# 1 Lesson 11: Antiderivatives of Elementary Functions

Size: px
Start display at page:

Transcription

1 1 Lesson 11: Antiderivatives of Elementary Functions Chapter 6 Material: pages in the textbook: The material in this lesson covers The definition of the antiderivative of a function of one variable. Development of algebraic formulas for computing antiderivatives. In particular, linearity when computing antiderivatives is presented. Development of simple formulas for antiderivatives based on differentiation formulas is presented The differentiation formula approach results in the determination of a general way to compute antiderivatives of polynomials. However, the use of differentiation formulas for all elementary functions will not work. Notation for the antiderivative using the integral sign if presented in this lesson. Prerequisite Content for Lesson 9 Students should understand all of the content through Chapter 4 of the textbook. The techniques developed in Chapter 4 are used development of general antiderivatives in the first two sections of this chapter. Students will need to use the linearity of the derivative operation when determining forms for antiderivatives. To be successful in this lesson, students must know how to algebraically manipulate polynomials, exponential and logarithm functions, hyperbolic trigonometric functions, and simple rational functions of polynomials. This lesson does not cover methods of integration. The lesson involves inverting the differentiation of elementary functions in the simplest of cases. So, the algebraic work and use of differentiation formulas involves noticing a relationship between a function and a derivative. Goals and Objectives for Lesson 11 Students who complete this lesson should be able to: identify the computation of an antiderivative as reversing differentiation, compute antiderivatives of many elementary functions, use the linearity of integration methods in the determination of general forms of antiderivatives, develop a table of antiderivatives of elementary functions, determine the general form of an antiderivative and understand the addition of a constant of integration, and determine the value of the constant of integration in the case when an additional or auxiliary condition is specified in the statement of the problem. The goals and objectives of this lesson involve a type of pattern matching due to the development of antiderivatives through differentiation formulas. The idea is that since students know certain formulas for differentiation, antiderivatives for some functions can be constructed using this relationship. 1

2 1.1 Lecture Notes for Lesson 11 - Day 1 Lesson 11 begins the process of un-doing differentiation. The techniques in this lesson are based on noticing relationships between functions and their derivatives. The lesson does not cover integration via substitution, integration by parts, or integration via partial fractions. These are covered in later lesson(s). The example (page 237-8) in the introduction to the chapter is simple, but illustrates the nonuniqueness of antiderivatives in that there are three different functions that result in the same derivative. The difference is that each of the three functions has a different constant value. The constant shows up in the definition of the antiderivative of a function and is an important part of the result. Definition of an Antiderivative The definition of an antiderivative given in the textbook is in terms of the relationship that if F (x) = f(x) then F (x) is an antiderivative of f(x). This seems to work best for the type of presentation used in a Business Calculus course like Math 1100 at USU. The antiderivative approach allows students the ability to use integration without the use of summations. Note that if an antiderivative is computed that the result can be tested by differentiating the result. If the differentiation produces the original function then the antiderivative is probably ok. Students should get in the habit of checking an antiderivative using differentiation. Antiderivatives Based on Differentiation Formulas The basic idea of this section is to use known formulas for derivatives to build tables of antiderivatives. Most of the initial formulas are easy to apply due to the simple forms. The initial results involve the computation of antiderivatives of polynomials and exponential and logarithm functions. Students should understand that more formulas will be developed in the next lessons. However, the methods of integration presented in the next lesson will require a lot more work. Algebraic Rules for Antiderivatives Students must understand the algebraic rules for computing antiderivatives. In this lesson the rules related to linearity are presented. That is a constant multiple of a derivative is the same as the derivative of the constant times the the function and the sum of derivatives of two functions is the derivative of the sum of the functions. 2

3 1.2 Lecture Notes for Lesson 11 - Day 2 The material in this lecture should include the determination of antiderivatives for basic elementary functions. The process used in each of the examples is to use differentiation formulas to directly determine an antiderivative from a given function. Note that initially, the formulas developed are relatively simple. The first formulas are simple monomials that are referred to as power functions. Once these formulas are developed, antiderivative formulas for polynomials can be developed. After this, more complicated formulas will continue to be developed until almost all types of elementary functions have been treated. Power Rules for Antiderivatives There is basically one rule for computing antiderivatives of powers. This is relatively easy and should be easy to explain to students. The rule does a basic inversion of the rule associated with differentiation of powers of an independent variable. Antiderivatives of Polynomials This is a simple, but important section. Students need to be able to work with polynomials in many applications in business and economics courses. Students should be able to compute antiderivatives of polynomials term by term very quickly. Exponential and Logarithm Rules for Antiderivatives Without the chain rule it is a bit difficult to do a lot with exponential and logarithm functions. Students should be able to guess the form of the antiderivative if the function is not too complicated. This is not the best way to compute antiderivatives. However, in the next lesson the use of substitution will be addressed which will take out all of the guess work in most cases. 3

4 1.3 Lecture Notes for Lesson 11 - Day 3 Hyperbolic Trigonometric Rules for Antiderivatives Use the tables of formulas for derivatives of the hyperbolic trigonometric functions to determine antiderivatives of simple combinations of these functions. As with exponential functions, unless the form is pretty simple, it is difficult to guess an antiderivative. The methods in the next lesson will help with a formal determination of antiderivatives of hyperbolic trigonometric functions. Antiderivative Forms for Simple Rational Functions There are only a few forms that can be used in this section. The only types of functions that can be used at this point involve simple linear factors in the denominator. However, once partial fraction decomposition is introduced, students will be able to do a lot more. A Simple Notation for Antiderivatives The notation to emphasize is F = f which tells us to compute an antiderivative of f. following. Students should also get used to using notation like the F (x) = x f(t) dt if F (x) = f(x). Not that t is a variable of integration and could be replaced with any integration variable. The notation will be reinforced in the next two lessons. Antiderivatives for Other Elementary Functions Students should be made aware that the determination of antiderivatives to this point has involved undoing simple differentiation formulas. There are a lot of elementary functions that cannot be integrated using the simple formulas. Make sure that students know that other methods of integration will also be covered. One of these methods is substitution or making a change of variables which is used to invert the chain rule in differentiation. In addition, integration by parts which inverts the product rule will be developed. Finally, partial fraction decomposition will be used to integrate all rational functions of polynomials in its most general form. 4

5 1.4 Homework Verifying the Antiderivative of a Function Problem 1: In each of the problems verify that the derivative of the function F (x) matches the given function, f(x). a. F (x) = x 3 x 2 + x 1, f(x) = 3 x 2 2 x + 1 b. F (x) = x e 2 x + x 1, f(x) = e 2 x (1 + 2 x) + 1 c. F (x) = ln(2 x) + cosh 2 (3 x) + 42, f(x) = sinh(3 x) cosh(3 x) 2 x 5

6 Nonuniqueness of Antiderivatives Problem 2: Show that each of the following functions is an antiderivative of the function, f(x) = 4x 3 2x 2 +x 1 a. F (x) = x x x2 x + 6 b. F (x) = x x x2 x 42 6

7 Specification of the Constant of Integration Problem 3: The marginal revenue function R (x) = x x 2 in a given mathematical model is given. If the revenue for producing x = 120 items is 10, 000, determine the unique antiderivative that includes this specific given condition. 7

8 Antiderivatives of Powers Problem 4: Compute antiderivatives of the following power functions. a. f(x) = x 38 b. f(x) = x 4/3 c. f(x) = x

9 Antiderivatives of Polynomials Problem 5: Compute antiderivatives for the following polynomials. a. f(x) = 5 x 3 6 x 7 b. f(x) = 15 4 x + 16 x 2 8 x 3 c. f(x) = e x 3 π x 2 9

10 Antiderivatives of Exponential Functions Problem 6: Compute antiderivatives for the following exponential functions a. f(x) = 2 x 10 x b. f(x) = e x+1 c. f(x) = e 2 x 10

11 Antiderivatives and Logarithm Functions Problem 7: For each of the functions given below, compute the antiderivative based on the logarithm differentiation formulas. Remember to use linearity and other properties. a. f(x) = 1 2x b. f(x) = x 11

12 Using a Formula to Compute Antiderivatives Problem 8: Given a function f(x) = ln(a x) + b a rule for computing the antiderivative of f is given as F (x) = x ln(a x) x + b x + B where B is the constant of integration. Verify that the formula is correct by computing the derivative of F (x) and comparing F (x) to f(x). Then compute the antiderivative of f(x) = ln(3 x) + 7 using the general formula. 12

13 Antiderivatives of Hyperbolic Trigonometric Functions Problem 9: Compute antiderivatives of the following hyperbolic trigonometric functions. a. f(x) = cosh(x) sinh(x) b. f(x) = sech 2 (x) + 42 c. f(x) = sech(x) cosh(x) 13

14 Antiderivatives of Simple Rational Functions Problem 10: Compute antiderivatives for each of the following simple rational functions of polynomials. a. f(x) = 7 x + 3 b. f(x) = 5 4 x c. f(x) = 2 x 11 14

### AskDrCallahan Calculus 1 Teacher s Guide

AskDrCallahan Calculus 1 Teacher s Guide 3rd Edition rev 080108 Dale Callahan, Ph.D., P.E. Lea Callahan, MSEE, P.E. Copyright 2008, AskDrCallahan, LLC v3-r080108 www.askdrcallahan.com 2 Welcome to AskDrCallahan

### DIFFERENTIATE SOMETHING AT THE VERY BEGINNING THE COURSE I'LL ADD YOU QUESTIONS USING THEM. BUT PARTICULAR QUESTIONS AS YOU'LL SEE

1 MATH 16A LECTURE. OCTOBER 28, 2008. PROFESSOR: SO LET ME START WITH SOMETHING I'M SURE YOU ALL WANT TO HEAR ABOUT WHICH IS THE MIDTERM. THE NEXT MIDTERM. IT'S COMING UP, NOT THIS WEEK BUT THE NEXT WEEK.

### Overview. Teacher s Manual and reproductions of student worksheets to support the following lesson objective:

Overview Lesson Plan #1 Title: Ace it! Lesson Nine Attached Supporting Documents for Plan #1: Teacher s Manual and reproductions of student worksheets to support the following lesson objective: Find products

EIGHTH GRADE RELIGION MORALITY ~ Your child knows that to be human we must be moral. knows there is a power of goodness in each of us. knows the purpose of moral life is happiness. knows a moral person

### 1-5 Square Roots and Real Numbers. Holt Algebra 1

1-5 Square Roots and Real Numbers Warm Up Lesson Presentation Lesson Quiz Bell Quiz 1-5 Evaluate 2 pts 1. 5 2 2 pts 2. 6 2 2 pts 3. 7 2 10 pts possible 2 pts 4. 8 2 2 pts 5. 9 2 Questions on 0-4/0-10/0-11

### MITOCW max_min_second_der_512kb-mp4

MITOCW max_min_second_der_512kb-mp4 PROFESSOR: Hi. Well, I hope you're ready for second derivatives. We don't go higher than that in many problems, but the second derivative is an important-- the derivative

### SEVENTH GRADE. Revised June Billings Public Schools Correlation and Pacing Guide Math - McDougal Littell Middle School Math 2004

SEVENTH GRADE June 2010 Billings Public Schools Correlation and Guide Math - McDougal Littell Middle School Math 2004 (Chapter Order: 1, 6, 2, 4, 5, 13, 3, 7, 8, 9, 10, 11, 12 Chapter 1 Number Sense, Patterns,

### Basic Information for MAT194F Calculus Engineering Science 2013

Basic Information for MAT194F Calculus Engineering Science 2013 1. Your Lecturers P.C. Stangeby Institute for Aerospace Studies To arrange a meeting, please email: pcs@starfire.utias.utoronto.ca D. Penneys

### BPS 7th Grade Pre-Algebra Revised summer 2014 Year at a Glance Unit Standards Practices Days

BPS 7th Grade Pre-Algebra Revised summer 2014 Year at a Glance Unit Standards Practices Days 1 All Operations with Integers 7.NS.1, 7.NS.2, 7.NS.3 1,4,6,8 7 2 All Operations with Rational Numbers 7.NS.1c,

### Note: Please use the actual date you accessed this material in your citation.

MIT OpenCourseWare http://ocw.mit.edu 18.03 Differential Equations, Spring 2006 Please use the following citation format: Arthur Mattuck and Haynes Miller, 18.03 Differential Equations, Spring 2006. (Massachusetts

### Proceedings of the Third International DERIVE/TI-92 Conference

Description of the TI-92 Plus Module Doing Advanced Mathematics with the TI-92 Plus Module Carl Leinbach Gettysburg College Bert Waits Ohio State University leinbach@cs.gettysburg.edu waitsb@math.ohio-state.edu

### MAT Practice (solutions) 1. Find an algebraic formula for a linear function that passes through the points ( 3, 7) and (6, 1).

MAT 110 - Practice (solutions) 1. Find an algebraic formula for a linear function that passes through the points ( 3, 7) and (6, 1). Answer: y = 2 3 + 5 2. Let f(x) = 8x 120 (a) What is the y intercept

### UNMANNED AERIAL SYSTEMS

Contributions and Editorial Correspondence Send article submissions with cover letters as e-mail attachments. No hard copy is necessary. Books are not solicited for review from authors or publishers. Those

### Matrix Mathematics: Theory, Facts, and Formulas

Matrix Mathematics: Theory, Facts, and Formulas Dennis S. Bernstein Click here if your download doesn"t start automatically Matrix Mathematics: Theory, Facts, and Formulas Dennis S. Bernstein Matrix Mathematics:

### Basic Information for MAT194F Calculus Engineering Science 2004

Basic Information for MAT194F Calculus Engineering Science 2004 1. Your Lecturers K. Consani Department of Mathematics Schedule: M 13-14 (MC252); T 11-12 (RS211); R 10-11 (BA1190). Kyu-Hwan Lee Department

### Elements of Style. Anders O.F. Hendrickson

Elements of Style Anders O.F. Hendrickson Years of elementary school math taught us incorrectly that the answer to a math problem is just a single number, the right answer. It is time to unlearn those

### Display Contest Submittals

Display Contest Submittals #1a ----- Original Message ----- From: Jim Horn To: rjnelsoncf@cox.net Sent: Tuesday, April 28, 2009 3:07 PM Subject: Interesting calculator display Hi, Richard Well, it takes

### MANAGEMENT (Also see Business) LIBRARY SCIENCE MARKETING COURSE DESCRIPTION 254 WEST LOS ANGELES COLLEGE CATALOG (3/2/2017)

92 Reading for College Success (3) Students improve their abilities to read a variety of texts at different levels of difficulty. They practice identifying topics, main ideas, and supporting details, as

### Mathematics Curriculum Document for Algebra 2

Unit Title: Square Root Functions Time Frame: 6 blocks Grading Period: 2 Unit Number: 4 Curriculum Enduring Understandings (Big Ideas): Representing relationships mathematically helps us to make predictions

### Laplace Transform: basic properties; functions of a complex variable; poles diagrams; s-shift law.

18.03 Lecture 26, April 14 Laplace Transform: basic properties; functions of a complex variable; poles diagrams; s-shift law. [1] The Laplace transform connects two worlds: The t domain t is real and positive

### MITOCW big_picture_integrals_512kb-mp4

MITOCW big_picture_integrals_512kb-mp4 PROFESSOR: Hi. Well, if you're ready, this will be the other big side of calculus. We still have two functions, as before. Let me call them the height and the slope:

### Math Assignment 10

Math 228 - Assignment 1 Dylan Zwic Spring 214 Section 7.4-1, 5, 1, 19, 31 Section 7.5-1, 6, 15, 21, 26 Section 7.6-1, 6, 11, 14, 15 1 Section 7.4 - Derivatives, Integrals, and Products of Transforms 7.4.1

### Note: Please use the actual date you accessed this material in your citation.

MIT OpenCourseWare http://ocw.mit.edu 18.06 Linear Algebra, Spring 2005 Please use the following citation format: Gilbert Strang, 18.06 Linear Algebra, Spring 2005. (Massachusetts Institute of Technology:

### Zdravko Cvetkovski. Inequalities. Theorems, Techniques and Selected Problems

Inequalities Zdravko Cvetkovski Inequalities Theorems, Techniques and Selected Problems Dipl. Math. Zdravko Cvetkovski Informatics Department European University-Republic of Macedonia Skopje, Macedonia

### Music. Associate in Science in Mathematics for Transfer (AS-T) Degree Major Code:

Explain and demonstrate mathematical concepts relevant to the course content. Analyze and construct proofs relevant to the course concepts. Create, interpret and analyze graphs relevant to the course content.

### So just by way of a little warm up exercise, I'd like you to look at that integration problem over there. The one

MITOCW Lec-02 What we're going to talk about today, is goals. So just by way of a little warm up exercise, I'd like you to look at that integration problem over there. The one that's disappeared. So the

### Math 8 Assignment Log. Finish Discussion on Course Outline. Activity Section 2.1 Congruent Figures Due Date: In-Class: Directions for Section 2.

08-23-17 08-24-17 Math 8 Log Discussion: Course Outline Assembly First Hour Finish Discussion on Course Outline Activity Section 2.1 Congruent Figures In-Class: Directions for Section 2.1 08-28-17 Activity

### Associate of Applied Science Songwriting. McLENNAN COMMUNITY COLLEGE

Associate of Applied Science Songwriting McLENNAN COMMUNITY COLLEGE 2015-2016 Degree Description Students completing this program are prepared to assume positions as composers, arrangers and songwriters.

### WJEC MATHEMATICS INTERMEDIATE ALGEBRA. SEQUENCES & Nth TERM

WJEC MATHEMATICS INTERMEDIATE ALGEBRA SEQUENCES & Nth TERM 1 Contents Number Machines Continuing a sequence Finding the nth term Writing terms using the nth term Picture Sequences Credits WJEC Question

### 8.3. Start Thinking! Warm Up. Find the area of the triangle Activity. Activity. 4 m. 14 in. 7 m. 9 in. 12 yd. 11 yd. 1 mm. 5.

Activity Start Thinking! For use before Activity You know how to find the area of squares, rectangles, triangles, trapezoids, and parallelograms. Describe three different methods you could use to estimate

### Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 1 1

11-8 Radical Radial Expressions Warm Up Lesson Presentation Lesson Quiz 1 Warm Up Simplify each expression. 1. 2. 3. 4. Objectives Multiply and divide radical expressions. Rationalize denominators. You

### AN INTRODUCTION TO CLASSICAL REAL ANALYSIS KARL R. STROMBERG. AMS CHELSEA PUBLISHING American Mathematical Society Providence, Rhode Island

AN INTRODUCTION TO CLASSICAL REAL ANALYSIS KARL R. STROMBERG AMS CHELSEA PUBLISHING American Mathematical Society Providence, Rhode Island AN INTRODUCTION TO CLASSICAL REAL ANALYSIS AN INTRODUCTION TO

### ELEC 310 Digital Signal Processing

ELEC 310 Digital Signal Processing Alexandra Branzan Albu 1 Instructor: Alexandra Branzan Albu email: aalbu@uvic.ca Course information Schedule: Tuesday, Wednesday, Friday 10:30-11:20 ECS 125 Office Hours:

### MITOCW ocw f08-lec19_300k

MITOCW ocw-18-085-f08-lec19_300k The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free.

### Course Catalog

The Zion Academy of Music Course Catalog 2014-2015 Fall/Spring 2 The Zion Academy of Music Course Catalog 2014-2015 Beginning Ensembles Beginning Jazz Ensemble Elementary Choir Junior Show Choir Strings

### Fourier Integral Representations Basic Formulas and facts

Engineering Mathematics II MAP 436-4768 Spring 22 Fourier Integral Representations Basic Formulas and facts 1. If f(t) is a function without too many horrible discontinuities; technically if f(t) is decent

### Correlation to the Common Core State Standards

Correlation to the Common Core State Standards Go Math! 2011 Grade 4 Common Core is a trademark of the National Governors Association Center for Best Practices and the Council of Chief State School Officers.

### 1-1 Variables and Expressions

1-1 Variables and Expressions Warm Up Lesson Presentation Lesson Quiz Warm Up Add or subtract. 1. 6 + 104 110 2. 12(9) 108 3. 23 8 15 4. Multiply or divide. 5. 324 18 18 6. 6 7. 13.5(10) 135 8. 18.2 2

### Appendix B. Elements of Style for Proofs

Appendix B Elements of Style for Proofs Years of elementary school math taught us incorrectly that the answer to a math problem is just a single number, the right answer. It is time to unlearn those lessons;

### BC Sequences and series 2015.notebook March 03, 2015

Feb 25 12:49 PM 1 Jan 27 9:13 AM 2 Jan 27 9:24 AM 3 Feb 25 12:49 PM 4 Warm up thoughts What notation is used for sequences? What notation is used for series? How are they related? Jan 13 8:01 AM 5 Limit

### Week. Intervals Major, Minor, Augmented, Diminished 4 Articulation, Dynamics, and Accidentals 14 Triads Major & Minor. 17 Triad Inversions

Week Marking Period 1 Week Marking Period 3 1 Intro.,, Theory 11 Intervals Major & Minor 2 Intro.,, Theory 12 Intervals Major, Minor, & Augmented 3 Music Theory meter, dots, mapping, etc. 13 Intervals

### Cryptanalysis of LILI-128

Cryptanalysis of LILI-128 Steve Babbage Vodafone Ltd, Newbury, UK 22 nd January 2001 Abstract: LILI-128 is a stream cipher that was submitted to NESSIE. Strangely, the designers do not really seem to have

### d. Could you represent the profit for n copies in other different ways?

Special Topics: U3. L3. Inv 1 Name: Homework: Math XL Unit 3 HW 9/28-10/2 (Due Friday, 10/2, by 11:59 pm) Lesson Target: Write multiple expressions to represent a variable quantity from a real world situation.

### Lecture 3: Nondeterministic Computation

IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 3: Nondeterministic Computation David Mix Barrington and Alexis Maciel July 19, 2000

### Calculus II For Dummies PDF

Calculus II For Dummies PDF An easy-to-understand primer on advanced calculus topics Calculus II is a prerequisite for many popular college majors, including pre-med, engineering, and physics. Calculus

### CPSC 121: Models of Computation Lab #5: Flip-Flops and Frequency Division

CPSC 121: Models of Computation Lab #5: Flip-Flops and Frequency Division Objectives In this lab, you will see two types of sequential circuits: latches and flip-flops. Latches and flip-flops can be used

### Recommended Operations

Category LMS Test.Lab Access Level End User Topic Rotating Machinery Publish Date 1-Aug-2016 Question: How to 'correctly' integrate time data within Time Domain Integration? Answer: While the most accurate

### Bite Size Brownies. Designed by: Jonathan Thompson George Mason University, COMPLETE Math

Bite Size Brownies Designed by: Jonathan Thompson George Mason University, COMPLETE Math The Task Mr. Brown E. Pan recently opened a new business making brownies called The Brown E. Pan. On his first day

### Punchline Algebra Book B Answer Key Marcy Mathworks

PUNCHLINE ALGEBRA BOOK B ANSWER KEY MARCY MATHWORKS PDF - Are you looking for punchline algebra book b answer key marcy mathworks Books? Now, you will be happy that at this time punchline algebra book

### North Carolina Standard Course of Study - Mathematics

A Correlation of To the North Carolina Standard Course of Study - Mathematics Grade 4 A Correlation of, Grade 4 Units Unit 1 - Arrays, Factors, and Multiplicative Comparison Unit 2 - Generating and Representing

### Lecture 1: What we hear when we hear music

Lecture 1: What we hear when we hear music What is music? What is sound? What makes us find some sounds pleasant (like a guitar chord) and others unpleasant (a chainsaw)? Sound is variation in air pressure.

### The Lazy Man Explains the Irrational. E. L. Lady

The Lazy Man Explains the Irrational E. L. Lady I ve been thinking about those numbers that you can t write as fractions, Mr. Tinker said. Irrational numbers, they re called, the Lazy Man answered. Well,

### Applied Microeconomics: Consumption, Production and Markets David L. Debertin

Applied Microeconomics: Consumption, Production and Markets David L. Debertin This is a microeconomic theory book designed for upper division undergraduate students in economics and agricultural economics.

### What Does Sticker Notation Mean For Math

What Does Sticker Mean For Math Free PDF ebook Download: What Does Sticker Mean For Math Download or Read Online ebook what does sticker notation mean for math in PDF Format From The Best User Guide Database

### Chapter 21. Margin of Error. Intervals. Asymmetric Boxes Interpretation Examples. Chapter 21. Margin of Error

Context Part VI Sampling Accuracy of Percentages Previously, we assumed that we knew the contents of the box and argued about chances for the draws based on this knowledge. In survey work, we frequently

### PUNCHLINE ALGEBRA BOOK B ANSWERS PDF

PUNCHLINE ALGEBRA BOOK B ANSWERS PDF ==> Download: PUNCHLINE ALGEBRA BOOK B ANSWERS PDF PUNCHLINE ALGEBRA BOOK B ANSWERS PDF - Are you searching for Punchline Algebra Book B Answers Books? Now, you will

### Where Are We Now? e.g., ADD \$S0 \$S1 \$S2?? Computed by digital circuit. CSCI 402: Computer Architectures. Some basics of Logic Design (Appendix B)

Where Are We Now? Chapter 1: computer systems overview and computer performance Chapter 2: ISA (machine-spoken language), different formats, and various instructions Chapter 3: We will learn how those

### Supplemental Spreadsheets, PowerPoint Files and Other Class Materials

University of Kentucky UKnowledge Agricultural Economics Textbook Gallery Agricultural Economics 4-2015 Supplemental Spreadsheets, PowerPoint Files and Other Class Materials David L. Debertin University

### The unbelievable musical magic of the number 12

The unbelievable musical magic of the number 12 This is an extraordinary tale. It s worth some good exploratory time. The students will encounter many things they already half know, and they will be enchanted

### Curriculum of B.Sc. Engineering Honours Degree Programme Textile & Clothing Technology

Name Category Lectures Credits Norm Evaluation (%) hrs/week Assignments GPA NGPA GPA NGPA CA WE Semester 1 MA1013 Mathematics C 3.0 1/1 3.0 20 80 CS1032 Programming Fundamentals C 2.0 3/1 3.0 20 80 ME1032

### CPSC 121: Models of Computation Lab #5: Flip-Flops and Frequency Division

CPSC 121: Models of Computation Lab #5: Flip-Flops and Frequency Division Objectives In this lab, we will see the sequential circuits latches and flip-flops. Latches and flip-flops can be used to build

### Writing maths, from Euclid to today

Writing maths, from Euclid to today ONE: EUCLID The first maths book of all time, and the maths book for most of the last 2300 years, was Euclid s Elements. Here the bit from it on Pythagoras s Theorem.

### 1/ 19 2/17 3/23 4/23 5/18 Total/100. Please do not write in the spaces above.

1/ 19 2/17 3/23 4/23 5/18 Total/100 Please do not write in the spaces above. Directions: You have 50 minutes in which to complete this exam. Please make sure that you read through this entire exam before

### What Is Sticker Notation

What Is Sticker Free PDF ebook Download: What Is Sticker Download or Read Online ebook what is sticker notation in PDF Format From The Best User Guide Database Mopeds do not have a sticker or placard outlining

### Our Musical, Mathematical Brains Ideas to turn STEM to STEAM

Debbie Char, St Louis Community College at Forest Park dchar@stlcc.edu Our Musical, Mathematical Brains Ideas to turn STEM to STEAM What you heard... Seasons of Love (the musical Rent) Upside Down, Inside

### Signals and Systems. Spring Room 324, Geology Palace, ,

Signals and Systems Spring 2013 Room 324, Geology Palace, 13756569051, zhukaiguang@jlu.edu.cn Chapter 7 Sampling 1) The Concept and Representation of Periodic Sampling of a CT Signal 2) Analysis of Sampling

### CALCULUS:EARLY TRANSCENDENTAL By Jon Rogawski

CALCULUS:EARLY TRANSCENDENTAL By Jon Rogawski Calculus: Early Transcendentals. William L. Briggs, University of Colorado, Denver. Lyle Cochran, Whitworth University. This product is part of the following

### db math Training materials for wireless trainers

db math Training materials for wireless trainers Goals To understand why we use db to make calculations on wireless links. To learn db math. To be able to solve some simple exercises. To understand what

### Algebra (2nd Edition) PDF

Algebra (2nd Edition) PDF Algebra, Second Edition, by Michael Artin, provides comprehensive coverage at the level of an honors-undergraduate or introductory-graduate course. The second edition of this

### _The_Power_of_Exponentials,_Big and Small_

_The_Power_of_Exponentials,_Big and Small_ Nataly, I just hate doing this homework. I know. Exponentials are a huge drag. Yeah, well, now that you mentioned it, let me tell you a story my grandmother once

### Why t? TEACHER NOTES MATH NSPIRED. Math Objectives. Vocabulary. About the Lesson

Math Objectives Students will recognize that when the population standard deviation is unknown, it must be estimated from the sample in order to calculate a standardized test statistic. Students will recognize

### Fourier Transforms 1D

Fourier Transforms 1D 3D Image Processing Torsten Möller Overview Recap Function representations shift-invariant spaces linear, time-invariant (LTI) systems complex numbers Fourier Transforms Transform

### MATH 195: Gödel, Escher, and Bach (Spring 2001) Notes and Study Questions for Tuesday, March 20

MATH 195: Gödel, Escher, and Bach (Spring 2001) Notes and Study Questions for Tuesday, March 20 Reading: Chapter VII Typographical Number Theory (pp.204 213; to Translation Puzzles) We ll also talk a bit

### (1) + 1(0.1) + 7(0.001)

Name: Quarterly 1 Study Guide The first quarterly test covers information from Modules 1, 2, and 3. If you complete this study guide and turn it in on Tuesday, you will receive 5 bonus points on your Quarterly

### McRuffy Press Fourth Grade Color Math Test 7

McRuffy Press Fourth Grade Color Math Test 7 Materials: Test pages (Resource pack, 3 sheets) Test Directions Page :. Problem solving: Solve the problems. 2. Fractions to decimals: Change the fractions

### Associate of Applied Science Venue & Talent Management (Music) McLENNAN COMMUNITY COLLEGE

Associate of Applied Science Venue & Talent Management (Music) McLENNAN COMMUNITY COLLEGE 2017-2018 Degree Description Students completing this program are prepared to assume management positions in the

### Sequential logic circuits

Computer Mathematics Week 10 Sequential logic circuits College of Information Science and Engineering Ritsumeikan University last week combinational digital circuits signals and busses logic gates and,

I visited friends in New York City during the summer. They took me to this HUGE Wal-Mart store. There was a display of cookie boxes that I could not believe! The display was in a pyramid shape with at

### COLLIN COUNTY COMMUNITY COLLEGE DISTRICT 2015 Core Curriculum Changes are requested to be effective beginning in the fall 2015 semester.

COLLIN COUNTY COMMUNITY COLLEGE DISTRICT 2015 Core Curriculum Changes are requested to be effective beginning in the fall 2015 semester. Collin County Community College District: 2015 Core Curriculum Revision

### Problem Weight Total 100

EE 350 Problem Set 4 Cover Sheet Fall 2016 Last Name (Print): First Name (Print): ID number (Last 4 digits): Section: Submission deadlines: Turn in the written solutions by 4:00 pm on Tuesday October 4

### Professor Weissman s Algebra Classroom

Combine Like Terms Unit #12 2007 Prof Weissman s Software Tel: 1-347-528-7837 mathprof@hotmail.com Professor Weissman s Algebra Classroom Martin Weissman, Jonathan S. Weissman, Tamara Farber, & Keith Monse

### PEP - Upper Elementary Report Card

PEP - Upper Elementary Report Card Student Name tical Life The student treats fellow students and adults with kindness and common courtesy. Lakeland Montessori Upper Elementary (9-) The student takes responsibility

CURRICULUM FOR ADVANCED PLACEMENT MUSIC THEORY GRADES 10-12 This curriculum is part of the Educational Program of Studies of the Rahway Public Schools. ACKNOWLEDGMENTS Frank G. Mauriello, Interim Assistant

### 2D ELEMENTARY CELLULAR AUTOMATA WITH FOUR NEIGHBORS

2D ELEMENTARY CELLULAR AUTOMATA WITH FOUR NEIGHBORS JOSÉ ANTÓNIO FREITAS Escola Secundária Caldas de Vizela, Rua Joaquim Costa Chicória 1, Caldas de Vizela, 4815-513 Vizela, Portugal RICARDO SEVERINO CIMA,

### Welcome Accelerated Algebra 2!

Welcome Accelerated Algebra 2! Tear-Out: Pg. 445-452 (Class notes) Pg. 461 (homework) U6H2: Pg. 390 #21-24 Pg. 448 #6-7, 9-11 Pg. 461 #6-8 Updates: U6Q1 will be February 15 th (Thursday) U6T will be March

### Lab P-6: Synthesis of Sinusoidal Signals A Music Illusion. A k cos.! k t C k / (1)

DSP First, 2e Signal Processing First Lab P-6: Synthesis of Sinusoidal Signals A Music Illusion Pre-Lab: Read the Pre-Lab and do all the exercises in the Pre-Lab section prior to attending lab. Verification:

### Senior Math Studies Lesson Planning Date Lesson Events

Senior Math Studies Lesson Planning 2014-2015 Date Lesson Events Aug 25 Style of Class: Student-led work teams (SLWTs) Facilitated (as opposed to taught) by the teacher Considering MS topics what areas

### A New General Class of Fuzzy Flip-Flop Based on Türkşen s Interval Valued Fuzzy Sets

Magyar Kutatók 7. Nemzetközi Szimpóziuma 7 th International Symposium of Hungarian Researchers on Computational Intelligence A New General Class of Fuzzy Flip-Flop Based on Türkşen s Interval Valued Fuzzy

### 2.2. Multiplying and Dividing Powers. INVESTIGATE the Math

. Multipling and Dividing Powers GOAL Develop and appl exponent principles to multipl and divide powers. INVESTIGATE the Math Amir thought there was a wa to simplif 1 6 1 9 1 without using a calculator.?

### Pattern Creator/Converter Software User Manual

Application Note: HFAN-9.5.0 Rev.1; 04/08 Pattern Creator/Converter Software User Manual Pattern Creator/Converter Software User Manual 1 Introduction The Pattern Creator/Converter software (available

### Algebra I Module 2 Lessons 1 19

Eureka Math 2015 2016 Algebra I Module 2 Lessons 1 19 Eureka Math, Published by the non-profit Great Minds. Copyright 2015 Great Minds. No part of this work may be reproduced, distributed, modified, sold,

### MITOCW ocw f07-lec02_300k

MITOCW ocw-18-01-f07-lec02_300k The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free.

### Musical Sound: A Mathematical Approach to Timbre

Sacred Heart University DigitalCommons@SHU Writing Across the Curriculum Writing Across the Curriculum (WAC) Fall 2016 Musical Sound: A Mathematical Approach to Timbre Timothy Weiss (Class of 2016) Sacred

### ECONOMICS 351* -- INTRODUCTORY ECONOMETRICS. Queen's University Department of Economics. ECONOMICS 351* -- Winter Term 2005 INTRODUCTORY ECONOMETRICS

Queen's University Department of Economics ECONOMICS 351* -- Winter Term 2005 INTRODUCTORY ECONOMETRICS Winter Term 2005 Instructor: Web Site: Mike Abbott Office: Room A521 Mackintosh-Corry Hall or Room

### CURIE Day 3: Frequency Domain Images

CURIE Day 3: Frequency Domain Images Curie Academy, July 15, 2015 NAME: NAME: TA SIGN-OFFS Exercise 7 Exercise 13 Exercise 17 Making 8x8 pictures Compressing a grayscale image Satellite image debanding

### Creating Color Combos

THE 2016 ROSENTHAL PRIZE for Innovation in Math Teaching Creating Color Combos Visual Modeling of Equivalent Ratios Traci Jackson Lesson Plan Grades 5-6 Table of Contents Creating Color Combos: Visual

### Different aspects of MAthematics

Different aspects of MAthematics Tushar Bhardwaj, Nitesh Rawat Department of Electronics and Computer Science Engineering Dronacharya College of Engineering, Khentawas, Farrukh Nagar, Gurgaon, Haryana

### Optimization In Operations Research (2nd Edition) PDF

Optimization In Operations Research (2nd Edition) PDF Developing skills and intuitions through accessible optimization models and analysis. Rardinâ s Optimization in Operations Research, Second Edition

### Text Calculus Early Transcendentals By Jon Rogawski W

TEXT CALCULUS EARLY TRANSCENDENTALS BY JON ROGAWSKI W PDF - Are you looking for text calculus early transcendentals by jon rogawski w Books? Now, you will be happy that at this time text calculus early