MATH 132: CALCULUS II SYLLABUS


 Jewel Bond
 3 years ago
 Views:
Transcription
1 MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early Transcendentals, vol., Customized for University of Massachusetts Amherst, by James Stewart, Brooks/ColeThompson Learning, General Course Description: Math 32 continues the study of singlevariable calculus. It deals with definite and indefinite integrals; infinite sequences and series; and plane curves whose x and y coordinates are functions of another variable such as time. The central concepts are: accumulated (net) change, as realized by the notion of definite integral; successively better approximations of functions by polynomials, as represented by the concept of power series. The emphasis is on on problemsolving and understanding concepts rather than on proving theorems. Learning goals for Calculus II (Math 32): () Continue to become a competent user of calculus. (2) Continue to develop problemsolving skills, especially in formulating verbal descriptions as mathematical problems and in constructing long, multistep solution. (3) Continue to develop ability to write wellorganized, coherent solutions to problems. (4) Become adept at computing indefinite integrals symbolically through use of basic methods. (5) Be able to formulate as a definite integral a problem about net change in a varying quantity. (6) Become familiar with the process of successive approximations to a quantity or a function. Required topics: A. The definite integral (Note: Items & 2 are now covered in Math 3, but are included here for completeness. This material should be quickly reviewed.). Areas and distances: approximation by sums, leading to The definite integral as a limit of Riemann sums. Leftendpoint, rightendpoint, and midpoint Riemann sums. Definition of definite integral as limit of Riemann sums. Applying the definition for a linear function. Calculating integrals of special functions by using geometry, e.g., b (m x + k) dx and a r r2 x r 2 dx. Linearity, endpointadditivity, and comparison properties of definite integrals. 3. The Fundamental Theorem of Calculus (FTC). Area and other functions of the form F (x) = x f(t) dt. a Statement of the FTC. At least an intuitive justification, or a plausibility argument, for the FTC. Using FTC to evaluate definite integrals. 4. Indefinite integrals. The f(x) dx notation. Indefinite integrals corresponding to derivatives of powers (including nonintegral powers) and of basic elementary transcendental functions. 5. The Net Change Theorem: statement and uses. 6. Integration by substitution (in both indefinite and definite integrals). B. Applications of integration. Applications to geometry. Area between curves. Volumes by slicing perpendicular to a line and, as a special case, volume of solid of revolution. (Not volume by shells.)
2 2 2. (If time permits.) One or two nongeometric applications, to be chosen by the course chair or individual instructors from topics such as those listed below. No more than one week should be devoted to this; this could even be done as individual or group projects. Owing to the conceptual physical understanding required, the fluid pressure and center of mass are not suggested here. Work (physics). Average value of a function (with concrete scientific instances). Consumer surplus (economics). Blood flow or cardiac output (biological science). Probability density functions (sciences and engineering). C. Methods of integration. Techniques of symbolic integration. Integration by parts, including repeated integration by parts and examples leading to equations of the form f(x) dx = g(x) + c f(x) dx. (If time permits. Trigonometric integrals and their application in trigonometric substitutions. Integration by partial fractions of a rational function whose denominator factors into two distinct linear factors. Use of recursion formulas for integrals. Note: It is suggested that calculator and/or computer technology capable of doing symbolic integration be demonstrated in order to show how integration is often done in practice and to indicate that named, nonelementary functions arise as antiderivatives. 2. Approximate integration. Midpoint, Trapezoidal, and Simpson s Rules. Qualitative comparison of the methods accuracy (but not bounds on the error). 3. Improper integrals: infinite endpoints and discontinuous integrands. D. Series and power series Note: This topic is placed here, before parametric equations, to ensure that the essential topics of power series expansions and approximation by Taylor polynomials are reached.. Sequences and limits of sequences: meaning of sequential limit; algebraic Limit Laws; the Squeeze Theorem and the Monotonic Convergence Theorem (statements and use). 2. Series Notions of convergence and sum of an infinite series. Geometric series and application to rational values of repeating decimals. The nth Term Test for divergence. 3. Testing series of constants for convergence Note: In this course, the important thing is power series and approximation by Taylor polynomials and not series of constants. So testing for convergence should introduce the notion of bounding the error in approximating the sum of a series by a partial sum and should emphasize those methods that are: (i) needed to establish convergence or divergence of standard examples such as harmonic, alternating harmonic, and pseries; (ii) most relevant to finding the radius of convergence of power series; and/or (iii) needed in order to justify the Ratio Test. The emphasis should be on examples that are simple rather than artificial or technically complicated. The Integral Test and bounds on the error. (Note: It is not clear that the Integral Test in its full generality is actually appropriate. It could suffice to use the argument behind that test to test pseries.) The Comparison Test (but not the Limit Comparison Test). The Alternating Series Test and bounding the error of the nth partial sum. Absolute convergence implies convergence (but omit terminology of absolute convergence and conditional convergence). The Ratio Test (but not the Root Test). 4. Power series. 5. Representation of functions as power series.
3 Examples derived from geometric series. Termbyterm differentiation and integration. 6. Taylor series (and Maclaurin series). Using the definition to find Taylor series. Uniqueness of Taylor series expansion: A power series expansion of a function is its Taylor series. Standard examples: Maclaurin series for e x, sin x, cos x, arctan x [and perhaps also for ln( + x)]. Approximating functions by Taylor polynomials; Taylor s Inequality for error bounds. Approximating values of functions using Taylor polynomials, with error bounds in some cases. E. Parametric equations and polar equations.. Curves defined by parametric equations. Graphs of parametric equations. Elimination of the parameter. Arc length of parametric curves. (But not tangents to parametric curves or area and surface area calculations involving parametric curves.) 2. Using polar coordinates. Polar equations of graphs. Conversion between polar and rectangular equations. Arc length in polar coordinates. (But not area in terms of polar coordinates.) 3
4 4 Representative problems to solve: These problems are intended strictly to suggest the level and coverage of the course; they are not meant as a template for exam questions. () Evaluate without technology or a Table of Integrals: (a) (cos x + 5 sin x) dx (f) ( ) x 2 + x dx (g) 2 (c) x + 4 dx (h) (d) x e 2 x dx (i) sin x (e) + cos 2 x dx (j) x + 3 x 2 3 x + 2 dx x 2 sin x dx 2 x 4 x 2 dx tan 3 x sec 5 x dx e 2 x e x 2 dx (2) (a) Approximate 0 e x2 dx by the Riemann sum with n = 4 subintervals and left endpoints as sample points. Is this Riemann sum an overestimate or an underestimate of the exact value? Approximate the same integral by using Simpson s Rule with n = 4 subintervals. (3) Find the area of the bounded region enclosed by the curves y = x 2 9 and y = 9 x 2. (4) Find the volume of the solid obtained by rotating around the xaxis the region bounded by the curve y = tan x and the lines y = 0 and x = π/4. (5) Determine the following derivatives. ( (a) sin ( 3 e t + t ) ) 32 d t d d t ( π d d t sin ( 3 e x + x ) ) 32 d x t 3 (6) The rate r at which people become ill with the flu at time t days after an epidemic begins is given by r = 000 t e t/20 people per day. How many people become ill with the flu during the first 0 days of this epidemic? (7) Does the improper integral converge or diverge, and why? If it converges, find its value. (a) 2 0 x 4 x 2 dx 0 arctan x x 2 + dx (8) Does the series converge? Why or why not? (a) n 3 n + n 4 (c) 4 n (d) (e) n = 2 ( ) n n 2 n (ln n) 2 (9) The sequence {S n } n= of partial sums of the series n= a n is given by S n = n/(5 + n) (for, 2, 3,... ). (a) Does the series n= a n converge and, if so, to what sum? Use the formula for S n to find an explicit formula for a n in terms of n. (0) Find the largest interval on which the power series n = 0 (x ) n (n + ) 3 n converges. () Find a power series representation for f(x) = 4/( + 2 x) around a = 0; write the power series using sigma ( ) notation. State for which x the power series actually has sum f(x).
5 5 (2) Calculate the degree 3 Taylor polynomial T 3 (x) of g(x) = / x around a =. (3) (a) Starting with the Maclaurin series for e x (which you may just write down), obtain a power series representation of e x2. Use your answer to (a) to express 0 e x2 dx as the sum of an infinite series. (c) Find a bound on the error if the first three (nonzero) terms of the series you obtained in were used to approximate 0 e x2 dx. (d) How many (nonzero) terms of the series obtained in would you need to use in order to approximate 0 e x2 dx with an error at most 0 5? (Do not actually obtain that approximation!) (4) Find all points at which the curve with parametric equations x = t 2 +, y = t 3 t has a horizontal tangent. (5) As the parameter t increases forever, starting at t = 0, the curve with parametric equations { x = e t cos t, y = e t sin t spirals inward toward the origin, getting ever closer to the origin (but never actually reaching it) as t. Find the length of this entire spiral curve. (6) (a) Find a Cartesiancoordinate equation for the curve having polar equation r = 2 cos θ. Sketch together the curves with polar equations r = 2 cos θ and r =. (c) Find some polar coordinates of each point where the curves intersect.
Student Performance Q&A:
Student Performance Q&A: 2008 AP Calculus AB and Calculus BC FreeResponse Questions The following comments on the 2008 freeresponse questions for AP Calculus AB and Calculus BC were written by the Chief
More informationAP Calculus BC 2012 FreeResponse Questions
AP Calculus BC 0 FreeResponse Questions About the College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded in
More informationAP Calculus AB Syllabus
Course Overview and Philosophy AP Calculus AB Syllabus The biggest idea in AP Calculus is the connections among the representations of the major concepts graphically, numerically, analytically, and verbally.
More informationAP Calculus BC. Course content and suggested texts and reference materials align with the College Board framework for AP Calculus BC.
AP Calculus BC Course Overview Topic Description AP Calculus BC Course Details In AP Calculus BC, students study functions, limits, derivatives, integrals, and infinite series This document details the
More informationPractice Final Math 122 Spring 12 Instructor: Jeff Lang
Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6
More informationCalculus AB 2014 Scoring Guidelines
P Calculus B 014 Scoring Guidelines 014 The College Board. College Board, dvanced Placement Program, P, P Central, and the acorn logo are registered trademarks of the College Board. P Central is the official
More information2008 AP Calculus AB Multiple Choice Exam
008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus
More informationEstimating the Average Value of a Function
Estimating the Average Value of a Function Problem: Determine the average value of the function f(x) over the interval [a, b]. Strategy: Choose sample points a = x 0 < x 1 < x 2 < < x n 1 < x n = b and
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationAP Calculus BC 2012 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded in 9, the College
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
More informationCreating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities
Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned
More informationCourse outline, MA 113, Spring 2014 Part A, Functions and limits. 1.1 1.2 Functions, domain and ranges, A1.11.2Review (9 problems)
Course outline, MA 113, Spring 2014 Part A, Functions and limits 1.1 1.2 Functions, domain and ranges, A1.11.2Review (9 problems) Functions, domain and range Domain and range of rational and algebraic
More informationInvestigating Area Under a Curve
Mathematics Investigating Area Under a Curve About this Lesson This lesson is an introduction to areas bounded by functions and the xaxis on a given interval. Since the functions in the beginning of the
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationAP CALCULUS AB 2008 SCORING GUIDELINES
AP CALCULUS AB 2008 SCORING GUIDELINES Question 1 Let R be the region bounded by the graphs of y = sin( π x) and y = x 4 x, as shown in the figure above. (a) Find the area of R. (b) The horizontal line
More informationPrecalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES
Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 01 Sets There are no statemandated Precalculus 02 Operations
More informationAlgebra I Credit Recovery
Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,
More informationcorrectchoice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
More informationMath Placement Test Sample Problems PREALGEBRA
Math Placement Test Sample Problems The Math Placement Test is an untimed, multiplechoice, computerbased test. The test is composed of four sections: prealgebra, algebra, college algebra, and trigonometry.
More informationDRAFT. Further mathematics. GCE AS and A level subject content
Further mathematics GCE AS and A level subject content July 2014 s Introduction Purpose Aims and objectives Subject content Structure Background knowledge Overarching themes Use of technology Detailed
More informationPCHS ALGEBRA PLACEMENT TEST
MATHEMATICS Students must pass all math courses with a C or better to advance to the next math level. Only classes passed with a C or better will count towards meeting college entrance requirements. If
More informationAP Calculus AB 2011 Scoring Guidelines
AP Calculus AB Scoring Guidelines The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity. Founded in 9, the
More informationMath Course Descriptions & Student Learning Outcomes
Math Course Descriptions & Student Learning Outcomes Table of Contents MAC 100: Business Math... 1 MAC 101: Technical Math... 3 MA 090: Basic Math... 4 MA 095: Introductory Algebra... 5 MA 098: Intermediate
More informationSequences and Series
Sequences and Series Consider the following sum: 2 + 4 + 8 + 6 + + 2 i + The dots at the end indicate that the sum goes on forever. Does this make sense? Can we assign a numerical value to an infinite
More informationAP Calculus AB 2012 FreeResponse Questions
AP Calculus AB 1 FreeResponse Questions About the College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded in
More informationMath 1B, lecture 5: area and volume
Math B, lecture 5: area and volume Nathan Pflueger 6 September 2 Introduction This lecture and the next will be concerned with the computation of areas of regions in the plane, and volumes of regions in
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationMATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab
MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring noncourse based remediation in developmental mathematics. This structure will
More informationMATH. ALGEBRA I HONORS 9 th Grade 12003200 ALGEBRA I HONORS
* Students who scored a Level 3 or above on the Florida Assessment Test Math Florida Standards (FSAMAFS) are strongly encouraged to make Advanced Placement and/or dual enrollment courses their first choices
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationRepresentation of functions as power series
Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions
More informationDerive 5: The Easiest... Just Got Better!
Liverpool John Moores University, 115 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de Technologie Supérieure, Canada Email; mbeaudin@seg.etsmtl.ca 1. Introduction Engineering
More informationMATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Prealgebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationExamples of Tasks from CCSS Edition Course 3, Unit 5
Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationThe Fourth International DERIVETI92/89 Conference Liverpool, U.K., 1215 July 2000. Derive 5: The Easiest... Just Got Better!
The Fourth International DERIVETI9/89 Conference Liverpool, U.K., 5 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de technologie supérieure 00, rue NotreDame Ouest Montréal
More informationName: ID: Discussion Section:
Math 28 Midterm 3 Spring 2009 Name: ID: Discussion Section: This exam consists of 6 questions: 4 multiple choice questions worth 5 points each 2 handgraded questions worth a total of 30 points. INSTRUCTIONS:
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
More informationAP Calculus BC 2006 FreeResponse Questions
AP Calculus BC 2006 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationSouth Carolina College and CareerReady (SCCCR) Algebra 1
South Carolina College and CareerReady (SCCCR) Algebra 1 South Carolina College and CareerReady Mathematical Process Standards The South Carolina College and CareerReady (SCCCR) Mathematical Process
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationAppendix 3 IB Diploma Programme Course Outlines
Appendix 3 IB Diploma Programme Course Outlines The following points should be addressed when preparing course outlines for each IB Diploma Programme subject to be taught. Please be sure to use IBO nomenclature
More informationPrentice Hall Algebra 2 2011 Correlated to: Colorado P12 Academic Standards for High School Mathematics, Adopted 12/2009
Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationAP Calculus AB 2007 FreeResponse Questions
AP Calculus AB 2007 FreeResponse Questions The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to
More informationMEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:
MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an
More informationMATHEMATICS & STATISTICS
Area: Mathematics Dean: Nancy Reitz, Interim Phone: (916) 4848215 Counseling: (916) 4848572 Mathematics Degree The A.S. degree in mathematics provides a foundation of mathematics for students in preparation
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:1512:05. Exam 1 will be based on: Sections 12.112.5, 14.114.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationThis unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationSchool of Mathematics, Computer Science and Engineering. Mathematics* Associate in Arts Degree COURSES, PROGRAMS AND MAJORS
Mathematics School of Mathematics, Computer Science and Engineering Dean: Lianna Zhao, MD Academic Chair: Miriam Castroconde Faculty: Miriam Castroconde; Terry Cheng; Howard Dachslager, PhD; Ilknur Erbas
More informationMATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!
MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Prealgebra Algebra Precalculus Calculus Statistics
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 20102011 School Year The timeframes listed on this calendar are estimates based on a fiftyminute class period. You may need to adjust some of them from time to
More informationMath 19B (Online) Calculus for Science Engineering and Mathematics University of California Santa Cruz
Math 19B (Online) Calculus for Science Engineering and Mathematics University of California Santa Cruz Instructors and Course Creators Tony Tromba, PhD, Distinguished Professor, Mathematics Department,
More information7.6 Approximation Errors and Simpson's Rule
WileyPLUS: Home Help Contact us Logout HughesHallett, Calculus: Single and Multivariable, 4/e Calculus I, II, and Vector Calculus Reading content Integration 7.1. Integration by Substitution 7.2. Integration
More informationMark Howell Gonzaga High School, Washington, D.C.
Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationAP Calculus AB 2011 FreeResponse Questions
AP Calculus AB 11 FreeResponse Questions About the College Board The College Board is a missiondriven notforprofit organization that connects students to college success and opportunity. Founded in
More informationMathematics INDIVIDUAL PROGRAM INFORMATION 2014 2015. 866.Macomb1 (866.622.6621) www.macomb.edu
Mathematics INDIVIDUAL PROGRAM INFORMATION 2014 2015 866.Macomb1 (866.622.6621) www.macomb.edu Mathematics PROGRAM OPTIONS CREDENTIAL TITLE CREDIT HOURS REQUIRED NOTES Associate of Arts Mathematics 62
More informationTrigonometric Functions and Equations
Contents Trigonometric Functions and Equations Lesson 1 Reasoning with Trigonometric Functions Investigations 1 Proving Trigonometric Identities... 271 2 Sum and Difference Identities... 276 3 Extending
More informationHIGH SCHOOL: GEOMETRY (Page 1 of 4)
HIGH SCHOOL: GEOMETRY (Page 1 of 4) Geometry is a complete college preparatory course of plane and solid geometry. It is recommended that there be a strand of algebra review woven throughout the course
More informationAP Calculus AB 2003 Scoring Guidelines Form B
AP Calculus AB Scoring Guidelines Form B The materials included in these files are intended for use by AP teachers for course and exam preparation; permission for any other use must be sought from the
More informationArea Under the Curve. Riemann Sums And the Trapezoidal Rule
Area Under the Curve Riemann Sums And the Trapezoidal Rule Who knew that D=R x T would connect to velocity, and now integration, and the area under a curve? Take a look at the attached applications. Let
More informationMath Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
More informationAn important theme in this book is to give constructive definitions of mathematical objects. Thus, for instance, if you needed to evaluate.
Chapter 10 Series and Approximations An important theme in this book is to give constructive definitions of mathematical objects. Thus, for instance, if you needed to evaluate 1 0 e x2 dx, you could set
More informationDiablo Valley College Catalog 20142015
Mathematics MATH Michael Norris, Interim Dean Math and Computer Science Division Math Building, Room 267 Possible career opportunities Mathematicians work in a variety of fields, among them statistics,
More informationCredit Number Lecture Lab / Shop Clinic / Coop Hours. MAC 224 Advanced CNC Milling 1 3 0 2. MAC 229 CNC Programming 2 0 0 2
MAC 224 Advanced CNC Milling 1 3 0 2 This course covers advanced methods in setup and operation of CNC machining centers. Emphasis is placed on programming and production of complex parts. Upon completion,
More informationLies My Calculator and Computer Told Me
Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing
More informationA power series about x = a is the series of the form
POWER SERIES AND THE USES OF POWER SERIES Elizabeth Wood Now we are finally going to start working with a topic that uses all of the information from the previous topics. The topic that we are going to
More informationPrentice Hall: Middle School Math, Course 1 2002 Correlated to: New York Mathematics Learning Standards (Intermediate)
New York Mathematics Learning Standards (Intermediate) Mathematical Reasoning Key Idea: Students use MATHEMATICAL REASONING to analyze mathematical situations, make conjectures, gather evidence, and construct
More informationAP Calculus AB 2006 Scoring Guidelines
AP Calculus AB 006 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a notforprofit membership association whose mission is to connect students to college
More informationPRECALCULUS GRADE 12
PRECALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More informationDELAWARE MATHEMATICS CONTENT STANDARDS GRADES 910. PAGE(S) WHERE TAUGHT (If submission is not a book, cite appropriate location(s))
Prentice Hall University of Chicago School Mathematics Project: Advanced Algebra 2002 Delaware Mathematics Content Standards (Grades 910) STANDARD #1 Students will develop their ability to SOLVE PROBLEMS
More informationMath Placement Test Study Guide. 2. The test consists entirely of multiple choice questions, each with five choices.
Math Placement Test Study Guide General Characteristics of the Test 1. All items are to be completed by all students. The items are roughly ordered from elementary to advanced. The expectation is that
More informationOverview. Essential Questions. Precalculus, Quarter 4, Unit 4.5 Build Arithmetic and Geometric Sequences and Series
Sequences and Series Overview Number of instruction days: 4 6 (1 day = 53 minutes) Content to Be Learned Write arithmetic and geometric sequences both recursively and with an explicit formula, use them
More informationGeorgia Standards of Excellence 20152016 Mathematics
Georgia Standards of Excellence 20152016 Mathematics Standards GSE Coordinate Algebra K12 Mathematics Introduction Georgia Mathematics focuses on actively engaging the student in the development of mathematical
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationPreCalculus Semester 1 Course Syllabus
PreCalculus Semester 1 Course Syllabus The Plano ISD eschool Mission is to create a borderless classroom based on a positive studentteacher relationship that fosters independent, innovative critical
More informationFor example, estimate the population of the United States as 3 times 10⁸ and the
CCSS: Mathematics The Number System CCSS: Grade 8 8.NS.A. Know that there are numbers that are not rational, and approximate them by rational numbers. 8.NS.A.1. Understand informally that every number
More informationIn mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.
MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target
More informationCourse Outlines. 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit)
Course Outlines 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit) This course will cover Algebra I concepts such as algebra as a language,
More informationPerformance Level Descriptors Grade 6 Mathematics
Performance Level Descriptors Grade 6 Mathematics Multiplying and Dividing with Fractions 6.NS.12 Grade 6 Math : SubClaim A The student solves problems involving the Major Content for grade/course with
More informationPHILOSOPHY OF THE MATHEMATICS DEPARTMENT
PHILOSOPHY OF THE MATHEMATICS DEPARTMENT The Lemont High School Mathematics Department believes that students should develop the following characteristics: Understanding of concepts and procedures Building
More informationMATHEMATICS (MATH) 3. Provides experiences that enable graduates to find employment in sciencerelated
194 / Department of Natural Sciences and Mathematics MATHEMATICS (MATH) The Mathematics Program: 1. Provides challenging experiences in Mathematics, Physics, and Physical Science, which prepare graduates
More informationAP Calculus BC 2001 FreeResponse Questions
AP Calculus BC 001 FreeResponse Questions The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must
More informationMathematics. Mathematics MATHEMATICS. 298 201516 Sacramento City College Catalog. Degree: A.S. Mathematics AST Mathematics for Transfer
MATH Degree: A.S. AST for Transfer Division of /Statistics & Engineering Anne E. Licciardi, Dean South Gym 220 9165582202 Associate in Science Degree Program Information The mathematics program provides
More informationOxford Cambridge and RSA Examinations
Oxford Cambridge and RSA Examinations OCR FREE STANDING MATHEMATICS QUALIFICATION (ADVANCED): ADDITIONAL MATHEMATICS 6993 Key Features replaces and (MEI); developed jointly by OCR and MEI; designed for
More informationMATH ADVISEMENT GUIDE
MATH ADVISEMENT GUIDE Recommendations for math courses are based on your placement results, degree program and career interests. Placement score: MAT 001 or MAT 00 You must complete required mathematics
More informationCOMPLEX NUMBERS. a bi c di a c b d i. a bi c di a c b d i For instance, 1 i 4 7i 1 4 1 7 i 5 6i
COMPLEX NUMBERS _4+i _i FIGURE Complex numbers as points in the Arg plane i _i +i i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with
More informationPolynomial Operations and Factoring
Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.
More information