Estimating the Average Value of a Function


 Mervyn Alfred Hoover
 1 years ago
 Views:
Transcription
1 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 estimate the average value of f(x) as follows. Let Avg[f(x)] denote the average value of f(x) on [a, b]. Let M i be the maximum value of the function f(x) on the subinterval [x i 1, x i ]. Then Avg[f(x)] 1 b a (M 1 x M n x n ). Let m i be the minimum value of f(x) on the subinterval [x i 1, x i ]. Then 1 b a (m 1 x m n x n ) Avg[f(x)]. Therefore, n m i x i Avg[f(x)] i=1 n M i x i. i=1 The hope is that the average value can be computed to an arbitrary degree of accuracy provided that sufficiently many sample points are chosen.
2 Example: Estimate the average value of f(x) = x 2 on the interval [0, 2]. Suppose one chooses the sample points x 0 = 0 < x 1 = 0.5 < x 2 = 1 < x 3 = 1.5 < x 4 = 2. Then the average value of is less than or equal to 1 b a (M 1 x M n x n ) 1 ( (0.5)2 (0.5) + (1) 2 (0.5) + (1.5) 2 (0.5) + (2) 2 (0.5) ) = That M 1 = (0.5) 2, M 2 = (1) 2, etc. can be observed from the graph of the function. The function f(x) = x 2 is increasing on the interval [0, 2], and so the maximum value of f(x) on each subinterval occurs at the right endpoint. (The maximum value would occur at the left endpoint in the case of a decreasing function. And the maximum value of an arbitrary function could occur almost anywhere within the subinterval.) Figure 1: The graph displays the function f(x) = x 2 on the interval [0, 2]. The values of M 1 x 1, M 2 x 2, etc. correspond to the areas of the blue shaded rectangles.
3 On the other hand, the average value of is greater than or equal to 1 b a (m 1 x m n x n ) 1 ( (0)2 (0.5) + (0.5) 2 (0.5) + (1) 2 (0.5) + (1.5) 2 (0.5) ) = That m 1 = (0) 2, m 2 = (0.5) 2, etc. can also be observed from the graph of the function. The function f(x) = x 2 is increasing on the interval [0, 2], and so the minimum value of f(x) on each subinterval occurs at the left endpoint. Figure 2: The graph displays the function f(x) = x 2 on the interval [0, 2]. The values of m 1 x 1, m 2 x 2, etc. correspond to the areas of the green shaded rectangles. Together, these two estimates yield Avg[f(x)] If the interval [0, 2] is divided into 100 subintervals instead of 4 subintervals, a computer calculation shows that Avg[f(x)]
4 Riemann Sums Definition: A sum of the form n y k x k, k=1 where a = x 0 < x 1 < < x n = b and each y k is some value of f(x) on the kth subinterval [x k 1, x k ] is called a Riemann sum. There are many kinds of Riemann sums depending upon the method used to choose the values of each y k. If we choose y k = M k, the maximum value of f(x) on the kth subinterval, then the resulting Riemann sum is called an upper Riemann sum. If y k = m k, then this is called a lower Riemann sum. The sample points a = x 0 < < x n = b are usually referred to as a partition of the interval [a, b]. It is convenient to record the points in the partition as a set of increasing values: P = {a = x 0 < < x n = b} and then refer to the set P as the partition.
5 Exercise: Let f(x) = x 3 x on the interval [0, 2]. Suppose that we choose sample points as follows 0 = x 0 < 1 = x 1 < 2 = x 2. Determine the values of M 1, M 2, m 1, and m 2. Use these values to compute the values of the upper and lower Riemann sums for this function using the given partition. Then estimate the average value of f(x) on [0, 2].
6 Solution: To find the maximum and minimum values, first find all of the critical points. f (x) = 3x 2 1 = critical points: x = ± 1 3. On the first subinterval, [x 0, x 1 ] = [0, 1], compute the values of f(x) at the end points and at the critical point (which lies in this interval): f(0) = 0, f(1) = 0, f(1/ 3) = (1/ 3) 3 1/ 3 = 1/ = 2/ 3. Therefore, the minimum value on the first subinterval is m 1 = 2/ 3, and the maximum value is M 1 = 0. On the second subinterval, [x 1, x 2 ] = [1, 2], compute the value of f(x) at the end points (and ignore the critical points since neither lies in this interval): f(1) = 0, f(2) = 6. Therefore, m 2 = 0 and M 2 = 6. Figure 3: The graph displays the function f(x) = x 3 x on the interval [0, 2]. The values of M 1 x 1, M 2 x 2, etc. correspond to the areas of the blue shaded rectangles. The first blue rectangle has a height of zero, and so it appears as a line segment from (0, 0) to (1, 0).
7 Figure 4: The graph displays the function f(x) = x 3 x on the interval [0, 2]. The values of m 1 x 1, m 2 x 2, etc. correspond to the green shaded rectangles. The area of the first rectangle is (m 1 x 1 ). since area is a positive quantity and m 1 < 0. The terms in the Riemann sum can (with care) be interpreted as a signed area. The second green rectangle has a height of zero, and so it appears as a line segment from (0, 0) to (1, 0). We now compute the Riemann sums. The upper Riemann sum is M 1 x 1 + M 2 x 2 = 0 (1 0) + 6 (2 1) = 6, and the lower Riemann sum is m 1 x 1 + m 2 x 2 = 2 3 (1 0) + 0 (2 1) = 2 3. Therefore, the average value of f(x) on [0, 2] can be estimated as follows: 1 3 Avg[f(x)] 3. (We multiplied each Riemann sum by 1/(b a) = 1/(2 0) = 1/2.) To improve this estimate, we should choose a finer partition of the interval [0, 2].
8 The Definite Integral The definite integral of f(x) on the interval [a, b] is essentially the limiting value of the upper and lower Riemann sums of a function. However, there are some complications that need to overcome to make this mathematically precise. Complication: Some functions do not have maximum or minimum values. Example: Define f(x) on the interval [0, 1] as follows: let f(x) = 1/x if x > 0 and let f(x) = 0 if x = 0. Then f(x) does not have a maximum value. Figure 5: The function above is defined at every point of the interval [0, 1], but this function does not have a maximum value on [0, 1]. Wild Complication: A function can have a maximum and a minimum on every subinterval, but choosing more sample points, may not improve the estimate. Example: Dirichlet s fuction is defined as follows: D(x) = 0 if x is rational and D(x) = 1 if x is irrational. On any interval, the maximum value of D(x) is one and the minimum value is zero.
9 A Way To Avoid Complications Assume that f(x) is continuous on the interval [a, b]. Then f(x) attains a maximum and a minimum value on any subinterval [x i 1, x i ] [a, b]. Moreover, if f(x) is continuous on the interval [a, b] and if for each positive integer n we choose the sample points a = x 0 < < x n = b so as to divide the interval [a, b] into n subintervals of equal length, then lim n k=1 n m i x i = lim n k=1 n M i x i. Definition: The common limiting value above is called the definite integral or the Riemann integral of f(x) from x = a to x = b. It is denoted by b a f(x) dx. In fact, more is true. If f(x) is continuous, and if n k=1 y k x k is any Riemann sum, then so long as all of the lengths of the subintervals tend to zero, the sequence of Riemann sums will approach the value of b a f(x) dx. This has the practical implication that you can numerically estimate the value of the definite integral using easier to compute Riemann sums (e.g. rightendpoint, leftendpoint, or midpoint Riemann sums). In practice, there are more sophisticated numerical techniques (e.g. Simpson s rule). Despite these nice answer, this is just the beginning of the story. To apply this theory to applications, functions which are not continous need to be considered. This necessitates a refining the theory of the definite integral. The modern approach is to study what is known as the Lebesgue integral.
10 Definite Integrals and Area The value of geometrically motivated definite integrals can be computed using the observation that if f(x) is nonnegative on [a, b], then the upper and lower Riemann sums estimate the area of the region below the curve y = f(x), above the xaxis and bounded on the left by the vertical line x = a and bounded on the right by the vertical line x = b. Figure 6: The function f(x) = xe x on the interval [0.5, 4] is displayed. The area below the curve refers to the region below y = f(x), above the xaxis, and between the vertical lines which pass through the endpoints of the interval. Exercise: Compute the value of each of the following definite integrals by sketching the graph of the integrand (the function inside the integral) and using geometrical formulas to compute the area x dx 2x dx 25 x 2 dx
11 Solutions to the exercise: x dx 2. This integral represents the area below the curve y = 2x. Since this curve is a line, the region below the curve is a triangle. The area of this triangle is (1/2) 5 10 = x dx This integral represents the area of a trapezoid. The area can be computed from the formula for the area of a trapezoid or by using the previous calculation and subtracting the area of the triangle below y = 2x on the interval [0, 3]. The value of the integral above is 25 (1/2) 3 6 = 16.
12 x 2 dx This integral represents the area of one quarter of a circle of radius 5. Therefore, the value of the integral is (1/4) π 5 2 = 25π/4.
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
More informationa b c d e You have two hours to do this exam. Please write your name on this page, and at the top of page three. GOOD LUCK! 3. a b c d e 12.
MA123 Elem. Calculus Fall 2015 Exam 3 20151119 Name: Sec.: Do not remove this answer page you will turn in the entire exam. No books or notes may be used. You may use an ACTapproved calculator during
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 informationMATH 132: CALCULUS II SYLLABUS
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
More informationUse finite approximations to estimate the area under the graph of the function. f(x) = x 3
5.1: 6 Use finite approximations to estimate the area under the graph of the function f(x) = x 3 between x = 0 and x = 1 using (a) a lower sum with two rectangles of equal width (b) a lower sum with four
More information2010 Solutions. a + b. a + b 1. (a + b)2 + (b a) 2. (b2 + a 2 ) 2 (a 2 b 2 ) 2
00 Problem If a and b are nonzero real numbers such that a b, compute the value of the expression ( ) ( b a + a a + b b b a + b a ) ( + ) a b b a + b a +. b a a b Answer: 8. Solution: Let s simplify the
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 information7. The Definite Integral
Section 7: The Definite Integral 7. The Definite Integral The Definite Integral has wide ranging applications in mathematics, the physical sciences and engineering. The theory and application of statistics,
More informationMath 1526 Consumer and Producer Surplus
Math 156 Consumer and Producer Surplus Scenario: In the grocery store, I find that twoliter sodas are on sale for 89. This is good news for me, because I was prepared to pay $1.9 for them. The store manager
More informationFunctions and Equations
Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c
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 informationAPPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS
APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS Now that we are starting to feel comfortable with the factoring process, the question becomes what do we use factoring to do? There are a variety of classic
More informationLearning 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
More informationLesson 3. Numerical Integration
Lesson 3 Numerical Integration Last Week Defined the definite integral as limit of Riemann sums. The definite integral of f(t) from t = a to t = b. LHS: RHS: Last Time Estimate using left and right hand
More informationNotes on Continuous Random Variables
Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes
More informationTaylor 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
More information15.1. Integration as the limit of a sum. Introduction. Prerequisites. Learning Outcomes. Learning Style
Integration as the limit of a sum 15.1 Introduction In Chapter 14, integration was introduced as the reverse of differentiation. A more rigorous treatment would show that integration is a process of adding
More informationArea and Arc Length in Polar Coordinates. Area of a Polar Region
46_5.qxd //4 :7 PM Page 79 SECTION.5 Area and Arc Length in Polar Coordinates 79 θ Section.5 r The area of a sector of a circle is A r. Figure.49 (a) β r = f( θ) α Area and Arc Length in Polar Coordinates
More informationChapter 3 RANDOM VARIATE GENERATION
Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.
More information(b)using the left hand end points of the subintervals ( lower sums ) we get the aprroximation
(1) Consider the function y = f(x) =e x on the interval [, 1]. (a) Find the area under the graph of this function over this interval using the Fundamental Theorem of Calculus. (b) Subdivide the interval
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n1 x n1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationStudent Performance Q&A:
Student Performance Q&A: AP Calculus AB and Calculus BC FreeResponse Questions The following comments on the freeresponse questions for AP Calculus AB and Calculus BC were written by the Chief Reader,
More informationList the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated
MATH 142 Review #1 (4717995) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Description This is the review for Exam #1. Please work as many problems as possible
More informationInformation Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay
Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture  17 ShannonFanoElias Coding and Introduction to Arithmetic Coding
More informationSection 6.4: Work. We illustrate with an example.
Section 6.4: Work 1. Work Performed by a Constant Force Riemann sums are useful in many aspects of mathematics and the physical sciences than just geometry. To illustrate one of its major uses in physics,
More informationMethod To Solve Linear, Polynomial, or Absolute Value Inequalities:
Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with
More informationChapter 4 One Dimensional Kinematics
Chapter 4 One Dimensional Kinematics 41 Introduction 1 4 Position, Time Interval, Displacement 41 Position 4 Time Interval 43 Displacement 43 Velocity 3 431 Average Velocity 3 433 Instantaneous Velocity
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 informationThis function is symmetric with respect to the yaxis, so I will let  /2 /2 and multiply the area by 2.
INTEGRATION IN POLAR COORDINATES One of the main reasons why we study polar coordinates is to help us to find the area of a region that cannot easily be integrated in terms of x. In this set of notes,
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 informationReview Sheet for Third Midterm Mathematics 1300, Calculus 1
Review Sheet for Third Midterm Mathematics 1300, Calculus 1 1. For f(x) = x 3 3x 2 on 1 x 3, find the critical points of f, the inflection points, the values of f at all these points and the endpoints,
More informationCalculus with Parametric Curves
Calculus with Parametric Curves Suppose f and g are differentiable functions and we want to find the tangent line at a point on the parametric curve x f(t), y g(t) where y is also a differentiable function
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 information15.3. Calculating centres of mass. Introduction. Prerequisites. Learning Outcomes. Learning Style
Calculating centres of mass 15.3 Introduction In this block we show how the idea of integration as the limit of a sum can be used to find the centre of mass of an object such as a thin plate, like a sheet
More informationIntegration. Topic: Trapezoidal Rule. Major: General Engineering. Author: Autar Kaw, Charlie Barker. http://numericalmethods.eng.usf.
Integration Topic: Trapezoidal Rule Major: General Engineering Author: Autar Kaw, Charlie Barker 1 What is Integration Integration: The process of measuring the area under a function plotted on a graph.
More informationNumerical integration of a function known only through data points
Numerical integration of a function known only through data points Suppose you are working on a project to determine the total amount of some quantity based on measurements of a rate. For example, you
More informationAsymptotes. Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if
Section 2.1: Vertical and Horizontal Asymptotes Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if, lim x a f(x) =, lim x a x a x a f(x) =, or. + + Definition.
More informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationStatistics Revision Sheet Question 6 of Paper 2
Statistics Revision Sheet Question 6 of Paper The Statistics question is concerned mainly with the following terms. The Mean and the Median and are two ways of measuring the average. sumof values no. of
More informationLecture 21 Integration: Left, Right and Trapezoid Rules
Lecture 1 Integration: Left, Right and Trapezoid Rules The Left and Right point rules In this section, we wish to approximate a definite integral b a f(x)dx, where f(x) is a continuous function. In calculus
More informationWEEK #22: PDFs and CDFs, Measures of Center and Spread
WEEK #22: PDFs and CDFs, Measures of Center and Spread Goals: Explore the effect of independent events in probability calculations. Present a number of ways to represent probability distributions. Textbook
More information2 Applications to Business and Economics
2 Applications to Business and Economics APPLYING THE DEFINITE INTEGRAL 442 Chapter 6 Further Topics in Integration In Section 6.1, you saw that area can be expressed as the limit of a sum, then evaluated
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
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 information10.2 Series and Convergence
10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and
More informationAP CALCULUS AB 2009 SCORING GUIDELINES
AP CALCULUS AB 2009 SCORING GUIDELINES Question 5 x 2 5 8 f ( x ) 1 4 2 6 Let f be a function that is twice differentiable for all real numbers. The table above gives values of f for selected points in
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
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 informationAP 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
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 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 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 informationFIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.
FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
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 informationAP Calculus AB 2003 Scoring Guidelines
AP Calculus AB Scoring Guidelines The materials included in these files are intended for use y AP teachers for course and exam preparation; permission for any other use must e sought from the Advanced
More informationMA123 Elem. Calculus Fall 2013 Exam
MA123 Elem Calculus Fall 2013 Exam 3 20131121 Name: Sec: Do not remove this answer page you will turn in the entire exam You have two hours to do this exam No books or notes may be used You may use a
More informationAP Calculus AB 2010 FreeResponse Questions Form B
AP Calculus AB 2010 FreeResponse Questions Form B The College Board The College Board is a notforprofit membership association whose mission is to connect students to college success and opportunity.
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationWeek #15  Word Problems & Differential Equations Section 8.1
Week #15  Word Problems & Differential Equations Section 8.1 From Calculus, Single Variable by HughesHallett, Gleason, McCallum et. al. Copyright 25 by John Wiley & Sons, Inc. This material is used by
More informationIntroduction. The Aims & Objectives of the Mathematical Portion of the IBA Entry Test
Introduction The career world is competitive. The competition and the opportunities in the career world become a serious problem for students if they do not do well in Mathematics, because then they are
More informationMEASURES OF VARIATION
NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are
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 informationIntroduction to Calculus
Introduction to Calculus Contents 1 Introduction to Calculus 3 11 Introduction 3 111 Origin of Calculus 3 112 The Two Branches of Calculus 4 12 Secant and Tangent Lines 5 13 Limits 10 14 The Derivative
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationRi and. i=1. S i N. and. R R i
The subset R of R n is a closed rectangle if there are n nonempty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an
More informationReal Numbers and Monotone Sequences
Real Numbers and Monotone Sequences. Introduction. Real numbers. Mathematical analysis depends on the properties of the set R of real numbers, so we should begin by saying something about it. There are
More informationSection 33 Approximating Real Zeros of Polynomials
 Approimating Real Zeros of Polynomials 9 Section  Approimating Real Zeros of Polynomials Locating Real Zeros The Bisection Method Approimating Multiple Zeros Application The methods for finding zeros
More informationMATHEMATICS Unit Pure Core 2
General Certificate of Education January 2008 Advanced Subsidiary Examination MATHEMATICS Unit Pure Core 2 MPC2 Wednesday 9 January 2008 1.30 pm to 3.00 pm For this paper you must have: an 8page answer
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More information9.2 Summation Notation
9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a
More informationLogo Symmetry Learning Task. Unit 5
Logo Symmetry Learning Task Unit 5 Course Mathematics I: Algebra, Geometry, Statistics Overview The Logo Symmetry Learning Task explores graph symmetry and odd and even functions. Students are asked to
More informationa. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F
FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all
More informationNorth Carolina Math 2
Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4.
More information2. THE xy PLANE 7 C7
2. THE xy PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real
More informationMATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL 3  NUMERICAL INTEGRATION METHODS
MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL  NUMERICAL INTEGRATION METHODS This tutorial is essential prerequisite material for anyone studying mechanical engineering. This tutorial uses the principle
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 informationMATH 21. College Algebra 1 Lecture Notes
MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a
More informationLINE INTEGRALS OF VECTOR FUNCTIONS: GREEN S THEOREM. Contents. 2. Green s Theorem 3
LINE INTEGRALS OF VETOR FUNTIONS: GREEN S THEOREM ontents 1. A differential criterion for conservative vector fields 1 2. Green s Theorem 3 1. A differential criterion for conservative vector fields We
More informationProbability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special DistributionsVI Today, I am going to introduce
More information4.3 Lagrange Approximation
206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
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 informationTeacher Page Key. Geometry / Day # 13 Composite Figures 45 Min.
Teacher Page Key Geometry / Day # 13 Composite Figures 45 Min. 91.G.1. Find the area and perimeter of a geometric figure composed of a combination of two or more rectangles, triangles, and/or semicircles
More informationsample median Sample quartiles sample deciles sample quantiles sample percentiles Exercise 1 five number summary # Create and view a sorted
Sample uartiles We have seen that the sample median of a data set {x 1, x, x,, x n }, sorted in increasing order, is a value that divides it in such a way, that exactly half (i.e., 50%) of the sample observations
More informationArea of Parallelograms, Triangles, and Trapezoids (pages 314 318)
Area of Parallelograms, Triangles, and Trapezoids (pages 34 38) Any side of a parallelogram or triangle can be used as a base. The altitude of a parallelogram is a line segment perpendicular to the base
More informationCurve Sketching GUIDELINES FOR SKETCHING A CURVE: A. Domain. B. Intercepts: x and yintercepts.
Curve Sketching GUIDELINES FOR SKETCHING A CURVE: A. Domain. B. Intercepts: x and yintercepts. C. Symmetry: even (f( x) = f(x)) or odd (f( x) = f(x)) function or neither, periodic function. ( ) ( ) D.
More informationStudent 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 informationNEW MEXICO Grade 6 MATHEMATICS STANDARDS
PROCESS STANDARDS To help New Mexico students achieve the Content Standards enumerated below, teachers are encouraged to base instruction on the following Process Standards: Problem Solving Build new mathematical
More informationChapter. Numerical Calculations
Chapter 3 Numerical Calculations 31 Before Performing a Calculation 32 Differential Calculations 33 Quadratic Differential Calculations 34 Integration Calculations 35 Maximum/Minimum Value Calculations
More informationAP 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
More informationSolve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers
More informationMATH 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
More informationSOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The OddRoot Property
498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1
More informationLESSON 4 Missing Numbers in Multiplication Missing Numbers in Division LESSON 5 Order of Operations, Part 1 LESSON 6 Fractional Parts LESSON 7 Lines,
Saxon Math 7/6 Class Description: Saxon mathematics is based on the principle of developing math skills incrementally and reviewing past skills daily. It also incorporates regular and cumulative assessments.
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 informationEngineering Math II Spring 2015 Solutions for Class Activity #2
Engineering Math II Spring 15 Solutions for Class Activity # Problem 1. Find the area of the region bounded by the parabola y = x, the tangent line to this parabola at 1, 1), and the xaxis. Then find
More informationLimit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)
SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f
More information