2.2 The derivative as a function

Slides:

Advertisements

Similar presentations

Section 3.3a. The Do Now Find the derivative of Does this make sense graphically???

Advertisements

The Derivative and the Tangent Line Problem. Local Linearity.

2.1 The derivative and the tangent line problem

Tangent lines Recall: tangent line is the limit of secant line The tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with slope.

The Derivative as a Function

If the derivative of a function is its slope, then for a constant function, the derivative must be zero. example: The derivative of a constant is zero.

The Derivative As A Function. 2 The First Derivative of f Interpretations f ′(a) is the value of the first derivative of f at x = a. f ′(x) is.

Derivative as a Function

Copyright © Cengage Learning. All rights reserved.

CHAPTER Continuity Derivatives Definition The derivative of a function f at a number a denoted by f’(a), is f’(a) = lim h  0 ( f(a + h) – f(a))

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.

DERIVATIVES The Derivative as a Function DERIVATIVES In this section, we will learn about: The derivative of a function f.

DERIVATIVES The Derivative as a Function DERIVATIVES In this section, we will learn about: The derivative of a function f.

LIMITS AND DERIVATIVES

SECTION 3.1 The Derivative and the Tangent Line Problem.

Calculus Warm-Up Find the derivative by the limit process, then find the equation of the line tangent to the graph at x=2.

3.1 –Tangents and the Derivative at a Point

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

Differentiability and Rates of Change. To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: cornercusp vertical.

Rules for Differentiation Colorado National Monument Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

The derivative of a function f at a fixed number a is In this lesson we let the number a vary. If we replace a in the equation by a variable x, we get.

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

2.3 Differentiation Rules Colorado National Monument Larson, Hostetler, & Edwards.

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.

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.

3.1 Definition of the Derivative & Graphing the Derivative

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

Definition of Derivative.  Definition   f‘(x): “f prime of x”  y‘ : “y prime” (what is a weakness of this notation?)  dy/dx : “dy dx” or, “the derivative.

3.3 Rules for Differentiation Colorado National Monument.

2.1 Day Differentiability.

3.2 Derivatives Great Sand Dunes National Monument, Colorado Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

The Derivative Obj: Students will be able to notate and evaluate derivatives.

3.1 Derivatives of a Function, p. 98 AP Calculus AB/BC.

Slide 3- 1 What you’ll learn about Definition of a Derivative Notation Relationship between the Graphs of f and f ' Graphing the Derivative from Data One-sided.

2.8 The Derivative As A Function. The Derivative if the limit exists. If f ’( a ) exists, we say f is differentiable at a. For y = f (x), we define the.

Derivatives Great Sand Dunes National Monument, Colorado Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

DO NOW:  Find the equation of the line tangent to the curve f(x) = 3x 2 + 4x at x = -2.

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.

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.

The Second Derivative Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon Siena College 2008 Photo by Vickie Kelly, 2003 Arches.

3.3 Differentiation Rules Colorado National Monument Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

2.2 Derivatives Great Sand Dunes National Monument, Colorado Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

Copyright © Cengage Learning. All rights reserved. Differentiation.

Unit 2 Lesson #1 Derivatives 1 Interpretations of the Derivative 1. As the slope of a tangent line to a curve. 2. As a rate of change. The (instantaneous)

Mean Value Theorem.

MTH1170 Higher Order Derivatives

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

Aim: How do we determine if a function is differential at a point?

The Derivative as a Function

3.2 Differentiability.

2.2 Rules for Differentiation

3.2: Rules for Differentiation

Derivatives Sec. 3.1.

3.1 Derivatives Part 2.

Differentiation Rules

3.2: Differentiability.

2.1 The Derivative & the Tangent Line Problem

The Derivative as a Function

Section 3.2 Differentiability.

Derivatives: definition and derivatives of various functions

3.1 Derivatives of a Function

Sec 2.8: The Derivative as a Function

3.2. Definition of Derivative.

2.4 The Derivative.

The Derivative as a Function

Presentation transcript:

2.2 The derivative as a function In Section 2.1 we considered the derivative at a fixed number a. Now let number a vary. If we replace number a by a variable x, then the derivative can be interpreted as a function of x : Alternative notations for the derivative: D and d / dx are called differentiation operators. dy / dx should not be regarded as a ratio.

The derivative is the slope of the original function.

Differentiable functions A function f is differentiable at a if f ′(a) exists. It is differentiable on an open interval (a,b) [ or (a,) or (- , a) or (- , ) ] if it is differentiable at every number in the interval. Theorem: If f is differentiable at a, then f is continuous at a. Note: The converse is false: there are functions that are continuous but not differentiable. Example: f(x) = | x |

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

Higher Order Derivatives: is the first derivative of y with respect to x. is the second derivative. (y double prime) is the third derivative. We will learn later what these higher order derivatives are used for. is the fourth derivative.