Section 3.2 Differentiability.

Slides:

Advertisements

Similar presentations

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

Advertisements

2.1 Tangent Line Problem. Tangent Line Problem The tangent line can be found by finding the slope of the secant line through the point of tangency and.

2.1 The derivative and the tangent line problem

THE DERIVATIVE AND THE TANGENT LINE PROBLEM

4.2 The Mean Value Theorem.

Derivative as a Function

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.

The Derivative. Def: The derivative of a function f at a number a, denoted f’(a) is: Provided this limit exists.

Differentiability, Local Linearity

SECTION 3.1 The Derivative and the Tangent Line Problem.

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

1 Discuss with your group. 2.1 Limit definition of the Derivative and Differentiability 2015 Devil’s Tower, Wyoming Greg Kelly, Hanford High School, Richland,

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.

Differentiability and Piecewise Functions

Unit 1 Limits. Slide Limits Limit – Assume that a function f(x) is defined for all x near c (in some open interval containing c) but not necessarily.

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

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.

Curve Sketching with Radical Functions

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

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

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

2.1 Day Differentiability.

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

Section 5.5 The Intermediate Value Theorem Rolle’s Theorem The Mean Value Theorem 3.6.

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

Differentiability Notes 3.2. I. Theorem If f(x) is differentiable at x = c, then f(x) is continuous at x = c. NOT VICE VERSA!!! A.) Ex. - Continuous at.

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

MTH 251 – Differential Calculus Chapter 3 – Differentiation Section 3.2 The Derivative as a Function Copyright © 2010 by Ron Wallace, all rights reserved.

Section 2.4 – Calculating the Derivative Numerically.

2.1 The Derivative and the Tangent Line Problem.

If f (x) is continuous over [ a, b ] and differentiable in (a,b), then at some point, c, between a and b : Mean Value Theorem for Derivatives.

Lesson 3.2 Differentiability AP Calculus Mrs. Mongold.

Calculus and Analytical Geometry Lecture # 15 MTH 104.

Section 3.2 Calculus Fall, Definition of Derivative We call this derivative of a function f. We use notation f’(x) (f prime of x)

4.2 The Mean Value Theorem.

3.2 Rolle’s Theorem and the

Hypothesis: Conclusion:

Lesson 3.2 Rolle’s Theorem Mean Value Theorem 12/7/16

Mean Value Theorem.

2.1 Tangent Line Problem.

Derivative Notation • The process of finding the derivative is called DIFFERENTIATION. • It is useful often to think of differentiation as an OPERATION.

Chapter 3 Derivatives Section 3.2 Differentiability.

The Derivative and the Tangent Line Problem (2.1)

3.2 Differentiability Arches National Park

The Derivative as a Function

3.2 Differentiability.

3.2 Differentiability, p. 109 Day 1

3.2 Rolle’s Theorem and the

Applications of Derivatives

THE DERIVATIVE AND THE TANGENT LINE PROBLEM

3.2 Differentiability Arches National Park - Park Avenue

Chapter 3 Derivatives Differentiability.

Chapter 3 Derivatives Section 3.2 Differentiability.

Differentiation Given the information about differentiable functions f(x) and g(x), determine the value.

3.2: Differentiability.

The Derivative (cont.) 3.1.

1. Be able to apply The Mean Value Theorem to various functions.

Section 2.1 Day 3 Derivatives

2.1B Derivative Graphs & Differentiability

Derivatives: definition and derivatives of various functions

Chapter 3 Derivatives Section 3.2 Differentiability.

The Intermediate Value Theorem

a) Find the local extrema

Chapter 3 Derivatives Section 3.2 Differentiability.

Rolle’s Theorem and the Mean Value Theorem

3.2. Definition of Derivative.

Rolle’s Theorem and the Mean Value Theorem

Tangent Line Approximations and Theorems

The Derivative and the Tangent Line Problem (2.1)

Presentation transcript:

Section 3.2 Differentiability

How f’(a) Might Fail to Exist A corner, where the one-sided derivatives differ

How f’(a) Might Fail to Exist A cusp, where the slopes of the secant lines approach  from one side and - from the other (an extreme case of a corner):

How f’(a) Might Fail to Exist A vertical tangent, where the slopes of the secant lines approach either  or - from both sides.

How f’(a) Might Fail to Exist A discontinuity, which will cause one or both of the one-sided derivatives to be nonexistent.

Example 1 Find all the points in the domain where f is not differentiable.

Example 2 Compute the numerical derivative.

Example 3 Compute the numerical derivative.

Example 4 Let f(x) = lnx. Use NDERIV to graph y = f’(x). Can you guess what function f’(x) is by analyzing its graph?

Differentiability Implies Continuity Theorem 1: Differentiability Implies Continuity: If f has a derivative at x = a, then f is continuous at x = a.

Intermediate Value Theorem for Derivatives Theorem 2: Intermediate Value Theorem for Derivatives: If a and b are any two points in an interval on which f is differentiable, then f’ takes on every value between f’(a) and f’(b).