Warm-Up: October 2, 2017 Find the slope of at.

Slides:

Advertisements

Similar presentations

3.1 Derivatives. Derivative A derivative of a function is the instantaneous rate of change of the function at any point in its domain. We say this is.

Advertisements

The Derivative and the Tangent Line Problem. Local Linearity.

The derivative and the tangent line problem (2.1) October 8th, 2012.

Determining secant slope and tangent slope. Slope is the rate of change or Secant slope the slope between two points tangent slope the slope at one point.

THE DERIVATIVE AND THE TANGENT LINE PROBLEM

1.4 – Differentiation Using Limits of Difference Quotients

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))

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.

SECTION 3.1 The Derivative and the Tangent Line Problem.

D EFINITION OF THE D ERIVATIVE Derivatives Review- 1.

3.1 –Tangents and the Derivative at a Point

M 112 Short Course in Calculus Chapter 2 – Rate of Change: The Derivative Sections 2.2 – The Derivative Function V. J. Motto.

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.

The Tangent Line Problem “And I dare say that this is not only the most useful and general problem in geometry that I know, but even that I ever desire.

Lesson 2-4 Tangent, Velocity and Rates of Change Revisited.

Assignment 4 Section 3.1 The Derivative and Tangent Line Problem.

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

Chapter 3.1 Tangents and the Derivative at a Point.

2.1 Day Differentiability.

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

2.1 The Derivative and The Tangent Line Problem

Chapter 3.2 The Derivative as a Function. If f ’ exists at a particular x then f is differentiable (has a derivative) at x Differentiation is the process.

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

2.1 The Derivative and the Tangent Line Problem.

Section 2.1 – Average and Instantaneous Velocity.

Calculus Section 3.1 Calculate the derivative of a function using the limit definition Recall: The slope of a line is given by the formula m = y 2 – y.

Warm Up a) What is the average rate of change from x = -2 to x = 2? b) What is the average rate of change over the interval [1, 4]? c) Approximate y’(2).

Implicit Differentiation

The Derivative and the Tangent Line Problem

Table of Contents 21. Section 4.3 Mean Value Theorem.

Wednesday, october 25th “Consider the postage stamp: its usefulness consists in the ability to stick to one thing till it gets there.”

Copyright © Cengage Learning. All rights reserved.

The Derivative and the Tangent Line Problem (2.1)

Derivative of a Function

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

Copyright © Cengage Learning. All rights reserved.

Arrival Activity: Put the answers to the following question in your notes. Use complete sentences so that you know what your answers mean when you review.

Sec 2.7: Derivative and Rates of Change

AP Calculus Chapter 2, Section 1

The Derivative and the Tangent Line Problems

Copyright © Cengage Learning. All rights reserved.

Applications of Derivatives

THE DERIVATIVE AND THE TANGENT LINE PROBLEM

The derivative and the tangent line problem (2.1)

Derivative of a Function

Derivative of a Function

Tangent Lines and Derivatives

Derivatives Sec. 3.1.

Tangent line to a curve Definition: line that passes through a given point and has a slope that is the same as the.

Derivative of a Function

2.1 The Derivative & the Tangent Line Problem

Derivative of a Function AP Calculus Honors Ms. Olifer

Tangent line to a curve Definition: line that passes through a given point and has a slope that is the same as the.

Tuesday, October 24 Lesson 3.1 Score 2.8

Copyright © Cengage Learning. All rights reserved.

Definition of a Derivative

2.7/2.8 Tangent Lines & Derivatives

Derivatives: definition and derivatives of various functions

Chapter 3: Differentiation Section 3.1: Definition of the Derivative

Section 2.1 – Average and Instantaneous Velocity

30 – Instantaneous Rate of Change No Calculator

Homework, Page Let f (x) = 3x2. Show that f (2+h) =3h2 + 12h Then show that and compute f ′(2) by taking the limit as h → 0. Rogawski Calculus.

3.1 Derivatives of a Function

Sec 2.8: The Derivative as a Function

Derivative of a Function

The Derivative and the Tangent Line Problem (2.1)

2.4 The Derivative.

Miss Battaglia AB Calculus

Presentation transcript:

Warm-Up: October 2, 2017 Find the slope of at

Derivative of a Function Section 3.1

Chapter 3 Overview We learned how to find the slope of a tangent to a curve as the limit of the slopes of secant lines. This gives us the instantaneous rate of change at a point. The study of rates of change of functions is called differential calculus.

Derivative The definition of derivative is the same as the function for slope. Provided the limit exists!!

Derivative Notation

Derivative Vocabulary If f’(x) exists, we say that f is differentiable at x. This is the same as saying “f has a derivative at x” A function that is differentiable at every point of its domain is a differentiable function. A function must be continuous in order to be differentiable.

Example Differentiate 𝑓 𝑥 = 𝑥 3

Definition (Alternate) The derivative of the function at a point a, is the limit 𝑓 ′ 𝑎 = lim 𝑥→𝑎 𝑓 𝑥 −𝑓 𝑎 𝑥−𝑎 Provided the limit exists.

Example Differentiate 𝑥 .

Warm-Up: October 3, 2017 Find the average rate of change of y=3x2 over the interval [-1, 4].

Graphing f’ from f NOTE: The domain of f’ may be smaller than the domain of f. Calculate/estimate the slope for various values of x. Plot the (x, y’) points Connect the points with a smooth curve for x-values where f is smooth and continuous.

Example 1

Graphing f from f’ You must be given at least one point. Start your graph at the given point.

Example 2 Sketch the graph that has the following properties 𝑓 0 =0 The graph of 𝑓′ is as shown in the figure. 𝑓 is continuous for all x.

One-Sided Derivatives Right-Hand Derivative Left-Hand Derivative Derivatives at endpoints are one-sided limits.

Example 3 𝑦= 𝑥 2 +1, 𝑥<2 −𝑥+3, 𝑥≥2

Derivatives from Data Open you textbook to page 103 Look at #20