# Estimating the Average Value of a Function

Save this PDF as:

Size: px
Start display at page:

## 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 sub-interval [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 sub-interval [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 sub-interval 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 sub-interval.) 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 sub-interval 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 sub-intervals instead of 4 sub-intervals, 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 k-th sub-interval [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 k-th sub-interval, 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 sub-interval, [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 sub-interval is m 1 = 2/ 3, and the maximum value is M 1 = 0. On the second sub-interval, [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 sub-intervals 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. right-endpoint, left-endpoint, or mid-point 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 non-negative on [a, b], then the upper and lower Riemann sums estimate the area of the region below the curve y = f(x), above the x-axis 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 x-axis, 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

### Name Calculus AP Chapter 7 Outline M. C.

Name Calculus AP Chapter 7 Outline M. C. A. AREA UNDER A CURVE: a. If y = f (x) is continuous and non-negative on [a, b], then the area under the curve of f from a to b is: A = f (x) dx b. If y = f (x)

### 7.5 Approximating Definite Integrals

WileyPLUS: Home Help Contact us Logout Hughes-Hallett, Calculus: Single and Multivariable, 4/e Calculus I, II, and Vector Calculus Reading content Integration 7.1. Integration by Substitution 7.2. Integration

### 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.

### a 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 2015-11-19 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 ACT-approved calculator during

### 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

### MATH 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

### Use 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

### Chapter V.G The Definite Integral and Area: Two Views

Chapter V.G The Definite Integral and Area: Two Views In is section we will look at e application of e definite integral to e problem of finding e area of a region in e plane. In particular e regions in

### 2010 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

### Appendix A: Numbers, Inequalities, and Absolute Values. Outline

Appendix A: Numbers, Inequalities, and Absolute Values Tom Lewis Fall Semester 2015 Outline Types of numbers Notation for intervals Inequalities Absolute value A hierarchy of numbers Whole numbers 1, 2,

### 7.6 Approximation Errors and Simpson's Rule

WileyPLUS: Home Help Contact us Logout Hughes-Hallett, Calculus: Single and Multivariable, 4/e Calculus I, II, and Vector Calculus Reading content Integration 7.1. Integration by Substitution 7.2. Integration

### II. Sketch the given region R and then find the area. 2. R is the region bounded by the curves y = 0, y = x 2 and x = 3.

Math 34 April I. It is estimated that t days from now a farmer s crop will be increasing at the rate of.5t +.4t + bushels per day. By how much will the value of the crop increase during the next 5 days

### Applications of the Integral

Chapter 6 Applications of the Integral Evaluating integrals can be tedious and difficult. Mathematica makes this work relatively easy. For example, when computing the area of a region the corresponding

### 7. 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,

### Finding Antiderivatives and Evaluating Integrals

Chapter 5 Finding Antiderivatives and Evaluating Integrals 5. Constructing Accurate Graphs of Antiderivatives Motivating Questions In this section, we strive to understand the ideas generated by the following

### MATH41112/61112 Ergodic Theory Lecture Measure spaces

8. Measure spaces 8.1 Background In Lecture 1 we remarked that ergodic theory is the study of the qualitative distributional properties of typical orbits of a dynamical system and that these properties

### Section 2.1 Rectangular Coordinate Systems

P a g e 1 Section 2.1 Rectangular Coordinate Systems 1. Pythagorean Theorem In a right triangle, the lengths of the sides are related by the equation where a and b are the lengths of the legs and c is

### Trapezoid Rule. y 2. y L

Trapezoid Rule and Simpson s Rule c 2002, 2008, 200 Donald Kreider and Dwigt Lar Trapezoid Rule Many applications of calculus involve definite integrals. If we can find an antiderivative for te integrand,

### Functions 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

### Math 1526 Consumer and Producer Surplus

Math 156 Consumer and Producer Surplus Scenario: In the grocery store, I find that two-liter 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

### 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

### Solid of Revolution - Finding Volume by Rotation

Solid of Revolution - Finding Volume by Rotation Finding the volume of a solid revolution is a method of calculating the volume of a 3D object formed by a rotated area of a 2D space. Finding the volume

### So here s the next version of Homework Help!!!

HOMEWORK HELP FOR MATH 52 So here s the next version of Homework Help!!! I am going to assume that no one had any great difficulties with the problems assigned this quarter from 4.3 and 4.4. However, if

### Lecture 33: Area by slicing

Lecture : Area by slicing Nathan Pflueger December 1 1 Introduction In this lecture and the next, we ll revisit the idea of Riemann sums, and show how it can be used to convert certain problems to computing

### South Carolina College- and Career-Ready (SCCCR) Algebra 1

South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process

### APPLICATIONS 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

### 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

### Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x b is given by

MATH 42, Fall 29 Examples from Section, Tue, 27 Oct 29 1 The First Hour Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x

### Lesson 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

### Zeros of Polynomial Functions

Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + 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

### 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

### Notes 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

### (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

### Area 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

### 15.1. 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

### Chapter 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.

### The Alternating Series Test

The Alternating Series Test So far we have considered mostly series all of whose terms are positive. If the signs of the terms alternate, then testing convergence is a much simpler matter. On the other

### Understanding 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

### Section 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

### Math 1206 Calculus Sec. 4.4: Estimating with Finite Sums

Math 1206 Calculus Sec. 4.4: Estimating with Finite Sums I. Consider the problem of finding the area under the curve on the fn y=-x 2 +5 over the domain [0, 2]. We can approximate this area by using a

### Method 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

### 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,

### This function is symmetric with respect to the y-axis, 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,

### Chapter 5 Applications of Integration

MA111 Application of Integration Asst.Prof.Dr.Supranee Lisawadi 1 Chapter 5 Applications of Integration Section 5.1 Area Between Two Curves In this section we use integrals to find areas of regions that

### List 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

### The Not-Formula Book for C1

Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes

### 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

### Chapter 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

### Section 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,

### Information 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 Shannon-Fano-Elias Coding and Introduction to Arithmetic Coding

### 1. AREA BETWEEN the CURVES

1 The area between two curves The Volume of the Solid of revolution (by slicing) 1. AREA BETWEEN the CURVES da = {( outer function ) ( inner )} dx function b b A = da = [y 1 (x) y (x)]dx a a d d A = da

### Review 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,

### North Carolina Math 1

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.

### Safe and Simple Calculus Activities

Safe and Simple Calculus Activities This paper focuses on a few easy-to-implement technology-enabled activities for a first year Calculus course. Specific topics will include the derivative at a point,

### x = y + 2, and the line

WS 8.: Areas between Curves Name Date Period Worksheet 8. Areas between Curves Show all work on a separate sheet of paper. No calculator unless stated. Multiple Choice. Let R be the region in the first

### Asymptotes. 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.

### Integration. 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.

### Chapter 17: Aggregation

Chapter 17: Aggregation 17.1: Introduction This is a technical chapter in the sense that we need the results contained in it for future work. It contains very little new economics and perhaps contains

### Chapter 2: Rocket Launch

Chapter 2: Rocket Launch Lesson 2.1.1. 2-1. Domain:!" x " Range: 2! y! " y-intercept! y = 2 no x-intercepts 2-2. Time Hours sitting Amount Earned 8PM 1 4 9PM 2 4*2hrs = 8 10PM 3 4*3hrs = 12 11:30PM 4.5

### 15.3. 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

### Area 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

### Numerical 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

### Calculus 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

### www.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

### 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

### Statistics 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

### 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

### Lecture 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

### ARE211, Fall2012. Contents. 2. Linear Algebra Preliminary: Level Sets, upper and lower contour sets and Gradient vectors 1

ARE11, Fall1 LINALGEBRA1: THU, SEP 13, 1 PRINTED: SEPTEMBER 19, 1 (LEC# 7) Contents. Linear Algebra 1.1. Preliminary: Level Sets, upper and lower contour sets and Gradient vectors 1.. Vectors as arrows.

### Area Under the Curve

Visit our website at www.cpm.org. Area Under the Curve For more information about the materials you find in this packet, contact: Chris Mikles (888) 808-4276 mikles@cpm.org Rocket Launch You have seen

### WEEK #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

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### f(x) = lim 2) = 2 2 = 0 (c) Provide a rough sketch of f(x). Be sure to include your scale, intercepts and label your axis.

Math 16 - Final Exam Solutions - Fall 211 - Jaimos F Skriletz 1 Answer each of the following questions to the best of your ability. To receive full credit, answers must be supported by a sufficient amount

### MASSACHUSETTS 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

### SECTION 0.11: SOLVING EQUATIONS. LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations.

(Section 0.11: Solving Equations) 0.11.1 SECTION 0.11: SOLVING EQUATIONS LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. PART A: DISCUSSION Much

### 10.2 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

### AP 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

### and at University of Georgia September 11, 2006 Sarah Donaldson

Situation 35: Solving Quadratic Equations Prepared at Penn State Mid-Atlantic Center for Mathematics Teaching and Learning June 30, 005 Jeanne Shimizu Prompt and at University of Georgia September 11,

### Math 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

### MEASURES 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

### ) + ˆf (n) sin( 2πnt. = 2 u x 2, t > 0, 0 < x < 1. u(0, t) = u(1, t) = 0, t 0. (x, 0) = 0 0 < x < 1.

Introduction to Fourier analysis This semester, we re going to study various aspects of Fourier analysis. In particular, we ll spend some time reviewing and strengthening the results from Math 425 on Fourier

### 2 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

### Supporting Australian Mathematics Project. A guide for teachers Years 11 and 12. Algebra and coordinate geometry: Module 11. Sequences and series

1 Supporting Australian Mathematics Project 3 4 5 6 7 8 9 10 11 1 A guide for teachers Years 11 and 1 Algebra and coordinate geometry: Module 11 Sequences and series Sequences and series A guide for teachers

### Probability Models for Continuous Random Variables

Density Probability Models for Continuous Random Variables At right you see a histogram of female length of life. (Births and deaths are recorded to the nearest minute. The data are essentially continuous.)

### Calculus II MAT 146 Integration Applications: Area Between Curves

Calculus II MAT 46 Integration Applications: Area Between Curves A fundamental application of integration involves determining the area under a curve for some interval on the x- or y-axis. In a previous

### 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

### Interactive Math Glossary Terms and Definitions

Terms and Definitions Absolute Value the magnitude of a number, or the distance from 0 on a real number line Additive Property of Area the process of finding an the area of a shape by totaling the areas

MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose

### Structure of Measurable Sets

Structure of Measurable Sets In these notes we discuss the structure of Lebesgue measurable subsets of R from several different points of view. Along the way, we will see several alternative characterizations

### 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

### 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

### 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

### with "a", "b" and "c" representing real numbers, and "a" is not equal to zero.

3.1 SOLVING QUADRATIC EQUATIONS: * A QUADRATIC is a polynomial whose highest exponent is. * The "standard form" of a quadratic equation is: ax + bx + c = 0 with "a", "b" and "c" representing real numbers,

### Math 181 Spring 2007 HW 1 Corrected

Math 181 Spring 2007 HW 1 Corrected February 1, 2007 Sec. 1.1 # 2 The graphs of f and g are given (see the graph in the book). (a) State the values of f( 4) and g(3). Find 4 on the x-axis (horizontal axis)

### Looking for Pythagoras: Homework Examples from ACE

Looking for Pythagoras: Homework Examples from ACE Investigation 1: Coordinate Grids, ACE #20, #37 Investigation 2: Squaring Off, ACE #16, #44, #65 Investigation 3: The Pythagorean Theorem, ACE #2, #9,

### MATH Area Between Curves

MATH - Area Between Curves Philippe Laval September, 8 Abstract This handout discusses techniques used to nd the area of regions which lie between two curves. Area Between Curves. Theor Given two functions

### Week #15 - Word Problems & Differential Equations Section 8.1

Week #15 - Word Problems & Differential Equations Section 8.1 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 25 by John Wiley & Sons, Inc. This material is used by