Definition of a Derivative

Slides:

Advertisements

Similar presentations

2.7 Tangents, Velocities, & Rates of Change

Advertisements

Definition of the Derivative Using Average Rate () a a+h f(a) Slope of the line = h f(a+h) Average Rate of Change = f(a+h) – f(a) h f(a+h) – f(a) h.

Sec 3.1: Tangents and the Derivative at a Point

2.1 Derivatives and Rates of Change. The slope of a line is given by: The slope of the tangent to f(x)=x 2 at (1,1) can be approximated by the slope of.

Slope and Equation of a line How to find the slop of a line? (x 1, y 1 ) (x 2, y 2 ) How to find the equation of a line? Sec 2.1: Rates of Change and.

Equations of Tangent Lines

The Derivative and the Tangent Line Problem. Local Linearity.

1 2.7 – Tangents, Velocity, and Other Rates of Change.

Find the slope of the tangent line to the graph of f at the point ( - 1, 10 ). f ( x ) = 6 - 4x

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 Chapter 3:. What is a derivative? A mathematical tool for studying the rate at which one quantity changes relative to another.

Derivatives - Equation of the Tangent Line Now that we can find the slope of the tangent line of a function at a given point, we need to find the equation.

Business Calculus Rates of Change Types of Change  Average rate of change: the average rate of change of y with respect to x is a ratio of.

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))

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.

The Tangent and Velocity Problem © John Seims (2008)

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

Lesson 2-8 Introduction to Derivatives. 2-7 Review Problem For a particle whose position at time t is f(t) = 6t 2 - 4t +1 ft.: b. Find the instantaneous.

3.1 Definition of the Derivative & Graphing the Derivative

2.4 Rates of Change and Tangent Lines Calculus. Finding average rate of change.

Tangents. The slope of the secant line is given by The tangent line’s slope at point a is given by ax.

Chapter 3.1 Tangents and the Derivative at a Point.

Finding the Derivative The Limit Process. What is the derivative of something? The derivative of a function f(x) is, mathematically speaking, the slope.

Aim: How do we find the derivative by limit process? Do Now: Find the slope of the secant line in terms of x and h. y x (x, f(x)) (x + h, f(x + h)) h.

§3.2 – The Derivative Function October 2, 2015.

AIM : How do we find tangent lines to a graph? Do Now: Find the slope of the line shown below that goes through points (0,6) and (3,0). HW2.1: p66-67 #9,10,19,22,23.

2.1 The Derivative and The Tangent Line Problem Slope of a Tangent Line.

UNIT 1B LESSON 7 USING LIMITS TO FIND TANGENTS 1.

Objectives Determine tangent lines to functions for a given point Compute the slope of curves Compute Instantaneous rate of change.

Business Calculus Derivative Definition. 1.4 The Derivative The mathematical name of the formula is the derivative of f with respect to x. This is the.

§3.1 – Tangent Lines, Velocity, Rate of Change October 1, 2015.

Section 1.4 The Tangent and Velocity Problems. WHAT IS A TANGENT LINE TO THE GRAPH OF A FUNCTION? A line l is said to be a tangent to a curve at a point.

Section 2.4 Rates of Change and Tangent Lines Calculus.

1. Definition of Derivative

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.

EXAMPLE FORMULA DEFINITION 1.

From previous sections

Table of Contents 9. Section 3.1 Definition of Derivative.

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).

2.1 Tangents & Velocities.

MTH1150 Tangents and Their Slopes

A Preview of Calculus Lesson 1.1.

The Tangent Line Problem

2.4 Rates of Change and Tangent Lines Day 1

2.1A Tangent Lines & Derivatives

2.7 Derivatives and Rates of Change

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

Slope at Point of Tangency

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

The Derivative and the Tangent Line Problems

Linear Models and Rates of Change

Applications of Derivatives

Derivatives by Definition

THE DERIVATIVE AND THE TANGENT LINE PROBLEM

Tangent Lines and Derivatives

2.1 The Derivative and the Slope of a Graph

Packet #4 Definition of the Derivative

Chapter 3: Differentiation Section 3.1: Definition of the Derivative

f(a+h) Slope of the line = Average Rate of Change = f(a+h) – f(a) h

2.2: Formal Definition of the Derivative

Rates of Change and Tangent Lines

30 – Instantaneous Rate of Change No Calculator

MATH 1314 Lesson 6: Derivatives.

Packet #32 Limits and More: A Calculus Preview!

Sec 2.7: Derivative and Rates of Change

Presentation transcript:

Definition of a Derivative Keeper 13 Honors Calculus

Understanding Functions Label the following on the picture below: 𝑓 𝑥 , 𝑓 𝑥+ℎ , ℎ, 𝑓 𝑥+ℎ −𝑓(𝑥) and a segment whose slope represents 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ and lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ

Slope of the Secant Line 𝑚 𝑠𝑒𝑐 = 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ This formula represents the AVERAGE rate of change

Slope of the Tangent Line 𝑚 𝑡𝑎𝑛 = lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ This formula represents the INSTANTANEOUS rate of change

Compare and Contrast the Average Rate of Change with the instantaneous rate of Change (derivative). Similarities Differences

Derivative Slope of a tangent line Rate of change at 1 point Instantaneous rate of change Denoted by 𝑦 ′ , f ′ x , or 𝑑𝑦 𝑑𝑥

Definition of a Derivative 𝑑𝑦 𝑑𝑥 = lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ

Find the Derivative using the definition 1. 𝑓 𝑥 =−3𝑥+2

Find the Derivative 2. 𝑓 𝑥 = 3𝑥 2 −2𝑥+4

Find the Derivative 3. 𝑓 𝑥 = 1 𝑥+1

Find the Derivative 4. 𝑓 𝑥 = 𝑥−2