AP Calculus Honors Ms. Olifer

Slides:

Advertisements

Similar presentations

Limit and Continuity.

Advertisements

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.

AP Calculus 1004 Continuity (2.3). C CONVERSATION: Voice level 0. No talking! H HELP: Raise your hand and wait to be called on. A ACTIVITY: Whole class.

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

Rates of Change and Limits

1 Continuity at a Point and on an Open Interval. 2 In mathematics, the term continuous has much the same meaning as it has in everyday usage. Informally,

AP CALCULUS 1003 Limits pt.3 Limits at Infinity and End Behavior.

Continuity Section 2.3.

Continuity 2.4.

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

AP Calculus 1005 Continuity (2.3). General Idea: General Idea: ________________________________________ We already know the continuity of many functions:

A rational function is a function whose rule can be written as a ratio of two polynomials. The parent rational function is f(x) = . Its graph is a.

2.1 Rates of Change and Limits. What you’ll learn about Average and Instantaneous Speed Definition of Limit Properties of Limits One-Sided and Two-Sided.

Continuity Chapter 2: Limits and Continuity.

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

CONTINUITY Mrs. Erickson Continuity lim f(x) = f(c) at every point c in its domain. To be continuous, lim f(x) = lim f(x) = lim f(c) x  c+x  c+ x 

2.3 Continuity.

2.4 Continuity and its Consequences and 2.8 IVT Tues Sept 15 Do Now Find the errors in the following and explain why it’s wrong:

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.

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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.1 Rates of Change and Limits.

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

AP Calculus BC Chapter 2: Limits and Continuity 2.3: Continuity.

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.

Continuity and One- Sided Limits (1.4) September 26th, 2012.

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

Suppose you drive 200 miles, and it takes you 4 hours. Then your average speed is: If you look at your speedometer during this trip, it might read 65 mph.

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

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

1.4 Continuity and One-Sided Limits Main Ideas Determine continuity at a point and continuity on an open interval. Determine one-sided limits and continuity.

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

1.4 Continuity Calculus.

Continuity Created by Mrs. King OCS Calculus Curriculum.

A rational function is a function whose rule can be written as a ratio of two polynomials. The parent rational function is f(x) = . Its graph is a.

Continuity and One-Sided Limits

Warm-Up: Use the graph of f (x) to find the domain and range of the function.

Continuity and One-Sided Limits (1.4)

Rates of Change and Limits

Rates of Change and Limits

Evaluating Limits Algebraically AP Calculus Ms. Olifer

AP Calculus September 6, 2016 Mrs. Agnew

Limits and Their Properties

LIMITS … oh yeah, Calculus has its limits!

AP Calculus Honors Ms. Olifer

AP Calculus Honors Ms. Olifer

1.6 Continuity Objectives:

Section 5.3 Definite Integrals and Antiderivatives

§2.5. Continuity In this section, we use limits to

Mean-Value Theorem for Integrals

2.5 Continuity In this section, we will:

1.4 Continuity and One-Sided Limits

Cornell Notes Section 1.3 Day 2 Section 1.4 Day 1

Section 2.2 Objective: To understand the meaning of continuous functions. Determine whether a function is continuous or discontinuous. To identify the.

Continuity and One-Sided Limits

Graph rational functions.

Chapter 2 Limits and Continuity Section 2.3 Continuity.

Sec 2.5: Continuity Continuous Function

1.4 Continuity and One-Sided Limits

Continuity A function is Continuous if it can be drawn without lifting the pencil, or writing utensil, from the paper. A continuous function has no breaks,

Chapter 2 Limits and Continuity Section 2.3 Continuity.

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.

2.3 Continuity.

Presentation transcript:

AP Calculus Honors Ms. Olifer Continuity AP Calculus Honors Ms. Olifer

Quick Review

Quick Review Solutions

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.

Continuity at a Point

Example Continuity at a Point

Continuity at a Point

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.

Removable discontinuity Continuity at a Point Continuous at x=0 Removable discontinuity Jump discontinuity Infinite disc. Oscillating disc.

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

One-sided Continuity A function f(x) is called: Left-continuous at x=c if Right-continuous at x=c if Pg. 82, Figure 6

More Practice…

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. Polynomial functions Rational functions (they have points of discontinuity at the zeros of their denominators) Absolute value functions Exponential functions Logarithmic functions Trigonometric functions Radical 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 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.