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

Slides:

Advertisements

Similar presentations

3.3 Rules for Differentiation

Advertisements

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

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.

4.2 The Mean Value Theorem.

Derivative as a Function

Sec 3.2: The Derivative as a Function If ƒ’(a) exists, we say that ƒ is differentiable at a. (has a derivative at a) DEFINITION.

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

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.

SECTION 3.1 The Derivative and the Tangent Line Problem.

Ch 4 - Logarithmic and Exponential Functions - Overview

MAT 125 – Applied Calculus 2.5 – One-Sided Limits & Continuity.

The Derivative Definition, Interpretations, and Rules.

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

3.2 Differentiability. Yes No All Reals To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: cornercusp.

Differentiability Arches National Park- Park Avenue.

GOAL: USE DEFINITION OF DERIVATIVE TO FIND SLOPE, RATE OF CHANGE, INSTANTANEOUS VELOCITY AT A POINT. 3.1 Definition of Derivative.

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

Curve Sketching with Radical Functions

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?

Sec. 3.3: Rules of Differentiation. The following rules allow you to find derivatives without the direct use of the limit definition. The Constant Rule.

1. Consider f(x) = x 2 What is the slope of the tangent at a=0?

Derivatives Derivatives By: Josh Hansen Brady Thompson Rex McArthur.

2.1 The Derivative and The Tangent Line Problem

Section 3.9 Antiderivatives

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.

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

Lesson 3.2 Differentiability AP Calculus Mrs. Mongold.

To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: cornercusp vertical tangent discontinuity.

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)

Section 3.2 The Derivative as a Function AP Calculus September 24, 2009 Berkley High School, D2B2.

2.1 The Derivative and the Tangent Line Problem Objectives: -Students will find the slope of the tangent line to a curve at a point -Students will use.

Theorems Lisa Brady Mrs. Pellissier Calculus AP 28 November 2008.

If f(x) is a continuous function on a closed interval x ∈ [a,b], then f(x) will have both an Absolute Maximum value and an Absolute Minimum value in the.

THE FUNDAMENTAL THEOREM OF CALCULUS Section 4.4. THE FUNDAMENTAL THEOREM OF CALCULUS Informally, the theorem states that differentiation and definite.

Differentiable vs. Continuous The process of finding the derivative of a function is called Differentiation. A function is called Differentiable at x if.

AP Calculus 3.2 Basic Differentiation Rules Objective: Know and apply the basic rules of differentiation Constant Rule Power Rule Sum and Difference Rule.

4.2 The Mean Value Theorem.

3.2 Rolle’s Theorem and the

Mean Value Theorem.

2.1 Tangent Line Problem.

3.3: Increasing/Decreasing Functions and the First Derivative Test

Bell Ringer Solve even #’s.

The Derivative as a Function

The Derivative Chapter 3.1 Continued.

3.2 Differentiability.

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

3.2 Differentiability, p. 109 Day 1

BASIC DIFFERENTIATION RULES AND RATES OF CHANGE

3.2 Rolle’s Theorem and the

3.2 Differentiability Arches National Park - Park Avenue

3.1: Increasing and Decreasing Functions

Exam2: Differentiation

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

3.2: Differentiability.

The Derivative (cont.) 3.1.

Section 3.2 Differentiability.

Find the derivative Find the derivative at the following point.

2.1B Derivative Graphs & Differentiability

Derivatives: definition and derivatives of various functions

Sec 2.8: The Derivative as a Function

3.2. Definition of Derivative.

BASIC DIFFERENTIATION RULES AND RATES OF CHANGE

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

Presentation transcript:

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

When f′(a) fails to exist Where the tangent does not exist: points of discontinuity (function does not exist) at corners at cusps Also at vertical tangents (slope is undefined)

Theorems If f has a derivative at x = a, then f is continuous at x = a. Intermediate Value Theorem for derivatives: If a & b are any two points in an interval on which f is differentiable, then f ′ takes on every value between f ′(a) & f ′(b).

Note: A function can be continuous but not differentiable but if it is differentiable it has to be continuous

Finding the Derivative

Derivative Rules Power rule: f(x)f′(x) 3x 7x x2x2 x3x3

Derivative Rules Sum and difference rules: If u and v are differentiable functions of x, then f(x)= 3x 2 +5x – 12

Examples