Chapter 4 The Exponential and Natural Logarithm Functions.

Slides:

Advertisements

Similar presentations

3.9 Derivatives of Exponential and Logarithmic Functions

Advertisements

Exponential & Logarithmic Equations

§ 4.4 The Natural Logarithm Function.

Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 55 Chapter 4 The Exponential and Natural Logarithm Functions.

Unit 6. For x  0 and 0  a  1, y = log a x if and only if x = a y. The function given by f (x) = log a x is called the logarithmic function with base.

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

DIFFERENTIATION & INTEGRATION CHAPTER 4.  Differentiation is the process of finding the derivative of a function.  Derivative of INTRODUCTION TO DIFFERENTIATION.

Logarithmic Equations Unknown Exponents Unknown Number Solving Logarithmic Equations Natural Logarithms.

Exponential and Logarithmic Functions 5 Exponential Functions Logarithmic Functions Differentiation of Exponential Functions Differentiation of Logarithmic.

Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 55 § 4.5 The Derivative of ln x.

Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 55 § 4.2 The Exponential Function e x.

Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 55 § 4.6 Properties of the Natural Logarithm Function.

Calculus Chapter 5 Day 1 1. The Natural Logarithmic Function and Differentiation The Natural Logarithmic Function- The number e- The Derivative of the.

Aim: Differentiating Natural Log Function Course: Calculus Do Now: Aim: How do we differentiate the natural logarithmic function? Power Rule.

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e– Slide 1 of 55 Chapter 4 The Exponential and Natural Logarithm.

3.9 Derivatives of Exponential & Logarithmic Functions

MAC 1105 Section 4.3 Logarithmic Functions. The Inverse of a Exponential Function 

The Derivative of a Logarithm. If f(x) = log a x, then Notice if a = e, then.

7.2The Natural Logarithmic and Exponential Function Math 6B Calculus II.

3.6 Derivatives of Logarithmic Functions 1Section 3.6 Derivatives of Log Functions.

3.9: Derivatives of Exponential and Log Functions Objective: To find and apply the derivatives of exponential and logarithmic functions.

Derivatives of Logarithmic Functions

Example 6 Solution of Exponential Equations Chapter 5.3 Solve the following exponential equations: a. b.  2009 PBLPathways.

Logarithmic and Exponential Equations

EQ: How do you use the properties of exponents and logarithms to solve equations?

Warm-up Solve: log3(x+3) + log32 = 2 log32(x+3) = 2 log3 2x + 6 = 2

Copyright © 2010 Pearson Education, Inc. All rights reserved Sec

Solving Exponential and Logarithmic Equations Section 6.6 beginning on page 334.

Unit 5: Modeling with Exponential & Logarithmic Functions Ms. C. Taylor.

Transcendental Functions Chapter 6. For x  0 and 0  a  1, y = log a x if and only if x = a y. The function given by f (x) = log a x is called the logarithmic.

Differentiating exponential functions.

Chapter 4 Techniques of Differentiation Sections 4.1, 4.2, and 4.3.

Derivatives of exponential and logarithmic functions

Session 6 : 9/221 Exponential and Logarithmic Functions.

Copyright © Cengage Learning. All rights reserved. Logarithmic, Exponential, and Other Transcendental Functions.

Derivatives of Exponential and Logarithmic Functions

Do Now (7.4 Practice): Graph. Determine domain and range.

Logarithms 1 Converting from Logarithmic Form to Exponential Form and Back 2 Solving Logarithmic Equations & Inequalities 3 Practice Problems.

NATURAL LOGARITHMS. The Constant: e e is a constant very similar to π. Π = … e = … Because it is a fixed number we can find e 2.

Logarithmic, Exponential, and Other Transcendental Functions

Solving Exponential Equations. We can solve exponential equations using logarithms. By converting to a logarithm, we can move the variable from the exponent.

BELL RINGER Write, in paragraph form, everything you remember about logarithmic and exponential functions including how to convert, solve logarithmic equations,

Solving Logarithmic Equations

Zooming In. Objectives  To define the slope of a function at a point by zooming in on that point.  To see examples where the slope is not defined. 

Direct Variation Chapter 5 Section 2. Objective  Students will write and graph an equation of a direct variation.

Solving Equations Exponential Logarithmic Applications.

Section 6.2* The Natural Logarithmic Function. THE NATURAL LOGARITHMIC FUNCTION.

§ 4.2 The Exponential Function e x.

Chapter 4 Logarithm Functions

Derivatives of exponentials and Logarithms

7 INVERSE FUNCTIONS.

Solving Exponential and Logarithmic Equations

Logarithmic Functions and Their Graphs

Work on Exp/Logs The following slides will help you to review the topic of exponential and logarithmic functions from College Algebra. Also, I am introducing.

§ 4.5 The Derivative of ln x.

Chapter 5 Logarithmic Functions.

§ 4.4 The Natural Logarithm Function.

5.4 Logarithmic Functions and Models

Techniques of Differentiation

Warm Up Six Chapter 5.5 Derivatives of base not e 11/20/2018

Exponential & Logarithmic Equations

Exponential Functions

Solve for x: 1) xln2 = ln3 2) (x – 1)ln4 = 2

Inverse, Exponential and Logarithmic Functions

4.3 – Differentiation of Exponential and Logarithmic Functions

3.4 Exponential and Logarithmic Equations

Tutorial 5 Logarithm & Exponential Functions and Applications

Exponential & Logarithmic Equations

Exponential & Logarithmic Equations

Chapter 8 Section 6 Solving Exponential & Logarithmic Equations

Presentation transcript:

Chapter 4 The Exponential and Natural Logarithm Functions

§ 4.1 Exponential Functions

Exponential Function DefinitionExample Exponential Function: A function whose exponent is the independent variable

Properties of Exponential Functions

Graphs of Exponential Functions Notice that, no matter what b is (except 1), the graph of y = b x has a y-intercept of 1. Also, if 0 1, then the function is increasing.

Solving Exponential EquationsEXAMPLE Solve the following equation for x.

§ 4.2 The Exponential Function e x

The Number e DefinitionExample e: An irrational number, approximately equal to , such that the function f (x) = b x has a slope of 1, at x = 0, when b = e

The Derivatives of a x and e x (a x )’ = a x Lna Example

§ 4.3 Differentiation of Exponential Functions

Chain Rule for e g ( x )

EXAMPLE Differentiate.

§ 4.4 The Natural Logarithm Function

The Natural Logarithm of x DefinitionExample Natural logarithm of x: Given the graph of y = e x, the reflection of that graph about the line y = x, denoted y = ln x

Properties of the Natural Logarithm

§ 4.5 The Derivative of ln x

Derivative Rules for Natural Logarithms

Differentiating Logarithmic ExpressionsEXAMPLE Differentiate.

Differentiating Logarithmic ExpressionsEXAMPLE The function has a relative extreme point for x > 0. Find the coordinates of the point. Is it a relative maximum point?