# MATH 132: CALCULUS II SYLLABUS

Save this PDF as:

Size: px
Start display at page:

## 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/Cole-Thompson Learning, General Course Description: Math 32 continues the study of single-variable 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 problem-solving 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 problem-solving skills, especially in formulating verbal descriptions as mathematical problems and in constructing long, multi-step solution. (3) Continue to develop ability to write well-organized, 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. Left-endpoint, right-endpoint, 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, endpoint-additivity, 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 non-integral 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 non-geometric 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, non-elementary 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 p-series; (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 p-series.) 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. Term-by-term 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 over-estimate or an under-estimate 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 x-axis 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 Cartesian-coordinate 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.

### AP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB.

AP Calculus BC Course Description: Advanced Placement Calculus BC is primarily concerned with developing the students understanding of the concepts of calculus and providing experiences with its methods

### Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

### Student Performance Q&A:

Student Performance Q&A: AP Calculus AB and Calculus BC Free-Response Questions The following comments on the free-response questions for AP Calculus AB and Calculus BC were written by the Chief Reader,

### Learning Objectives for Math 165

Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### Student Name: Instructor:

Math 30 - Final Exam Version Student Name: Instructor: INSTRUCTIONS READ THIS NOW Print your name in CAPITAL letters. It is your responsibility that your test paper is submitted to your professor. Please

### Jefferson College CTL GUIDELINES TO CONSIDER WHEN CREATING EXPECTED LEARNING OUTCOMES

GUIDELINES TO CONSIDER WHEN CREATING EXPECTED LEARNING OUTCOMES Source: Assessing Student Learning, by Linda Suskie. The book is available for checkout in the Center for Teaching and Learning. Aim for

### f (x) = x 2 (x) = (1 + x 2 ) 2 (x) = (1 + x 2 ) 2 + 8x 2 (1 + x 2 ) 3 8x (1 + x 2 ) x (1 + x 2 ) 3 48x3 f(0) = tan 1 (0) = 0 f (0) = 2

Math 5 Exam # Practice Problem Solutions. Find the Maclaurin series for tan (x (feel free just to write out the first few terms. Answer: Let f(x = tan (x. Then the first few derivatives of f are: f (x

### AP 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.

### ROCHESTER INSTITUTE OF TECHNOLOGY COURSE OUTLINE FORM COLLEGE OF SCIENCE. School of Mathematical Sciences. Revised COURSE: COS-MATH-173 Calculus C

! ROCHESTER INSTITUTE OF TECHNOLOGY COURSE OUTLINE FORM COLLEGE OF SCIENCE School of Mathematical Sciences New Revised COURSE: COS-MATH-173 Calculus C 1.0 Course designations and approvals: Required Course

### Diploma Plus in Certificate in Advanced Engineering

Diploma Plus in Certificate in Advanced Engineering Mathematics New Syllabus from April 2011 Ngee Ann Polytechnic / School of Interdisciplinary Studies 1 I. SYNOPSIS APPENDIX A This course of advanced

### Student Performance Q&A:

Student Performance Q&A: 2008 AP Calculus AB and Calculus BC Free-Response Questions The following comments on the 2008 free-response questions for AP Calculus AB and Calculus BC were written by the Chief

### MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.

MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin

### Math Department Student Learning Objectives Updated April, 2014

Math Department Student Learning Objectives Updated April, 2014 Institutional Level Outcomes: Victor Valley College has adopted the following institutional outcomes to define the learning that all students

### AP 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

### Common Curriculum Map. Discipline: Math Course: College Algebra

Common Curriculum Map Discipline: Math Course: College Algebra August/September: 6A.5 Perform additions, subtraction and multiplication of complex numbers and graph the results in the complex plane 8a.4a

### PURE MATHEMATICS AM 27

AM Syllabus (015): Pure Mathematics AM SYLLABUS (015) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (015): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)

### PURE MATHEMATICS AM 27

AM SYLLABUS (013) PURE MATHEMATICS AM 7 SYLLABUS 1 Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics and

### Maplets for Calculus Tutoring without the Tutor 1

Maplets for Calculus Tutoring without the Tutor 1 Douglas B. Meade meade@math.sc.edu Department of Mathematics University of South Carolina Columbia, SC U.S.A. Philip B. Yasskin yasskin@math.tamu.edu Department

### Prep for Calculus. Curriculum

Prep for Calculus This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular

### Advanced Higher Mathematics Course Assessment Specification (C747 77)

Advanced Higher Mathematics Course Assessment Specification (C747 77) Valid from August 2015 This edition: April 2016, version 2.4 This specification may be reproduced in whole or in part for educational

### MATHEMATICS 31 A. COURSE OVERVIEW RATIONALE

MATHEMATICS 31 A. COURSE OVERVIEW RATIONALE To set goals and make informed choices, students need an array of thinking and problem-solving skills. Fundamental to this is an understanding of mathematical

### Estimating 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

### Curriculum Map. Discipline: Math Course: AP Calculus AB Teacher: Louis Beuschlein

Curriculum Map Discipline: Math Course: AP Calculus AB Teacher: Louis Beuschlein August/September: State: 8.B.5, 8.C.5, 8.D.5 What is a limit? What is a derivative? What role do derivatives and limits

### AP Calculus AB 1998 Scoring Guidelines

AP Calculus AB 1998 Scoring Guidelines These materials are intended for non-commercial use by AP teachers for course and exam preparation; permission for any other use must be sought from the Advanced

### Practice 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

### Lecture 24: Series finale

Lecture 24: Series finale Nathan Pflueger 2 November 20 Introduction This lecture will discuss one final convergence test and error bound for series, which concerns alternating series. A general summary

### Georgia 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

### Course Syllabus. Math Calculus I. Revision Date: 8/15/2016

Course Syllabus Math 2413- Calculus I Revision Date: 8/15/2016 Catalog Description: Limits and continuity; the Fundamental Theorem of Calculus; definition of the derivative of a function and techniques

### AP Calculus BC 2012 Free-Response Questions

AP Calculus BC 0 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in

### AP Calculus AB. Practice Exam. Advanced Placement Program

Advanced Placement Program AP Calculus AB Practice Exam The questions contained in this AP Calculus AB Practice Exam are written to the content specifications of AP Exams for this subject. Taking this

### Power Series. We already know how to express one function as a series. Take a look at following equation: 1 1 r = r n. n=0

Math -0 Calc Power, Taylor, and Maclaurin Series Survival Guide One of the harder concepts that we have to become comfortable with during this semester is that of sequences and series We are working with

### Taylor 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

### 6.8 Taylor and Maclaurin s Series

6.8. TAYLOR AND MACLAURIN S SERIES 357 6.8 Taylor and Maclaurin s Series 6.8.1 Introduction The previous section showed us how to find the series representation of some functions by using the series representation

### MATH FINAL EXAMINATION - 3/22/2012

MATH 22 - FINAL EXAMINATION - /22/22 Name: Section number: About this exam: Partial credit will be given on the free response questions. To get full credit you must show all of your work. This is a closed

### find the instantaneous rate of change of a function and connect it with the slope of the tangent line

page of 9 Unit : Limits and Continuity 20 00 5 9 Totals Always Include 2 blocks for Review & Test chapter 2 District Google Documents site What is a limit? 4 2 What does it mean for a function to be continuous?

### 2008 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

### Solutions 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

### EQ: How can regression models be used to display and analyze the data in our everyday lives?

School of the Future Math Department Comprehensive Curriculum Plan: ALGEBRA II (Holt Algebra 2 textbook as reference guide) 2016-2017 Instructor: Diane Thole EU for the year: How do mathematical models

### MATH 2300 review problems for Exam 3 ANSWERS

MATH 300 review problems for Exam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justify your answer by either computing the sum or by by showing which convergence test

### Creating, 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

### Infinite series, improper integrals, and Taylor series

Chapter Infinite series, improper integrals, and Taylor series. Introduction This chapter has several important and challenging goals. The first of these is to understand how concepts that were discussed

### Definition. Sequences. Sequence As A Function. Finding the n th Term. Converging Sequences. Divergent Sequences 12/13/2010. A of numbers. Lesson 9.

Definition Sequences Lesson 9.1 A of numbers Listed according to a given Typically written as a 1, a 2, a n Often shortened to { a n } Example 1, 3, 5, 7, 9, A sequence of numbers Finding the n th Term

### AP Calculus BC 2012 Scoring Guidelines

AP Calculus BC Scoring Guidelines The College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 9, the College

### Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012

Some series converge, some diverge. Series Convergence Tests Math 22 Calculus III D Joyce, Fall 202 Geometric series. We ve already looked at these. We know when a geometric series converges and what it

### Investigating 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 x-axis on a given interval. Since the functions in the beginning of the

### 1 Mathematical Induction

Extra Credit Homework Problems Note: these problems are of varying difficulty, so you might want to assign different point values for the different problems. I have suggested the point values each problem

### CHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often

7 CHAPTER Definite Integrals Since integration can be used in a practical sense in many applications it is often useful to have integrals evaluated for different values of the variable of integration.

### Precalculus 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 0-1 Sets There are no state-mandated Precalculus 0-2 Operations

### Calculus 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

### x 2 would be a solution to d y

CATHOLIC JUNIOR COLLEGE H MATHEMATICS JC PRELIMINARY EXAMINATION PAPER I 0 System of Linear Equations Assessment Objectives Solution Feedback To use a system of linear c equations to model and solve y

### correct-choice 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

### Course outline, MA 113, Spring 2014 Part A, Functions and limits. 1.1 1.2 Functions, domain and ranges, A1.1-1.2-Review (9 problems)

Course outline, MA 113, Spring 2014 Part A, Functions and limits 1.1 1.2 Functions, domain and ranges, A1.1-1.2-Review (9 problems) Functions, domain and range Domain and range of rational and algebraic

### South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus 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

### If f is continuous on [a, b], then the function g defined by. f (t) dt. is continuous on [a, b] and differentiable on (a, b), and g (x) = f (x).

The Fundamental Theorem of Calculus, Part 1 If f is continuous on [a, b], then the function g defined by g(x) = x a f (t) dt a x b is continuous on [a, b] and differentiable on (a, b), and g (x) = f (x).

Algebra I 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, with an emphasis

### Math 41: Calculus Final Exam December 7, 2009

Math 41: Calculus Final Exam December 7, 2009 Name: SUID#: Select your section: Atoshi Chowdhury Yuncheng Lin Ian Petrow Ha Pham Yu-jong Tzeng 02 (11-11:50am) 08 (10-10:50am) 04 (1:15-2:05pm) 03 (11-11:50am)

### Applications of Integration Day 1

Applications of Integration Day 1 Area Under Curves and Between Curves Example 1 Find the area under the curve y = x2 from x = 1 to x = 5. (What does it mean to take a slice?) Example 2 Find the area under

### Algebra 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,

### MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

### EXERCISES FOR CHAPTER 6: Taylor and Maclaurin Series

EXERCISES FOR CHAPTER 6: Taylor and Maclaurin Series. Find the first 4 terms of the Taylor series for the following functions: (a) ln centered at a=, (b) centered at a=, (c) sin centered at a = 4. (a)

### Number and Numeracy SE/TE: 43, 49, 140-145, 367-369, 457, 459, 479

Ohio Proficiency Test for Mathematics, New Graduation Test, (Grade 10) Mathematics Competencies Competency in mathematics includes understanding of mathematical concepts, facility with mathematical skills,

### Objectives. Materials

Activity 5 Exploring Infinite Series Objectives Identify a geometric series Determine convergence and sum of geometric series Identify a series that satisfies the alternating series test Use a graphing

### High School Algebra 1 Common Core Standards & Learning Targets

High School Algebra 1 Common Core Standards & Learning Targets Unit 1: Relationships between Quantities and Reasoning with Equations CCS Standards: Quantities N-Q.1. Use units as a way to understand problems

### Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below.

Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School

### ALGEBRA 1/ALGEBRA 1 HONORS

ALGEBRA 1/ALGEBRA 1 HONORS CREDIT HOURS: 1.0 COURSE LENGTH: 2 Semesters COURSE DESCRIPTION The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical

### NEW 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

### Chapter 1 Quadratic Equations in One Unknown (I)

Tin Ka Ping Secondary School 015-016 F. Mathematics Compulsory Part Teaching Syllabus Chapter 1 Quadratic in One Unknown (I) 1 1.1 Real Number System A Integers B nal Numbers C Irrational Numbers D Real

### Items related to expected use of graphing technology appear in bold italics.

- 1 - Items related to expected use of graphing technology appear in bold italics. Investigating the Graphs of Polynomial Functions determine, through investigation, using graphing calculators or graphing

### Prep for College Algebra

Prep for College Algebra This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet

### CALCULUS 2. 0 Repetition. tutorials 2015/ Find limits of the following sequences or prove that they are divergent.

CALCULUS tutorials 5/6 Repetition. Find limits of the following sequences or prove that they are divergent. a n = n( ) n, a n = n 3 7 n 5 n +, a n = ( n n 4n + 7 ), a n = n3 5n + 3 4n 7 3n, 3 ( ) 3n 6n

### MATHEMATICS DEPARTMENT

MATHEMATICS DEPARTMENT All students are required to take four credits in mathematics, including one credit in Algebra and one credit in Geometry. Students advancing to a Maryland State College or University

### 1. Students will demonstrate an understanding of the real number system as evidenced by classroom activities and objective tests

MATH 102/102L Inter-Algebra/Lab Properties of the real number system, factoring, linear and quadratic equations polynomial and rational expressions, inequalities, systems of equations, exponents, radicals,

### The graphs of f and g intersect at (0, 0) and one other point. Find that point: f(y) = g(y) y 2 4y 2y 2 6y = = 2y y 2. 2y(y 3) = 0

. Compute the area between the curves x y 4y and x y y. Let f(y) y 4y y(y 4). f(y) when y or y 4. Let g(y) y y y( y). g(y) when y or y. x 3 y? The graphs of f and g intersect at (, ) and one other point.

### Definite Integrals and Riemann Sums

MTH229 Project 1 Exercises Definite Integrals and Riemann Sums NAME: SECTION: INSTRUCTOR: Exercise 1: a. Create a script m-file containing the content above, only change the value of n to 2,5. What is

### PowerTeaching i3: Algebra I Mathematics

PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I Key Ideas and

### Representation 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

### PCHS 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

### Georgia Department of Education. Calculus

K-12 Mathematics Introduction Calculus The Georgia Mathematics Curriculum focuses on actively engaging the students in the development of mathematical understanding by using manipulatives and a variety

### MATH. 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 (FSA-MAFS) are strongly encouraged to make Advanced Placement and/or dual enrollment courses their first choices

### DRAFT. 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

### Middle Grades Mathematics 5 9

Middle Grades Mathematics 5 9 Section 25 1 Knowledge of mathematics through problem solving 1. Identify appropriate mathematical problems from real-world situations. 2. Apply problem-solving strategies

### STUDENT LEARNING OUTCOMES FOR THE SANTIAGO CANYON COLLEGE MATHEMATICS DEPARTMENT (Last Revised 8/20/14)

STUDENT LEARNING OUTCOMES FOR THE SANTIAGO CANYON COLLEGE MATHEMATICS DEPARTMENT (Last Revised 8/20/14) Department SLOs: Upon completion of any course in Mathematics the student will be able to: 1. Create

### Sequences 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

### Thnkwell 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

### Math 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

### WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS

Visit this link to read the introductory text for this syllabus. 1. Circular Measure Lengths of Arcs of circles and Radians Perimeters of Sectors and Segments measure in radians 2. Trigonometry (i) Sine,

### Introduction to Series and Sequences Math 121 Calculus II D Joyce, Spring 2013

Introduction to Series and Sequences Math Calculus II D Joyce, Spring 03 The goal. The main purpose of our study of series and sequences is to understand power series. A power series is like a polynomial

### ANALYTICAL MATHEMATICS FOR APPLICATIONS 2016 LECTURE NOTES Series

ANALYTICAL MATHEMATICS FOR APPLICATIONS 206 LECTURE NOTES 8 ISSUED 24 APRIL 206 A series is a formal sum. Series a + a 2 + a 3 + + + where { } is a sequence of real numbers. Here formal means that we don

### MATH 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 non-course based remediation in developmental mathematics. This structure will

### Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

### Course Title: Honors Algebra Course Level: Honors Textbook: Algebra 1 Publisher: McDougall Littell

Course Title: Honors Algebra Course Level: Honors Textbook: Algebra Publisher: McDougall Littell The following is a list of key topics studied in Honors Algebra. Identify and use the properties of operations

### The Natural Logarithmic Function: Integration. Log Rule for Integration

6_5.qd // :58 PM Page CHAPTER 5 Logarithmic, Eponential, and Other Transcendental Functions Section 5. EXPLORATION Integrating Rational Functions Earl in Chapter, ou learned rules that allowed ou to integrate

### Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4)

Chapter 2: Functions and Linear Functions 1. Know the definition of a relation. Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4) 2. Know the definition of a function. 3. What

### Examples 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

### MVE041 Flervariabelanalys

MVE041 Flervariabelanalys 2015-16 This document contains the learning goals for this course. The goals are organized by subject, with reference to the course textbook Calculus: A Complete Course 8th ed.

### Derive 5: The Easiest... Just Got Better!

Liverpool John Moores University, 1-15 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de Technologie Supérieure, Canada Email; mbeaudin@seg.etsmtl.ca 1. Introduction Engineering