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.

Slides:

Advertisements

Similar presentations

Sec 3.1: Tangents and the Derivative at a Point

Advertisements

Aim: What do slope, tangent and the derivative have to do with each other? Do Now: What is the equation of the line tangent to the circle at point (7,

Equations of Tangent Lines

Limits and an Introduction to Calculus

Calculus 2413 Ch 3 Section 1 Slope, Tangent Lines, and Derivatives.

Sec. 2.1: The Derivative and the Tangent Line

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.

THE DERIVATIVE AND THE TANGENT LINE PROBLEM

The Derivative. Objectives Students will be able to Use the “Newton’s Quotient and limits” process to calculate the derivative of a function. Determine.

1.4 – Differentiation Using Limits of Difference Quotients

1.2 – Algebraic Limits and Continuity. Find the following limits using the Limit Properties: 1.2 – Algebraic Limits and Continuity.

Limit Definition of the Derivative. Objective  To use the limit definition to find the derivative of a function.  TS: Devoloping a capacity for working.

The Secant-Line Calculation of the Derivative

 Total Change/Length of the interval  Δy/Δx  Average rate of change is the slope of the secant line.

1.6 – Tangent Lines and Slopes Slope of Secant Line Slope of Tangent Line Equation of Tangent Line Equation of Normal Line Slope of Tangent =

Copyright © Cengage Learning. All rights reserved. 12 Limits and an Introduction to Calculus.

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.

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.

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.

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.

Powerpoint Templates Page 1 Powerpoint Templates Review Calculus.

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

Derivatives Using the Limit Definition

§3.2 – The Derivative Function October 2, 2015.

Chain Rule 3.5. Consider: These can all be thought of as composite functions F(g(x)) Derivatives of Composite functions are found using the chain rule.

Warm Up Determine a) ∞ b) 0 c) ½ d) 3/10 e) – Rates of Change and Tangent Lines.

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

2.1 The Derivative and the Tangent Line Problem.

3.3 Product Rule Tues Oct 27 Do Now Evaluate each 1) 2)

Derivative Shortcuts -Power Rule -Product Rule -Quotient Rule -Chain Rule.

The Tangent Line The Secant Line. The Tangent Line The Secant Line.

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.

DIFFERENCE QUOTIENT, DERIVATIVE RULES, AND TANGENT LINES RIZZI – CALC BC.

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.

Calculating Derivatives From first principles!. 2.1 The Derivative as a Limit See the gsp demo demodemo Let P be any point on the graph of the function.

§ 4.2 The Exponential Function e x.

Mean Value Theorem.

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).

2-4 Rates of change & tangent lines

MTH1150 Tangents and Their Slopes

Ch. 11 – Limits and an Introduction to Calculus

Section 11.3A Introduction to Derivatives

2.1 Tangent Line Problem.

The Tangent Line Problem

The Derivative and the Tangent Line Problem (2.1)

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

Techniques of Differentiation

Slope at Point of Tangency

Sec 2.7: Derivative and Rates of Change

Definition of the Derivative

Derivatives by Definition

THE DERIVATIVE AND THE TANGENT LINE PROBLEM

2.4 The Chain Rule.

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

F’ means derivative or in this case slope.

2.1 The Derivative and the Slope of a Graph

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

Definition of a Derivative

Tangent Line Recall from geometry

Derivatives: definition and derivatives of various functions

Chapter 2 Differentiation.

MATH 1314 Lesson 6: Derivatives.

The Derivative and the Tangent Line Problem (2.1)

Sec 2.7: Derivative and Rates of Change

2-1: The Derivative Objectives: Explore the tangent line problem

Presentation transcript:

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 f(x + h) – f(x)

y x (x, f(x)) (x + h, f(x + h)) h f(x + h) – f(x) As h approaches 0, the secant line approaches the tangent line (x, f(x))

Definition of the Slope of a Graph The slope m of the graph of f at the point (x, f(x)) is equal to the slope of its tangent line at (x, f(x)), and is given by provided this limit exists Use the limit process, we can find the exact slope of the tangent line This quotient is called different of quotient

Find the slope of the graph f(x) = x 2 at the point (–2, 4) Begin by find an expression that represents the slope of a secant line at the point (–2, 4). Set up different quotient Use f(x) = x 2 Expand terms simplify Factor and cancel simplify The graph has a slope of – 4 at the point (–2, 4).

Find the formula for the slope of the graph of f(x) = x What is the slope at the points (-1,2) and (2,5)? Use the formula m = 2x to find the slope at point (-1,2) the slope is m = 2(-1) = – 2. at the point (2,5), the slope is m = 2(2) = 4

Definition of the Derivative The derivative of f at x is given by Remember that the derivative f’(x) is a formula for the slope of the tangent line to the graph of f at the point (x, f(x)).

Find the derivative of f(x) = 3x 2 – 2x f’(x) = 6x – 2 Find f’(x) for. Then find the slope of the graph of f at the points (1,1) and (4,2). At the point (1,1), the slope is f’(1) = 1/2. At the point (4,2), the slope is f’(4) = 1/4