MATH Area Between Curves

Size: px
Start display at page:

Download "MATH Area Between Curves"

Transcription

1 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 f () and g () we are interested in nding the area of the region bounded b these two curves, assuming such a region eists. In some instances, we ma need to specif additional parameters such as vertical lines to more clearl de ne the region. Let us assume we want to nd the area of the region shown in gure. The region is delimited b the curves f (), g (), and the vertical lines a and b. We use a technique similar to the technique used when we de ned the de nite integral. First, we divide the interval [a; b] into n subintervals of equal length. The length of each subinterval will be b a n. This introduces n + points we label a,,, :::, n b. Each interval is of the form [ i ; i+ ] and has width i. In each interval [ i ; i+ ] we pick a point we call i and we form the rectangle of width and height f ( i ) g ( i ). This is shown on gure. We can then approimate the area of the region b adding the area of each rectangle. This gives us AREA nx f ( i ) i g ( i )

2 Figure : Finding the area of the region Figure : Approimating the area of the region using rectangles

3 As for the de nite integral, the eact area is obtained b taking the limit as n!. In other words, we have AREA lim n! nx f ( i ) i g ( i ) This is precisel the de nition of the de nite integral of the function f () g () between a and b. In other words, we have AREA Z b a (f () g ()) d For this to work, it is important that f and g be such that the graph of f is above the graph of g. For this reason, we prefer to use the notation top and bottom. In the case of gure, we would have top f () and bottom g (). Therefore, we have the formula: AREA Z b a ( top bottom ) d where a and b correspond to the range of values the variable must have to cover the region. Both top and bottom are functions of. The region should be graphed so that we know which function is the top function ( top ) and which function is the bottom function ( bottom ). If the graphs of f () and g () intersect, the top and bottom functions ma switch. In this case, we would epress the area of the region as several integrals. One wa to understand what the integral represent is to picture covering the entire area with those vertical rectangles of width d and height top bottom. The product ( top bottom ) d gives the area of each rectangle. We add the area of these rectangles for between a and b. In some instances, this wa of covering a region ma not be ver practical. Consider the region shown in gure. In this case, one can see that nding top and bottom is going to be di cult. The two functions will change several times as we tr to cover the whole region. Instead, it would be easier to cover the region with thin horizontal rectangles as shown in gure. The technique used is similar to the previous case. What we did for, we now do for. We have to epress the curves bounding the region in terms of. We see that the region is bounded b the curves F () and G (),

4 Figure : Finding the area of the region Figure : Approimating the area of the region

5 and the horizontal lines c and d. We divide the interval [c; d] into n subintervals of equal length. The length of each subinterval will be. We approimate the area of the region b adding the area of each rectangle. Thus we have nx AREA F (i ) G (i ) i To get the eact area, we then take the limit as n!. This gives us AREA lim n! nx F (i ) i G ( i ) This is precisel the de nition of the de nite integral of the function F () G () between c and d. In other words, we have AREA Z d c (F () G ()) d For this to work, it is important that F and G be such that the graph of F is to the right of the graph of G. For this reason, we prefer to use the notation right and left. In the case of gure, we would have right F () and left G (). Therefore, we have the formula: AREA Z d c ( right left ) d where c and d correspond to the range of values must have to cover the region. This time, right and left will be functions of. Remark In some instances, either integral can be used. In general, what determines which integral to use is whether it is easier to cover the region with vertical narrow strips or horizontal narrow strips. The eamples below should help clarif this. Remark In all the eamples below, we need to gure out two things:. If we use a narrow vertical strip ( d integral) (a) What is the height of a tpical rectangle in other words what are top and bottom? The will be epressed as functions of. 5

6 Figure 5: Graph of sin and cos (b) What is the range of values for?. If we use a narrow horizontal strip ( d integral) (a) What is the width of a tpical rectangle in other words what are right and left? The will be epressed as functions of. (b) What is the range of values for? Eample Epress the area of the region bounded b sin, cos,, and using one or more d integrals, then evaluate the integral(s). The curves are shown in Figure 5. The point where sin and cos meet has coordinates d integral, we have AREA Z ; p!. If we use a ( top bottom ) d. Since top and bottom 6

7 change whether we are to the left or to the right of the point of intersection, we need to use two integrals to epress the area. AREA Z Z ( top bottom ) d (cos sin ) d + Z (sin cos ) d (sin + cos )j + ( cos sin )j p p!! + ( + ) + ( ) p p p!! Eample Epress the area of the region bounded b the line and the parabola + 6, using one or more d integrals, then evaluate the integral(s). The graph of the two curves is shown below: First, we nd the points of intersection b solving + 6 From the second equation, we get +. If we substitute in the rst equation, we obtain + 8 or 8 which we can factor as ( ) ( + ). Therefore, the solutions are or. Hence, the points of intersection are ( ; ) and (5; ). Using a d integral, we see that: AREA Z ( right left ) d. Looking at the picture, we see that right corresponds to the line. Since on 7

8 the line ; we have that right +. Also, left corresponds to the parabola. On the parabola + 6, therefore left We replace in the integral to nd the area. Z AREA + + d Z + + d Eample Epress the area of the region bounded b f () and g () + using one or more integrals, then evaluate the integrals. First, we draw a picture of the region.. A tpical rectangle of width d and height g () f () can be used to cover the region. We still need to nd the limits of integration. For this, we need 8

9 to nd the points at which the two curves intersect. This is done b solving + Therefore, the area of the region is A Z Z Z Z + + d + d + d + d 8 Eample Epress the area of the region bounded b, e,, and using one or more integrals, then evaluate the integrals. First, we draw a picture of the region

10 A tpical rectangle of width d and height e can be used to cover the region. This time, we are given the values of. The area is A e e Z (e ) d e Eample 5 Epress the area of the region bounded b and using one or more integrals, then evaluate the integrals. First, we draw a picture of the region. A tpical rectangle of width d and height can be used to cover the region. This time, we are not given the limit values for, we need to nd them. The correspond to the -coordinates of the points of intersection between and. To nd these, we solve

11 Thus, the area is: A Z Z d d Eample 6 Show that the area of a circle of radius r is r. First, we draw a picture. Since ever circle of radius r has the same area, we consider the circle of radius r centered at the origin The upper half of the circle is the function p r. The lower half is the function p r. The picture shows a circle of radius. A tpical rectangle of width d and height top bottom p p r r p r

12 To cover the circle, ranges from r to r. Thus, the area is A Z r r Z r p r d p r d p r + r sin r r r sin r r Eample 7 Epress the area of the region bounded b + 5,,, and using one or more integrals, then evaluate the integrals. First, we draw the region If we tr to proceed as before, we see that we will have to use several integrals as top and bottom change throughout the region. Instead, we use a thin horizontal rectangle of height d to cover the region. The area of such a rectangle will be ( right left ) d. We need to epress right and left as a function of (since we now have a d integral). This is eas, on the right, the

13 rectangle ends on the curve, so right. On the left, the rectangle ends on the line + 5, therefore 5, thus left 5. Finall, since we have a d integral, we need thew range of values for (not ). The are given to us, ranges from to. It follows that the area is: A Z Z 8 ( right left ) d + 5 d Things to know Be able to graph the region delimited b a set of given curves and epress its area using d or d integrals. Related problems assigned #,, 5, 7,,, 5 on pages 6, 7

Representation of functions as power series

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

More information

Partial Derivatives. @x f (x; y) = @ x f (x; y) @x x2 y + @ @x y2 and then we evaluate the derivative as if y is a constant.

Partial Derivatives. @x f (x; y) = @ x f (x; y) @x x2 y + @ @x y2 and then we evaluate the derivative as if y is a constant. Partial Derivatives Partial Derivatives Just as derivatives can be used to eplore the properties of functions of 1 variable, so also derivatives can be used to eplore functions of 2 variables. In this

More information

1 Maximizing pro ts when marginal costs are increasing

1 Maximizing pro ts when marginal costs are increasing BEE12 Basic Mathematical Economics Week 1, Lecture Tuesda 12.1. Pro t maimization 1 Maimizing pro ts when marginal costs are increasing We consider in this section a rm in a perfectl competitive market

More information

Quadratic Functions and Parabolas

Quadratic Functions and Parabolas MATH 11 Quadratic Functions and Parabolas A quadratic function has the form Dr. Neal, Fall 2008 f () = a 2 + b + c where a 0. The graph of the function is a parabola that opens upward if a > 0, and opens

More information

Math 2443, Section 16.3

Math 2443, Section 16.3 Math 44, Section 6. Review These notes will supplement not replace) the lectures based on Section 6. Section 6. i) ouble integrals over general regions: We defined double integrals over rectangles in the

More information

Partial Fractions Decomposition

Partial Fractions Decomposition Partial Fractions Decomposition Dr. Philippe B. Laval Kennesaw State University August 6, 008 Abstract This handout describes partial fractions decomposition and how it can be used when integrating rational

More information

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving

More information

Alex and Morgan were asked to graph the equation y = 2x + 1

Alex and Morgan were asked to graph the equation y = 2x + 1 Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and -intercept wa First, I made a table. I chose some -values, then plugged

More information

The Graph of a Linear Equation

The Graph of a Linear Equation 4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

More information

7.7 Solving Rational Equations

7.7 Solving Rational Equations Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate

More information

TI-84/83 graphing calculator

TI-84/83 graphing calculator TI-8/83 graphing calculator Let us use the table feature to approimate limits and then solve the problems the old fashion way. Find the following limits: Eample 1.1 lim f () Turn on your TI-8 graphing

More information

Double Integrals in Polar Coordinates

Double Integrals in Polar Coordinates Double Integrals in Polar Coordinates. A flat plate is in the shape of the region in the first quadrant ling between the circles + and +. The densit of the plate at point, is + kilograms per square meter

More information

Core Maths C2. Revision Notes

Core Maths C2. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

More information

2008 AP Calculus AB Multiple Choice Exam

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

More information

C3: Functions. Learning objectives

C3: Functions. Learning objectives CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the

More information

SL Calculus Practice Problems

SL Calculus Practice Problems Alei - Desert Academ SL Calculus Practice Problems. The point P (, ) lies on the graph of the curve of = sin ( ). Find the gradient of the tangent to the curve at P. Working:... (Total marks). The diagram

More information

Objectives. By the time the student is finished with this section of the workbook, he/she should be able

Objectives. By the time the student is finished with this section of the workbook, he/she should be able QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a

More information

The small increase in x is. and the corresponding increase in y is. Therefore

The small increase in x is. and the corresponding increase in y is. Therefore Differentials For a while now, we have been using the notation dy to mean the derivative of y with respect to. Here is any variable, and y is a variable whose value depends on. One of the reasons that

More information

PROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS

PROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS PROBLEM SET Practice Problems for Exam # Math 352, Fall 24 Oct., 24 ANSWERS i Problem. vlet R be the region bounded by the curves x = y 2 and y = x. A. Find the volume of the solid generated by revolving

More information

Using the data above Height range: 6 1to 74 inches Weight range: 95 to 205

Using the data above Height range: 6 1to 74 inches Weight range: 95 to 205 Plotting When plotting data, ou will normall be using two numbers, one for the coordinate, another for the coordinate. In some cases, like the first assignment, ou ma have onl one value. There, the second

More information

Use finite approximations to estimate the area under the graph of the function. f(x) = x 3

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

More information

Methods to Solve Quadratic Equations

Methods to Solve Quadratic Equations Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a second-degree

More information

Section 5-9 Inverse Trigonometric Functions

Section 5-9 Inverse Trigonometric Functions 46 5 TRIGONOMETRIC FUNCTIONS Section 5-9 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions

More information

Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X

Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus

More information

Homework 2 Solutions

Homework 2 Solutions Homework Solutions 1. (a) Find the area of a regular heagon inscribed in a circle of radius 1. Then, find the area of a regular heagon circumscribed about a circle of radius 1. Use these calculations to

More information

Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

More information

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

More information

Section 10-5 Parametric Equations

Section 10-5 Parametric Equations 88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative

More information

4.1 Piecewise-Defined Functions

4.1 Piecewise-Defined Functions Section 4.1 Piecewise-Defined Functions 335 4.1 Piecewise-Defined Functions In preparation for the definition of the absolute value function, it is etremel important to have a good grasp of the concept

More information

Section 6-3 Double-Angle and Half-Angle Identities

Section 6-3 Double-Angle and Half-Angle Identities 6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities

More information

7.3 Volumes Calculus

7.3 Volumes Calculus 7. VOLUMES Just like in the last section where we found the area of one arbitrary rectangular strip and used an integral to add up the areas of an infinite number of infinitely thin rectangles, we are

More information

Work as the Area Under a Graph of Force vs. Displacement

Work as the Area Under a Graph of Force vs. Displacement Work as the Area Under a Graph of vs. Displacement Situation A. Consider a situation where an object of mass, m, is lifted at constant velocity in a uniform gravitational field, g. The applied force is

More information

36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?

36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous? 36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this

More information

Implicit Differentiation

Implicit Differentiation Revision Notes 2 Calculus 1270 Fall 2007 INSTRUCTOR: Peter Roper OFFICE: LCB 313 [EMAIL: roper@math.utah.edu] Standard Disclaimer These notes are not a complete review of the course thus far, and some

More information

Core Maths C3. Revision Notes

Core Maths C3. Revision Notes Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...

More information

Connecting Transformational Geometry and Transformations of Functions

Connecting Transformational Geometry and Transformations of Functions Connecting Transformational Geometr and Transformations of Functions Introductor Statements and Assumptions Isometries are rigid transformations that preserve distance and angles and therefore shapes.

More information

6.8 Taylor and Maclaurin s Series

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

More information

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

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

More information

Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science

Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking

More information

PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS

PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS A ver important set of curves which has received considerabl attention in recent ears in connection with the factoring of large numbers

More information

6.3 Parametric Equations and Motion

6.3 Parametric Equations and Motion SECTION 6.3 Parametric Equations and Motion 475 What ou ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher... and wh

More information

Higher. Polynomials and Quadratics 64

Higher. Polynomials and Quadratics 64 hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining

More information

4.4 Concavity and Curve Sketching

4.4 Concavity and Curve Sketching Concavity and Curve Sketching Section Notes Page We can use the second derivative to tell us if a graph is concave up or concave down To see if something is concave down or concave up we need to look at

More information

When I was 3.1 POLYNOMIAL FUNCTIONS

When I was 3.1 POLYNOMIAL FUNCTIONS 146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we

More information

c 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint.

c 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint. Lecture 2b: Utility c 2008 Je rey A. Miron Outline: 1. Introduction 2. Utility: A De nition 3. Monotonic Transformations 4. Cardinal Utility 5. Constructing a Utility Function 6. Examples of Utility Functions

More information

Section 2.4: Equations of Lines and Planes

Section 2.4: Equations of Lines and Planes Section.4: Equations of Lines and Planes An equation of three variable F (x, y, z) 0 is called an equation of a surface S if For instance, (x 1, y 1, z 1 ) S if and only if F (x 1, y 1, z 1 ) 0. x + y

More information

1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model

1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model . Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses

More information

1.2 Solving a System of Linear Equations

1.2 Solving a System of Linear Equations 1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables

More information

7.3 Solving Systems by Elimination

7.3 Solving Systems by Elimination 7. Solving Sstems b Elimination In the last section we saw the Substitution Method. It turns out there is another method for solving a sstem of linear equations that is also ver good. First, we will need

More information

D.3. Angles and Degree Measure. Review of Trigonometric Functions

D.3. Angles and Degree Measure. Review of Trigonometric Functions APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric

More information

Polynomial Degree and Finite Differences

Polynomial Degree and Finite Differences CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

More information

Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123

Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123 Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from

More information

Estimating the Average Value of a Function

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

More information

1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered

1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,

More information

Midterm 2 Review Problems (the first 7 pages) Math 123-5116 Intermediate Algebra Online Spring 2013

Midterm 2 Review Problems (the first 7 pages) Math 123-5116 Intermediate Algebra Online Spring 2013 Midterm Review Problems (the first 7 pages) Math 1-5116 Intermediate Algebra Online Spring 01 Please note that these review problems are due on the day of the midterm, Friday, April 1, 01 at 6 p.m. in

More information

CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES

CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES 6 LESSON CALCULUS 1: LIMITS, AVERAGE GRADIENT AND FIRST PRINCIPLES DERIVATIVES Learning Outcome : Functions and Algebra Assessment Standard 1..7 (a) In this section: The limit concept and solving for limits

More information

ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )

ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 ) SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as

More information

5.2 Inverse Functions

5.2 Inverse Functions 78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,

More information

The Distance Formula and the Circle

The Distance Formula and the Circle 10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,

More information

3.5 Summary of Curve Sketching

3.5 Summary of Curve Sketching 3.5 Summary of Curve Sketching Follow these steps to sketch the curve. 1. Domain of f() 2. and y intercepts (a) -intercepts occur when f() = 0 (b) y-intercept occurs when = 0 3. Symmetry: Is it even or

More information

STRAND: ALGEBRA Unit 3 Solving Equations

STRAND: ALGEBRA Unit 3 Solving Equations CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic

More information

This function is symmetric with respect to the y-axis, so I will let - /2 /2 and multiply the area by 2.

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,

More information

SAT Subject Test Practice Test II: Math Level II Time 60 minutes, 50 Questions

SAT Subject Test Practice Test II: Math Level II Time 60 minutes, 50 Questions SAT Subject Test Practice Test II: Math Level II Time 60 minutes, 50 Questions All questions in the Math Level 1 and Math Level Tests are multiple-choice questions in which you are asked to choose the

More information

x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1 Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs

More information

Contents. 6 Graph Sketching 87. 6.1 Increasing Functions and Decreasing Functions... 87. 6.2 Intervals Monotonically Increasing or Decreasing...

Contents. 6 Graph Sketching 87. 6.1 Increasing Functions and Decreasing Functions... 87. 6.2 Intervals Monotonically Increasing or Decreasing... Contents 6 Graph Sketching 87 6.1 Increasing Functions and Decreasing Functions.......................... 87 6.2 Intervals Monotonically Increasing or Decreasing....................... 88 6.3 Etrema Maima

More information

6.3 Polar Coordinates

6.3 Polar Coordinates 6 Polar Coordinates Section 6 Notes Page 1 In this section we will learn a new coordinate sstem In this sstem we plot a point in the form r, As shown in the picture below ou first draw angle in standard

More information

Week #15 - Word Problems & Differential Equations Section 8.1

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

More information

FACTORING QUADRATICS 8.1.1 through 8.1.4

FACTORING QUADRATICS 8.1.1 through 8.1.4 Chapter 8 FACTORING QUADRATICS 8.. through 8..4 Chapter 8 introduces students to rewriting quadratic epressions and solving quadratic equations. Quadratic functions are any function which can be rewritten

More information

15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors

15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential

More information

Geometry Notes RIGHT TRIANGLE TRIGONOMETRY

Geometry Notes RIGHT TRIANGLE TRIGONOMETRY Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right

More information

Curve Sketching. MATH 1310 Lecture 26 1 of 14 Ronald Brent 2016 All rights reserved.

Curve Sketching. MATH 1310 Lecture 26 1 of 14 Ronald Brent 2016 All rights reserved. Curve Sketching 1. Domain. Intercepts. Symmetry. Asymptotes 5. Intervals of Increase or Decrease 6. Local Maimum and Minimum Values 7. Concavity and Points of Inflection 8. Sketch the curve MATH 110 Lecture

More information

1.2 GRAPHS OF EQUATIONS

1.2 GRAPHS OF EQUATIONS 000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the - and -intercepts of graphs of equations. Write the standard forms of equations of

More information

MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

More information

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its

More information

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1 Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.

More information

AP Calculus AB First Semester Final Exam Practice Test Content covers chapters 1-3 Name: Date: Period:

AP Calculus AB First Semester Final Exam Practice Test Content covers chapters 1-3 Name: Date: Period: AP Calculus AB First Semester Final Eam Practice Test Content covers chapters 1- Name: Date: Period: This is a big tamale review for the final eam. Of the 69 questions on this review, questions will be

More information

Triple integrals in Cartesian coordinates (Sect. 15.4) Review: Triple integrals in arbitrary domains.

Triple integrals in Cartesian coordinates (Sect. 15.4) Review: Triple integrals in arbitrary domains. Triple integrals in Cartesian coordinates (Sect. 5.4 Review: Triple integrals in arbitrar domains. s: Changing the order of integration. The average value of a function in a region in space. Triple integrals

More information

7.3 Parabolas. 7.3 Parabolas 505

7.3 Parabolas. 7.3 Parabolas 505 7. Parabolas 0 7. Parabolas We have alread learned that the graph of a quadratic function f() = a + b + c (a 0) is called a parabola. To our surprise and delight, we ma also define parabolas in terms of

More information

C1: Coordinate geometry of straight lines

C1: Coordinate geometry of straight lines B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the

More information

LIMITS AND CONTINUITY

LIMITS AND CONTINUITY LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from

More information

COMPLEX STRESS TUTORIAL 3 COMPLEX STRESS AND STRAIN

COMPLEX STRESS TUTORIAL 3 COMPLEX STRESS AND STRAIN COMPLX STRSS TUTORIAL COMPLX STRSS AND STRAIN This tutorial is not part of the decel unit mechanical Principles but covers elements of the following sllabi. o Parts of the ngineering Council eam subject

More information

SECTION 2.2. Distance and Midpoint Formulas; Circles

SECTION 2.2. Distance and Midpoint Formulas; Circles SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation

More information

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the

More information

Math 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) =

Math 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) = Math 50 Eam # Practice Problem Solutions. Find the vertical asymptotes (if any) of the functions g() = + 4, h() = 4. Answer: The only number not in the domain of g is = 0, so the only place where g could

More information

Rotated Ellipses. And Their Intersections With Lines. Mark C. Hendricks, Ph.D. Copyright March 8, 2012

Rotated Ellipses. And Their Intersections With Lines. Mark C. Hendricks, Ph.D. Copyright March 8, 2012 Rotated Ellipses And Their Intersections With Lines b Mark C. Hendricks, Ph.D. Copright March 8, 0 Abstract: This paper addresses the mathematical equations for ellipses rotated at an angle and how to

More information

Systems of Linear Equations: Solving by Substitution

Systems of Linear Equations: Solving by Substitution 8.3 Sstems of Linear Equations: Solving b Substitution 8.3 OBJECTIVES 1. Solve sstems using the substitution method 2. Solve applications of sstems of equations In Sections 8.1 and 8.2, we looked at graphing

More information

LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0

LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0 LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( )

More information

Solutions to Homework 10

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 information

10.2 The Unit Circle: Cosine and Sine

10.2 The Unit Circle: Cosine and Sine 0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on

More information

Chapter 6 Quadratic Functions

Chapter 6 Quadratic Functions Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where

More information

x 2 if 2 x < 0 4 x if 2 x 6

x 2 if 2 x < 0 4 x if 2 x 6 Piecewise-defined Functions Example Consider the function f defined by x if x < 0 f (x) = x if 0 x < 4 x if x 6 Piecewise-defined Functions Example Consider the function f defined by x if x < 0 f (x) =

More information

Double integrals. Notice: this material must not be used as a substitute for attending the lectures

Double integrals. Notice: this material must not be used as a substitute for attending the lectures ouble integrals Notice: this material must not be used as a substitute for attending the lectures . What is a double integral? Recall that a single integral is something of the form b a f(x) A double integral

More information

Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.

Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t. REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()

More information

7.3 Graphing Rational Functions

7.3 Graphing Rational Functions Section 7.3 Graphing Rational Functions 639 7.3 Graphing Rational Functions We ve seen that the denominator of a rational function is never allowed to equal zero; division b zero is not defined. So, with

More information

Math 1B, lecture 5: area and volume

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

More information

Mathematics 31 Pre-calculus and Limits

Mathematics 31 Pre-calculus and Limits Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals

More information

4.6 Null Space, Column Space, Row Space

4.6 Null Space, Column Space, Row Space NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear

More information

GRAPH OF A RATIONAL FUNCTION

GRAPH OF A RATIONAL FUNCTION GRAPH OF A RATIONAL FUNCTION Find vertical asmptotes and draw them. Look for common factors first. Vertical asmptotes occur where the denominator becomes zero as long as there are no common factors. Find

More information

THE PARABOLA 13.2. section

THE PARABOLA 13.2. section 698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.

More information

Calculus with Parametric Curves

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

More information