Value of a Function Mean Value Theorem for.

LIMITS OF FUNCTIONS. CONTINUITY Definition (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.

Limits and Continuity Unit 1 Day 4.

CALCULUS CHAPTER 1 Section 1.4: Continuity and One-Sided Limits Calculus Chapter 1 Section 4 1.

Definition: Continuous A continuous process is one that takes place gradually, without interruption or abrupt change.

Copyright © 2016, 2012, and 2010 Pearson Education, Inc. 1 What you ’ ll learn about Definition of continuity at a point Types of discontinuities Sums,

1 Limits and Continuity. 2 Intro to Continuity As we have seen some graphs have holes in them, some have breaks and some have other irregularities. We.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 1.

Calculus 1-4 Continuity and One-Sided Limits. Vocabulary Continuous: A function that is uninterrupted in the given interval None of these is continuous.

Chapter 2.3 Continuity. Objectives Continuity at a Point Continuous Functions Algebraic Combinations Composites Intermediate Value Theorem for Continuous.

1.4 Continuity Calculus.

Chapter 3 Derivatives Section 3.2 Differentiability.

Continuity and One-Sided Limits (1.4)

Rates of Change and Limits

Rates of Change and Limits

Rules for Differentiation

Rules for Differentiation

Continuity Sec. 2.3.

Derivative of a Function

Algebraic Limits and Continuity

INFINITE LIMITS Section 1.5.

AP Calculus Honors Ms. Olifer

Chapter 3 Derivatives Section 3.2 Differentiability.

Testing Convergence at Endpoints

8.4 Improper Integrals.

Connecting f′ and f″ with the graph of f

Limits Involving Infinity

Chapter 8 Section 8.2 Applications of Definite Integral

Chapter 3 Derivatives Section 3.2 Differentiability.

Definite Integrals and Antiderivatives

Chapter 2 Limits and Continuity Section 2.3 Continuity.

Sec 2.5: Continuity Continuous Function

Limits Involving Infinity

Chapter 3 Derivatives Section 3.2 Differentiability.

Connecting f′ and f″ with the graph of f

Rules for Differentiation

Chapter 6 Applications of Derivatives Section 6.2 Definite Integrals.

Chapter 5 Applications of Derivatives Section 5.2 Mean Value Theorem.

Definite Integrals & Antiderivatives

INFINITE LIMITS Section 1.5.

Rates of Change and Limits

Sec 2.5: Continuity Continuous Function

Limits and Continuity Section 2.3 Continuity.

Chapter 2 Limits and Continuity Section 2.3 Continuity.

Limits and Their Properties

Presentation transcript:

Chapter 2 Limits and Continuity Section 2.3 Continuity

Quick Review

Quick Review

Quick Review

Quick Review

Quick Review Solutions

Quick Review Solutions

Quick Review Solutions

Quick Review Solutions

What you’ll learn about Definition of continuity at a point Types of discontinuities Sums, differences, products, quotients, and compositions of continuous functions Common continuous functions Continuity and the Intermediate Value Theorem …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.

Continuity at a Point

Example Continuity at a Point

Continuity at a Point

Continuity at a Point

Continuity at a Point The typical discontinuity types are: Removable (2.21b and 2.21c) Jump (2.21d) Infinite (2.21e) Oscillating (2.21f)

Continuity at a Point

Example Continuity at a Point [5,5] by [5,10]

Continuous Functions

Continuous Functions [5,5] by [5,10]

Properties of Continuous Functions

Composite of Continuous Functions

Intermediate Value Theorem for Continuous Functions

Intermediate Value Theorem for Continuous Functions

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.