Chapter 3 Derivatives Section 3.2 Differentiability.

Slides:

Advertisements

Similar presentations

3.2 Differentiability Photo by Vickie Kelly, 2003 Arches National Park Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry.

Advertisements

DERIVATIVE OF A FUNCTION 1.5. DEFINITION OF A DERIVATIVE OTHER FORMS: OPERATOR:,,,

The Derivative and the Tangent Line Problem

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 1.

Section 3.2b. The “Do Now” Find all values of x for which the given function is differentiable. This function is differentiable except possibly where.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Chapter 3 Review Limits and Continuity.

Section 6.1: Euler’s Method. Local Linearity and Differential Equations Slope at (2,0): Tangent line at (2,0): Not a good approximation. Consider smaller.

Local Linearity The idea is that as you zoom in on a graph at a specific point, the graph should eventually look linear. This linear portion approximates.

Chapter 2 Section 2 The Derivative!. Definition The derivative of a function f(x) at x = a is defined as f’(a) = lim f(a+h) – f(a) h->0 h Given that a.

Differentiability, Local Linearity

Chapter 3 The Derivative. 3.2 The Derivative Function.

Today in Calculus Go over homework Derivatives by limit definition Power rule and constant rules for derivatives Homework.

3.3 Rules for Differentiation AKA “Shortcuts”. Review from places derivatives do not exist: ▫Corner ▫Cusp ▫Vertical tangent (where derivative is.

AP CALCULUS AB Chapter 3: Derivatives Section 3.2: Differentiability.

Section 2.8 The Derivative as a Function Goals Goals View the derivative f ´(x) as a function of x View the derivative f ´(x) as a function of x Study.

3.2 Differentiability Objectives Students will be able to: 1)Find where a function is not differentiable 2)Distinguish between corners, cusps, discontinuities,

Homework Quiz Page 105 Exercises 2 & 8 Show work!.

Drill Tell whether the limit could be used to define f’(a).

AP Calculus AB/BC 3.2 Differentiability, p. 109 Day 1.

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003 Arches National Park 2.1 The Derivative and the Tangent Line Problem (Part.

Derivatives Test Review Calculus. What is the limit equation used to calculate the derivative of a function?

Section 3.2 Differentiability. Ways a function might not be differentiable.  1. a corner  Often occurs with absolute value functions.

Local Linear Approximation Objective: To estimate values using a local linear approximation.

Lesson 3.2 Differentiability AP Calculus Mrs. Mongold.

2.2 The Derivative as a Function. 22 We have considered the derivative of a function f at a fixed number a: Here we change our point of view and let the.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 1.

1. Definition of Derivative

Linear Approximations. In this section we’re going to take a look at an application not of derivatives but of the tangent line to a function. Of course,

Chapter 10 Limits and the Derivative

2.7 and 2.8 Derivatives Great Sand Dunes National Monument, Colorado

Bell Ringer Solve even #’s.

Chapter 3 Derivatives Section 3.2 Differentiability.

Derivative of a Function

3.2 Differentiability Arches National Park

The Derivative as a Function

3.2 Differentiability.

2.1 The Derivative and the Tangent Line Problem (Part 2)

3.2 Differentiability, p. 109 Day 1

Derivative of a Function

Derivative of a Function

3.2 Differentiability Arches National Park - Park Avenue

Chapter 3 Derivatives Differentiability.

Chapter 3 Derivatives Section 3.2 Differentiability.

Exam2: Differentiation

3.2: Differentiability.

The Derivative (cont.) 3.1.

2.1 The Derivative and the Tangent Line Problem

Section 3.2 Differentiability.

Section 2.1 Day 3 Derivatives

Find the derivative Find the derivative at the following point.

2.1B Derivative Graphs & Differentiability

Chapter 3 Derivatives.

Derivatives: definition and derivatives of various functions

Chapter 2 Limits and Continuity Section 2.3 Continuity.

Differentiability and Continuity

Linearization and Newton’s Method

Chapter 3 Derivatives Section 3.2 Differentiability.

3.2 Differentiability Arches National Park

Chapter 6 Applications of Derivatives Section 6.2 Definite Integrals.

Linearization and Newton’s Method

Chapter 2 Limits and Continuity Section 2.3 Continuity.

3.2 Differentiability Arches National Park

Unit 2 - Derivatives.

Differentiation Using Limits of Difference Quotients

Derivative of a Function

Chapter 2 Limits and Continuity Section 2.3 Continuity.

2.1 The Derivative and the Tangent Line Problem (Part 2)

Chapter 2 Limits and the Derivative

2.4 The Derivative.

The Derivative as a Function

Presentation transcript:

Chapter 3 Derivatives Section 3.2 Differentiability

Quick Review

Quick Review

Quick Review Solutions

Quick Review Solutions

What you’ll learn about Why f ′(a) might fail to exist at x = a Differentiability implies local linearity Numerical derivatives on a calculator Differentiability implies continuity Intermediate Value Theorem for derivatives … and why Graphs of differentiable functions can be approximated by their tangent lines at points where the derivative exists.

How f ′(a) Might Fail to Exist

How f ′(a) Might Fail to Exist

How f ′(a) Might Fail to Exist

How f ′(a) Might Fail to Exist

How f ′(a) Might Fail to Exist

Example How f ′(a) Might Fail to Exist

How f ′(a) Might Fail to Exist Most of the functions we encounter in calculus are differentiable wherever they are defined, which means they will not have corners, cusps, vertical tangent lines or points of discontinuity within their domains. Their graphs will be unbroken and smooth, with a well-defined slope at each point.

Differentiability Implies Local Linearity A good way to think of differentiable functions is that they are locally linear; that is, a function that is differentiable at a closely resembles its own tangent line very close to a. In the jargon of graphing calculators, differentiable curves will “straighten out” when we zoom in on them at a point of differentiability.

Differentiability Implies Local Linearity

Numerical Derivatives on a Calculator

Numerical Derivatives on a Calculator The numerical derivative of f at a, which we will denote NDER is the number The numerical derivative of f, which we will denote NDER is the function

Example Derivatives on a Calculator

Derivatives on a Calculator Because of the method used internally by the calculator, you will sometimes get a derivative value at a nondifferentiable point. This is a case of where you must be “smarter” than the calculator.

Differentiability Implies Continuity The converse of Theorem 1 is false. A continuous functions might have a corner, a cusp or a vertical tangent line, and hence not be differentiable at a given point.

Intermediate Value Theorem for Derivatives Not every function can be a derivative.