Differentiation of Logarithmic Functions
|
|
|
- Basil Bradley
- 5 years ago
- Views:
Transcription
1 Differentiation of Logarithmic Functions The rule for finding the derivative of a logarithmic function is given as: If y log a then dy or y. d a ( ln This rule can be proven by rewriting the logarithmic function in eponential form and then using the eponential derivative rule covered in the last section. y log a Begin with logarithmic function y a ln a a y ln a Convert into eponential form Differentiate both sides of the equation Substitute for the eponential function a y Solve for y by dividing each side by (ln a a ( ln As with the eponential rules, the derivative of a logarithmic function can be simplified if the base of the logarithm is e. y log ln Logarithmic function with base e e Apply the logarithm derivative rule above e ( ln Use properties of logarithms to simplify ( Gerald Manahan SLAC, San Antonio College, 008
2 Eample : Find the derivative of log6 f log 6 ( f ( ln 6 f Eample : Find the derivative of f ( log( Right now the only derivative rule we have for logarithms is for log. So in order to take the derivative of this function we must first use the properties of logarithms to rewrite the function. First use the product property of ( y log( log( + log( f log log + log y r Net use the eponent property of log( r log( log( + log f log( log + Now we can find the derivative. log( + log f f ( 0+ ln0 ( ln0 Even though we were able to find the derivative of this function, it would be easier to find the derivative if we had a rule that dealt with the situation where a function is equal to the log of another function. In order to do this, we would have to combine the chain rule with the logarithm rule. Gerald Manahan SLAC, San Antonio College, 008
3 Lets look at an eample similar to the first function that we looked at when proving the logarithm derivative rule. [ ] g f g log b Composite logarithmic function b [ ] f g g Convert to eponential form [ ] [ ] f g ln b b f g( g Differentiate both sides of the equation ( ln b g( f [ g( ] g ( Substitute g( for the eponential function [ g( ] f g ( ( ln b g( Divide each side by (ln b g( Derivative of g( log b If y g( log b then ( g ln b g If y ln g( then g ( g( Now lets go back to eample and find the derivative again but this time using the above rule. Gerald Manahan SLAC, San Antonio College, 008
4 Eample : Find the derivative of f ( log( Note: Remember if a base is not shown it is understood to be 0. log( f D ( f ( ln0 8 ln0 ln0 ( ln0 As you can see we get the same derivative as before but in an shorter and easier process. Eample : Find the derivative of y ln ( t + 5t In this problem we can simplify the derivative process by first applying the properties of logarithms to move the eponent out in front of the function as a coefficient. ( t t y ln t + 5t ln + 5 Now we can find the derivative y ln t + 5 t D ln ( t + 5t Gerald Manahan SLAC, San Antonio College, 008
5 Eample (Continued: ( + 5 ( t + t D t t 5 ( t + 5 ( t + t 5 Now lets look at a more comple function that will require the use of several derivative rules. Remember when finding the derivative of a comple function take it step by step. Don t try to do it all at once. Eample 5: Find the derivative of y ( + e ( + ln 5 Since the function is in the form of a fraction we must begin by applying the quotient rule. When you go to find the derivative of the numerator you are will have to use both the product and eponential rules. The derivative of the denominator will require the use of the logarithm rule. First, apply the quotient rule. y ( + e ( + ln 5 ( + D ( + e ( + e D ( + ln 5 ln 5 ln ( 5 + Gerald Manahan SLAC, San Antonio College, 008 5
6 Eample 5 (Continued: Net, apply the product and eponential rule to the derivative of the numerator. ( + D ( + e ( + e D ( + ln 5 ln 5 ln ( 5 + ( + ( + D e + e D ( + ( + e D ( + ln 5 ln 5 ln ( 5 + ( + ( + e D ( + e ( ( + e D ( + ln 5 ln 5 ln ( 5 + ln 5+ + e + e + e D ln 5+ ln ( 5 + Now, apply the logarithmic rule to find the derivative of the denominator. ( + ( + ( e + e ( + e D ( + ln 5 ln 5 ln ( 5 + ( 5+ ( 5 D ln ( 5+ ( ( e e + + ( + e + ln ( ln ( 5+ ( ( e e + + ( + e + ln ( 5 + ( 5 The only thing left to do now is to simplify the derivative using the properties of algebra. Gerald Manahan SLAC, San Antonio College, 008 6
7 Eample 5 (Continued: 5 ln ( 5+ ( ( e e + + ( + e + ln ( 5 + e + + ln 5+ + e ( + ln 5 ( + + ( + ( + ( + ( 5 + e 5 ln 5 5e ( 5+ ln( 5+ ( + + ( + ( + ( + e 5 ln 5 5 ( 5+ ln( 5+ 5 ( 5 Gerald Manahan SLAC, San Antonio College, 008 7
The Product and Quotient Rules
The Product and Quotient Rules In this section, you will learn how to find the derivative of a product of functions and the derivative of a quotient of functions. A function that is the product of functions
Higher-Order Derivatives
Higher-Order Derivatives Higher-order derivatives are simply the derivative of a derivative. You would use the same derivative rules that you learned for finding the first derivative of a function. The
Differentiation of Exponential Functions
Differentiation of Eponential Fnctions The net derivative rles that o will learn involve eponential fnctions. An eponential fnction is a fnction in the form of a constant raised to a variable power. The
DIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES 3.6 Derivatives of Logarithmic Functions In this section, we: use implicit differentiation to find the derivatives of the logarithmic functions and, in particular,
Page 1. These are all fairly simple functions in that wherever the variable appears it is by itself. What about functions like the following, ( ) ( )
Chain Rule Page We ve taken a lot of derivatives over the course of the last few sections. However, if you look back they have all been functions similar to the following kinds of functions. 0 w ( ( tan
Optimization II. Now lets look at a few examples of the applications of extrema.
Optimization II So far you have learned how to find the relative and absolute etrema of a function. This is an important concept because of how it can be applied to real life situations. In many situations
Curve Sketching. The process of curve sketching can be performed in the following steps:
Curve Sketching So ar you have learned how to ind st and nd derivatives o unctions and use these derivatives to determine where a unction is:. Increasing/decreasing. Relative extrema 3. Concavity 4. Points
5. Find the slope intercept equation of the line parallel to y = 3x + 1 through the point (4, 5).
Rewrite using rational eponents. 2 1. 2. 5 5. 8 4 4. 4 5. Find the slope intercept equation of the line parallel to y = + 1 through the point (4, 5). 6. Use the limit definition to find the derivative
3.7 Indeterminate Forms - l Hôpital s Rule
3.7. INDETERMINATE FORMS - L HÔPITAL S RULE 4 3.7 Indeterminate Forms - l Hôpital s Rule 3.7. Introduction An indeterminate form is a form for which the answer is not predictable. From the chapter on lits,
Exponential and Logarithmic Functions
Lesson 6 Eponential and Logarithmic Fu tions Lesson 6 Eponential and Logarithmic Functions Eponential functions are of the form y = a where a is a constant greater than zero and not equal to one and is
Adding and Subtracting Rational Expressions
Adding and Subtracting Rational Epressions As a review, adding and subtracting fractions requires the fractions to have the same denominator. If they already have the same denominator, combine the numerators
1 Rational Exponents and Radicals
Introductory Algebra Page 1 of 11 1 Rational Eponents and Radicals 1.1 Rules of Eponents The rules for eponents are the same as what you saw earlier. Memorize these rules if you haven t already done so.
Logarithmic differentiation
Roberto s Notes on Differential Calculus Chapter 5: Derivatives of transcendental functions Section Logarithmic differentiation What you need to know already: All basic differentiation rules, implicit
DIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES 3.2 The Product and Quotient Rules In this section, we will learn about: Formulas that enable us to differentiate new functions formed from old functions by
Example 1: What do you know about the graph of the function
Section 1.5 Analyzing of Functions In this section, we ll look briefly at four types of functions: polynomial functions, rational functions, eponential functions and logarithmic functions. Eample 1: What
Section 4.5 Graphs of Logarithmic Functions
6 Chapter 4 Section 4. Graphs of Logarithmic Functions Recall that the eponential function f ( ) would produce this table of values -3 - -1 0 1 3 f() 1/8 ¼ ½ 1 4 8 Since the arithmic function is an inverse
Summary sheet: Exponentials and logarithms
F Know and use the function a and its graph, where a is positive Know and use the function e and its graph F2 Know that the gradient of e k is equal to ke k and hence understand why the eponential model
Pre-Algebra 8 Notes Exponents and Scientific Notation
Pre-Algebra 8 Notes Eponents and Scientific Notation Rules of Eponents CCSS 8.EE.A.: Know and apply the properties of integer eponents to generate equivalent numerical epressions. Review with students
AP Calculus AB Summer Assignment
AP Calculus AB Summer Assignment Name: When you come back to school, it is my epectation that you will have this packet completed. You will be way behind at the beginning of the year if you haven t attempted
Differentials. In previous sections when you were taking the derivative of a function it was shown that the derivative notations f ( x)
Differentials This section will show how differentials can be used for linear approximation, marginal analysis, and error estimation. In previous math courses you learned that the slope of a line is found
C6-2 Differentiation 3
chain, product and quotient rules C6- Differentiation Pre-requisites: C6- Estimate Time: 8 hours Summary Learn Solve Revise Answers Summary The chain rule is used to differentiate a function of a function.
AP Calculus AB Summer Assignment
AP Calculus AB Summer Assignment Name: When you come back to school, you will be epected to have attempted every problem. These skills are all different tools that you will pull out of your toolbo this
Math RE - Calculus I Exponential & Logarithmic Functions Page 1 of 9. y = f(x) = 2 x. y = f(x)
Math 20-0-RE - Calculus I Eponential & Logarithmic Functions Page of 9 Eponential Function The general form of the eponential function equation is = f) = a where a is a real number called the base of the
Algebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation?
Algebra Concepts Equation Solving Flow Chart Page of 6 How Do I Solve This Equation? First, simplify both sides of the equation as much as possible by: combining like terms, removing parentheses using
Calculus 1 (AP, Honors, Academic) Summer Assignment 2018
Calculus (AP, Honors, Academic) Summer Assignment 08 The summer assignments for Calculus will reinforce some necessary Algebra and Precalculus skills. In order to be successful in Calculus, you must have
Outline. 1 Integration by Substitution: The Technique. 2 Integration by Substitution: Worked Examples. 3 Integration by Parts: The Technique
MS2: IT Mathematics Integration Two Techniques of Integration John Carroll School of Mathematical Sciences Dublin City University Integration by Substitution: The Technique Integration by Substitution:
Differentiation. The main problem of differential calculus deals with finding the slope of the tangent line at a point on a curve.
Dierentiation The main problem o dierential calculus deals with inding the slope o the tangent line at a point on a curve. deinition() : The slope o a curve at a point p is the slope, i it eists, o the
Intermediate Algebra Section 9.3 Logarithmic Functions
Intermediate Algebra Section 9.3 Logarithmic Functions We have studied inverse functions, learning when they eist and how to find them. If we look at the graph of the eponential function, f ( ) = a, where
QUADRATIC EQUATIONS. + 6 = 0 This is a quadratic equation written in standard form. x x = 0 (standard form with c=0). 2 = 9
QUADRATIC EQUATIONS A quadratic equation is always written in the form of: a + b + c = where a The form a + b + c = is called the standard form of a quadratic equation. Eamples: 5 + 6 = This is a quadratic
A summary of factoring methods
Roberto s Notes on Prerequisites for Calculus Chapter 1: Algebra Section 1 A summary of factoring methods What you need to know already: Basic algebra notation and facts. What you can learn here: What
Define a rational expression: a quotient of two polynomials. ..( 3 10) (3 2) Rational expressions have the same properties as rational numbers:
1 UNIT 7 RATIONAL EXPRESSIONS & EQUATIONS Simplifying Rational Epressions Define a rational epression: a quotient of two polynomials. A rational epression always indicates division EX: 10 means..( 10)
Complex fraction: - a fraction which has rational expressions in the numerator and/or denominator
Comple fraction: - a fraction which has rational epressions in the numerator and/or denominator o 2 2 4 y 2 + y 2 y 2 2 Steps for Simplifying Comple Fractions. simplify the numerator and/or the denominator
EXPONENT REVIEW!!! Concept Byte (Review): Properties of Exponents. Property of Exponents: Product of Powers. x m x n = x m + n
Algebra B: Chapter 6 Notes 1 EXPONENT REVIEW!!! Concept Byte (Review): Properties of Eponents Recall from Algebra 1, the Properties (Rules) of Eponents. Property of Eponents: Product of Powers m n = m
C. Finding roots of trinomials: 1st Example: x 2 5x = 14 x 2 5x 14 = 0 (x 7)(x + 2) = 0 Answer: x = 7 or x = -2
AP Calculus Students: Welcome to AP Calculus. Class begins in approimately - months. In this packet, you will find numerous topics that were covered in your Algebra and Pre-Calculus courses. These are
Calculus I Practice Test Problems for Chapter 2 Page 1 of 7
Calculus I Practice Test Problems for Chapter Page of 7 This is a set of practice test problems for Chapter This is in no way an inclusive set of problems there can be other types of problems on the actual
Math Calculus f. Business and Management - Worksheet 12. Solutions for Worksheet 12 - Limits as x approaches infinity
Math 0 - Calculus f. Business and Management - Worksheet 1 Solutions for Worksheet 1 - Limits as approaches infinity Simple Limits Eercise 1: Compute the following its: 1a : + 4 1b : 5 + 8 1c : 5 + 8 Solution
Day 3: Section P-6 Rational Expressions; Section P-7 Equations. Rational Expressions
1 Day : Section P-6 Rational Epressions; Section P-7 Equations Rational Epressions A rational epression (Fractions) is the quotient of two polynomials. The set of real numbers for which an algebraic epression
±. Then. . x. lim g( x) = lim. cos x 1 sin x. and (ii) lim
MATH 36 L'H ˆ o pital s Rule Si of the indeterminate forms of its may be algebraically determined using L H ˆ o pital's Rule. This rule is only stated for the / and ± /± indeterminate forms, but four other
2017 AP Calculus AB Summer Assignment
07 AP Calculus AB Summer Assignment Mrs. Peck ( kapeck@spotsylvania.k.va.us) This assignment is designed to help prepare you to start Calculus on day and be successful. I recommend that you take off the
College Algebra. Chapter 5 Review Created by: Lauren Atkinson. Math Coordinator, Mary Stangler Center for Academic Success
College Algebra Chapter 5 Review Created by: Lauren Atkinson Math Coordinator, Mary Stangler Center for Academic Success Note: This review is composed of questions from the chapter review at the end of
Algebra/Trigonometry Review Notes
Algebra/Trigonometry Review Notes MAC 41 Calculus for Life Sciences Instructor: Brooke Quinlan Hillsborough Community College ALGEBRA REVIEW FOR CALCULUS 1 TOPIC 1: POLYNOMIAL BASICS, POLYNOMIAL END BEHAVIOR,
We all learn new things in different ways. In. Properties of Logarithms. Group Exercise. Critical Thinking Exercises
Section 4.3 Properties of Logarithms 437 34. Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowly to the one
is on the graph of y = f 1 (x).
Objective 2 Inverse Functions Illustrate the idea of inverse functions. f() = 2 + f() = Two one-to-one functions are inverses of each other if (f g)() = of g, and (g f)() = for all in the domain of f.
AP Calculus BC Summer Assignment 2018
AP Calculus BC Summer Assignment 018 Name: When you come back to school, I will epect you to have attempted every problem. These skills are all different tools that we will pull out of our toolbo at different
Fox Lane High School Department of Mathematics
Fo Lane High School Department of Mathematics June 08 Hello Future AP Calculus AB Student! This is the summer assignment for all students taking AP Calculus AB net school year. It contains a set of problems
8.3 Zero, Negative, and Fractional Exponents
www.ck2.org Chapter 8. Eponents and Polynomials 8.3 Zero, Negative, and Fractional Eponents Learning Objectives Simplify epressions with zero eponents. Simplify epressions with negative eponents. Simplify
Algebra Review C H A P T E R. To solve an algebraic equation with one variable, find the value of the unknown variable.
C H A P T E R 6 Algebra Review This chapter reviews key skills and concepts of algebra that you need to know for the SAT. Throughout the chapter are sample questions in the style of SAT questions. Each
UNIT 4A MATHEMATICAL MODELING OF INVERSE, LOGARITHMIC, AND TRIGONOMETRIC FUNCTIONS Lesson 2: Modeling Logarithmic Functions
Lesson : Modeling Logarithmic Functions Lesson A..1: Logarithmic Functions as Inverses Utah Core State Standards F BF. Warm-Up A..1 Debrief In the metric system, sound intensity is measured in watts per
1 Basic algebra. A few formulas: (a+b) 2 = a 2 +2ab+b 2 (a+b) 3 = a 3 +3a 2 b+3ab 2 +b 3.
Basic algebra A few formulas: (a+b) 2 = a 2 +2ab+b 2 (a+b) 3 = a 3 +3a 2 b+3ab 2 +b 3. The coefficients follow from PASCAL S TRIANGLE, the epansion is called BINOMIAL. Also: a 2 b 2 = (a b)(a+b) a 3 b
Algebra Review. Unit 7 Polynomials
Algebra Review Below is a list of topics and practice problems you have covered so far this semester. You do not need to work out every question on the review. Skip around and work the types of questions
Section 4.3: Quadratic Formula
Objective: Solve quadratic equations using the quadratic formula. In this section we will develop a formula to solve any quadratic equation ab c 0 where a b and c are real numbers and a 0. Solve for this
9.5 HONORS Determine Odd and Even Functions Graphically and Algebraically
9.5 HONORS Determine Odd and Even Functions Graphically and Algebraically Use this blank page to compile the most important things you want to remember for cycle 9.5: 181 Even and Odd Functions Even Functions:
Introduction. So, why did I even bother to write this?
Introduction This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The review contains the occasional
In economics, the amount of a good x demanded is a function of the price of that good. In other words,
I. UNIVARIATE CALCULUS Given two sets X and Y, a function is a rule that associates each member of X with eactly one member of Y. That is, some goes in, and some y comes out. These notations are used to
Ph.D. Katarína Bellová Page 1 Mathematics 1 (10-PHY-BIPMA1) RETAKE EXAM, 4 April 2018, 10:00 12:00
Ph.D. Katarína Bellová Page Mathematics (0-PHY-BIPMA) RETAKE EXAM, 4 April 08, 0:00 :00 Problem [4 points]: Prove that for any positive integer n, the following equality holds: + 4 + 7 + + (3n ) = n(3n
Exponential, Logarithmic and Inverse Functions
Chapter Review Sec.1 and. Eponential, Logarithmic and Inverse Functions I. Review o Inverrse I Functti ions A. Identiying One-to-One Functions is one-to-one i every element in the range corresponds to
Solutions to the Math 1051 Sample Final Exam (from Spring 2003) Page 1
Solutions to the Math 0 Sample Final Eam (from Spring 00) Page Part : Multiple Choice Questions. Here ou work out the problems and then select the answer that matches our answer. No partial credit is given
Math Review and Lessons in Calculus
Math Review and Lessons in Calculus Agenda Rules o Eponents Functions Inverses Limits Calculus Rules o Eponents 0 Zero Eponent Rule a * b ab Product Rule * 3 5 a / b a-b Quotient Rule 5 / 3 -a / a Negative
MATH 1220 Midterm 1 Thurs., Sept. 20, 2007
MATH 220 Midterm Thurs., Sept. 20, 2007 Write your name and ID number at the top of this page. Show all your work. You may refer to one double-sided sheet of notes during the eam and nothing else. Calculators
Basic ALGEBRA 2 SUMMER PACKET
Name Basic ALGEBRA SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Algebra II. We will use these concepts on a regular basis throughout
APPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.4 Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at
Math 112 Section 10 Lecture notes, 1/7/04
Math 11 Section 10 Lecture notes, 1/7/04 Section 7. Integration by parts To integrate the product of two functions, integration by parts is used when simpler methods such as substitution or simplifying
Avon High School Name AP Calculus AB Summer Review Packet Score Period
Avon High School Name AP Calculus AB Summer Review Packet Score Period f 4, find:.) If a.) f 4 f 4 b.) Topic A: Functions f c.) f h f h 4 V r r a.) V 4.) If, find: b.) V r V r c.) V r V r.) If f and g
7.3 Adding and Subtracting Rational Expressions
7.3 Adding and Subtracting Rational Epressions LEARNING OBJECTIVES. Add and subtract rational epressions with common denominators. 2. Add and subtract rational epressions with unlike denominators. 3. Add
4.8 Partial Fraction Decomposition
8 CHAPTER 4. INTEGRALS 4.8 Partial Fraction Decomposition 4.8. Need to Know The following material is assumed to be known for this section. If this is not the case, you will need to review it.. When are
Core Mathematics 3 A2 compulsory unit for GCE Mathematics and GCE Pure Mathematics Mathematics. Unit C3. C3.1 Unit description
Unit C3 Core Mathematics 3 A2 compulsory unit for GCE Mathematics and GCE Pure Mathematics Mathematics C3. Unit description Algebra and functions; trigonometry; eponentials and logarithms; differentiation;
3.2 Logarithmic Functions and Their Graphs
96 Chapter 3 Eponential and Logarithmic Functions 3.2 Logarithmic Functions and Their Graphs Logarithmic Functions In Section.6, you studied the concept of an inverse function. There, you learned that
Pre-Calculus Summer Packet
Pre-Calculus Summer Packet Name ALLEN PARK HIGH SCHOOL Summer Assessment Pre-Calculus Summer Packet For Students Entering Pre-Calculus Summer 05 This summer packet is intended to be completed by the FIRST
is on the graph of y = f 1 (x).
Objective 2 Inverse Functions Illustrate the idea of inverse functions. f() = 2 + f() = Two one-to-one functions are inverses of each other if (f g)() = of g, and (g f)() = for all in the domain of f.
Chapter 5: Integrals
Chapter 5: Integrals Section 5.5 The Substitution Rule (u-substitution) Sec. 5.5: The Substitution Rule We know how to find the derivative of any combination of functions Sum rule Difference rule Constant
5 3w. Unit 2 Function Operations and Equivalence Standard 4.1 Add, Subtract, & Multiply Polynomials
Unit Function Operations and Equivalence This document is meant to be used as an eample guide for each of the skills we will be holding students accountable for with Standard 4.1. This document should
ACCUPLACER MATH 0310
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to
1 Implicit Differentiation
1 Implicit Differentiation In logarithmic differentiation, we begin with an equation y = f(x) and then take the logarithm of both sides to get ln y = ln f(x). In this equation, y is not explicitly expressed
Math-2 Lesson 2-4. Radicals
Math- Lesson - Radicals = What number is equivalent to the square root of? Square both sides of the equation ( ) ( ) = = = is an equivalent statement to = 1.7 1.71 1.70 1.701 1.7008... There is no equivalent
(2.5) 1. Solve the following compound inequality and graph the solution set.
Intermediate Algebra Practice Final Math 0 (7 th ed.) (Ch. -) (.5). Solve the following compound inequalit and graph the solution set. 0 and and > or or (.7). Solve the following absolute value inequalities.
R3.6 Solving Linear Inequalities. 3) Solve: 2(x 4) - 3 > 3x ) Solve: 3(x 2) > 7-4x. R8.7 Rational Exponents
Level D Review Packet - MMT This packet briefly reviews the topics covered on the Level D Math Skills Assessment. If you need additional study resources and/or assistance with any of the topics below,
Section II: Exponential and Logarithmic Functions. Module 6: Solving Exponential Equations and More
Haberman MTH 111c Section II: Eponential and Logarithmic Functions Module 6: Solving Eponential Equations and More EXAMPLE: Solve the equation 10 = 100 for. Obtain an eact solution. This equation is so
Lecture 7: Indeterminate forms; L Hôpitals rule; Relative rates of growth. If we try to simply substitute x = 1 into the expression, we get
Lecture 7: Indeterminate forms; L Hôpitals rule; Relative rates of growth 1. Indeterminate Forms. Eample 1: Consider the it 1 1 1. If we try to simply substitute = 1 into the epression, we get. This is
Math Review Packet #5 Algebra II (Part 2) Notes
SCIE 0, Spring 0 Miller Math Review Packet #5 Algebra II (Part ) Notes Quadratic Functions (cont.) So far, we have onl looked at quadratic functions in which the term is squared. A more general form of
Algebra I Notes Unit Nine: Exponential Expressions
Algera I Notes Unit Nine: Eponential Epressions Syllaus Ojectives: 7. The student will determine the value of eponential epressions using a variety of methods. 7. The student will simplify algeraic epressions
Chapter 6 Logarithmic and Exponential Functions
Chapter 6 Logarithmic and Eponential Functions 6.1 The Definition of e Eample 1 Consider the eponential function f 1 ( ) 1 What happens to f() as gets very large? Type the function into Y=. Let Y 1 1 1/.
2.2 THE DERIVATIVE 2.3 COMPUTATION OF DERIVATIVES: THE POWER RULE 2.4 THE PRODUCT AND QUOTIENT RULES 2.6 DERIVATIVES OF TRIGONOMETRIC FUNCTIONS
Differentiation CHAPTER 2 2.1 TANGENT LINES AND VELOCITY 2.2 THE DERIVATIVE 2.3 COMPUTATION OF DERIVATIVES: THE POWER RULE 2.4 THE PRODUCT AND QUOTIENT RULES 25 2.5 THE CHAIN RULE 2.6 DERIVATIVES OF TRIGONOMETRIC
APPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.4 Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at
Chapter 5: Integrals
Chapter 5: Integrals Section 5.3 The Fundamental Theorem of Calculus Sec. 5.3: The Fundamental Theorem of Calculus Fundamental Theorem of Calculus: Sec. 5.3: The Fundamental Theorem of Calculus Fundamental
CALCULUS I. Integrals. Paul Dawkins
CALCULUS I Integrals Paul Dawkins Table of Contents Preface... ii Integrals... Introduction... Indefinite Integrals... Computing Indefinite Integrals... Substitution Rule for Indefinite Integrals... More
Math 120. x x 4 x. . In this problem, we are combining fractions. To do this, we must have
Math 10 Final Eam Review 1. 4 5 6 5 4 4 4 7 5 Worked out solutions. In this problem, we are subtracting one polynomial from another. When adding or subtracting polynomials, we combine like terms. Remember
A.P. Calculus Summer Packet
A.P. Calculus Summer Packet Going into AP calculus, there are certain skills that have been taught to you over the previous years that we assume you have. If you do not have these skills, you will find
AP Calculus AB Summer Assignment
Name: AP Calculus AB Summer Assignment Due Date: The beginning of class on the last class day of the first week of school. The purpose of this assignment is to have you practice the mathematical skills
arb where a A, b B and we say a is related to b. Howdowewritea is not related to b? 2Rw 1Ro A B = {(a, b) a A, b B}
Functions Functions play an important role in mathematics as well as computer science. A function is a special type of relation. So what s a relation? A relation, R, from set A to set B is defined as arb
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS. Introduction It is possible to integrate any rational function, constructed as the ratio of polynomials by epressing it as a sum of simpler fractions
Answer the following questions using a fraction and a percent (round to the nearest tenth of a percent).
ALGEBRA 1 Ch 10 Closure Solving Comple Equations Name: Two-Way Tables: A simple random sample of adults in a metropolitan area was selected and a survey was administered to determine the relationship between
Basic methods to solve equations
Roberto s Notes on Prerequisites for Calculus Chapter 1: Algebra Section 1 Basic methods to solve equations What you need to know already: How to factor an algebraic epression. What you can learn here:
COUNCIL ROCK HIGH SCHOOL MATHEMATICS. A Note Guideline of Algebraic Concepts. Designed to assist students in A Summer Review of Algebra
COUNCIL ROCK HIGH SCHOOL MATHEMATICS A Note Guideline of Algebraic Concepts Designed to assist students in A Summer Review of Algebra [A teacher prepared compilation of the 7 Algebraic concepts deemed
Algebra Final Exam Review Packet
Algebra 1 00 Final Eam Review Packet UNIT 1 EXPONENTS / RADICALS Eponents Degree of a monomial: Add the degrees of all the in the monomial together. o Eample - Find the degree of 5 7 yz Degree of a polynomial:
(1) Assignment # 1 Absolute Value. (2) Assignment # 2 Compound Absolute Values. (3) Assignment # 3 Exponents. (4) Assignment # 4 Simplifying Radicals
Alg_0 Packet # The beginning of our Quest () Assignment # Absolute Value () Assignment # Compound Absolute Values () Assignment # Eponents () Assignment # Simplifying Radicals (5) Assignment # 5 Fractional
THE INSTITUTE OF FINANCE MANAGEMENT (IFM) Department of Mathematics. Mathematics 01 MTU Elements of Calculus in Economics
THE INSTITUTE OF FINANCE MANAGEMENT (IFM) Department of Mathematics Mathematics 0 MTU 070 Elements of Calculus in Economics Calculus Calculus deals with rate of change of quantity with respect to another
LESSON 2 Negative exponents Product and power theorems for exponents Circle relationships
9.A negative eponents LESSON Negative eponents Product and power theorems for eponents Circle relationships.a negative eponents Negative eponents cannot be understood because they are the result of a definition,
Chapter 8 Notes SN AA U2C8
Chapter 8 Notes SN AA U2C8 Name Period Section 8-: Eploring Eponential Models Section 8-2: Properties of Eponential Functions In Chapter 7, we used properties of eponents to determine roots and some of
( ) is a point on a function f, then the point ( y, x) is on the
L. Inverses No topic in math confuses students more than inverses. If a function is a rule that maps to y, an inverse is a rule that brings y back to the original. If a point, y ( ) is a point on a function
Exponential and Logarithmic Equations and Models. College Algebra
Exponential and Logarithmic Equations and Models College Algebra Product Rule for Logarithms The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of