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

Size: px
Start display at page:

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

Transcription

1 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 our discussion are bounded by eier lines or e graphs of functions. There are two ways to approach is problem as in many oer applications of e definite integral. We will consider e area problem in some detail because it is typical. Our first approach for finding an area will center on e approximation of e area as a sum of areas of rectangles at are determined by e given region. This follows our earlier treatment of e definite integral as a limit of sums: What's good for one is good for all and e more e merrier. These estimating sums will have a limiting value which by definition is a definite integral. A second contrasting approach to e same problem follows e eme at information about a rate of change (a differential equation) can help determine e value of a variable which might oerwise be considered constant. In is approach we'll consider e area as a function of a single variable, much like in e derivative form of e fundamental eorem of calculus. The key here is to estimate e rate at which e area is changing as a function of one variable. The area of a rectangle forms e basis for an estimate of e change in area of e region. 1. Area as a Limit of Sums Our initial work on e definition of e definite integral used an interpretation of e Euler sums as e area of rectangles which approximated e area of a region enclosed by e graph of a non-negative function, e X -axis, and lines X = a and X = b. The next example generalizes is situation. Keep in mind e definition of e definite integral, namely: This definition allows us to recognize e limit of estimates of is form as definite integrals. The advantage of seeing is form is at e definite integral in many cases can be evaluated exactly by using e Fundamental Theorem of Calculus. This connection will be more clear after you read e next example. Example V.G.1. In is example we will a) express e area of is region as e difference of two areas; b) estimate e area of e region in e first quadrant enclosed by e Y-axis, e graph of g(x)=x and f(x)=-x ; and c) express e area of is region as a definite integral, explaining how is integral is related to e difference in part a) and e estimate in part b). Discussion: a) First graph e region as described and note at e curves intersect in e first quadrant when x = -x, i.e., when x = 1, so e x values for is region are between 0 and 1. See Figure ***.The region can be considered as being obtained from e region benea e graph of Y = - X and above e X-axis between e Y - axis and e line X = 1, wi area A1, after cutting away e region benea e graph of Figure 1 Figure Chapter V.G 6/15/14 1

2 Y= X and above e X-axis between e Y-axis and e line X = 1, wi area A. The area can erefore be described as e difference of e areas of e two regions, i.e., A1-A. Each of ese areas can be computed from a definite integral, and. So e area we want to find is 5/3-1/3 = 4/3. b) For a first estimate we cut e interval in half and estimate e area by e areas of two rectangles wi base 1/ and altitudes determined by left hand endpoints, 0 and 1/. Wi ese heights of =-0 and 7/4-1/4 = 3/, e estimate for e area is 1/+ 3/ 1/ = 7/4. See Figure ***. Of course a finer partition of e interval and a correspondingly larger number of rectangles would give a better estimate of e area. In general if we let x=1/n and x = k/n, en e estimate for e area is given by k. Larger values for N will give a better estimate for e area as Figure *** illustrates. Wi N = 10 we find an estimate for e area of Notice at from e figure it should clear at any of e estimates obtained using ese sums will be an overestimate of e actual area. c)the areas in part a) were expressed as integrals, and Figure 3 Figure 4 while e area in question was e difference A1-A.. This area can be expressed also as a single integral by using e linearity property of e definite integral,. Notice at is is precisely e definite integral at e sum in part b) estimates by e definition of e integral. Thus we have two approaches to finding e area: i) recognize e region as being composed of regions for which e areas are computable and ii) estimate e area wi sums. Wi each approach e area can be found by evaluating a single definite integral using e Fundamental Theorem of Calculus,. Generalization. The work in e last example can be generalized wiout much difficulty if we follow e estimation approach but continue to assume at for all x in [a,b], g(x) f(x). In is situation e region will be contained between e graphs of y=f(x) and y=g(x) and e lines X = a and X = b wi a < b and where f and g are continuous functions on [a,b]. We partition e interval [a,b] into N pieces of leng wi x = a + k x. Above e k subinterval we consider a rectangle wi its height h k determined by e difference in e function values at e left hand endpoint of e interval, i.e.,. The sum of areas of e individual rectangles, given by k, is our estimate for e area of e region. As N, ese sums give better estimates for e area of e region, but ey also approach (by definition) a definite integral. Therefore e area of e region must be given by Chapter V.G 6/15/14

3 . Comment. In many situations e functions whose graphs bound a region change over e controlling domain interval. It is not uncommon in fact for two function's graphs to cross so at for a part of e interval f(x) g(x) while for anoer part g(x) f(x). In ese situations e region needs to be broken down for analysis into regions over intervals where e upper and lower curves do not change. Finding e values for x where ese changes happen, for example, where f(x) = g(x), is crucial to is investigation. For convenience it is possible to denote e result in situations where only two functions, f and g, are involved wiout ese complications by noting at in eier case. However, in practice is notation does not actually simplify e computation since to evaluate an integral wi an absolute value in e integrand requires furer analysis of e integrand. Summary: The area of a region between two continuous functions f and g and e lines X=a and X=b is given by. The next example follows e eme of estimation to find an area. It differs from e previous example primarily because e estimating rectangles are controlled by e use of e vertical axis, giving an alternative for regions where e boundary curves are more easily described wi e horizontal (X) component of e points on e curves being deter function of eir vertical (Y) component. We continue to follow e principles of What's Good For One Is Good For All and The More The Merrier! Example V.G.. Consider e region bounded by e curve Y = X and e line wi equation Y = X -. a) Estimate e area of is region. b) Express e area of is region as a definite integral. c) Use part b to find e area of e given region. Solution. a) First we graph e region under discussion and note at e curves intersect when Y = Y +, i.e., e points on e curve where Y =-1 and Y = so e Y values for is region are between -1 and. As a first estimate we cut is interval of Y values [-1,] on e Y-axis into ree equal pieces of leng 1 and estimate e area by e areas of ree horizontal "rectangles" wi wid 1 and lengs of 0, and 3-1 =, giving e estimate of (0 + + ) 1 = 4. Of course a finer partition of e interval and a correspondingly larger number of horizontal rectangles would give a better estimate of e area. In general we let y =3/N and y k = -1+ k y. We use e horizontal segment connecting e points on e curves wi Y coordinates y to determine e side of e k rectangle. This gives e k rectangle dimensions by y so en e estimate for e area of e k rectangle is given by to an estimate for e total area given by e sum Chapter V.G 6/15/14 3 or k-1. This leads. As you might expect larger values for N will give a better estimate for e area. [Draw a figure at illustrates is situation wi rectangles.] For example wi N=15, we find an estimate of 4.48.

4 b) Here e pattern of estimating an area wi sums shows its payoff. You should be able to recognize at e sum in part a) estimates e definite integral:. Since ese sums can estimate only one number as eir limit when N, it must be at e area is e number described by. c) From part b) e area can be found by evaluating e definite integral using e Evaluation Form of e Fundamental Theorem of Calculus, Comment: The generalization of is example is to regions at have eir boundaries determined by curves described wi e X-coordinate of a point on e curve determined as a function of its corresponding Y-coordinate. In e example we had X=f(Y)=+Y and X=g(Y)=Y and an interval of Y values [c,d] which controlled e boundary curves values. The result for regions of is type is at e area can be expressed as.. Area as a Solution to a Differential Equation In our first treatment of e Fundamental Theorem of Calculus we saw at an area function was e solution to a differential equation. We will find e same kind of analysis useful in finding e area between two curves as e next example illustrates by resolving our first example wi a quite different approach using e derivative and e fundamental eorem of calculus for differential equations. Example V.G.3. a) Let A be e function determined by e statement at A(t) denotes e area of e region in e first quadrant enclosed by e Y - axis, e graph of g(x)=x, f(x)=-x, and e line X = t when t is between 0 and 1. Estimate e instantaneous rate of change of A(t) at t where t is in e interval [0,1]. b) Find e instantaneous rate of change of e area A of is region as a function of t. c) Explain how e differential equation in part b) shows e area in question must be. Solution. a) An estimate of e rate of change at t is given by. But is approximately e area of a rectangle wi altitude g(t)-f(t) and base t, so at Chapter V.G 6/15/14 4

5 . Draw a figure at illustrates is for e functions f and g. b) From part a) we have. c) The initial condition for A(t) is A(0)=0. By e Differential Equation form of e Fundamental Theorem of Calculus, e solution to e differential equation A'(t)=f(t)-g(t)=-t wi A(0)=0 is.therefore. Exercises. In each of e following problems a) sketch a graph of e region described, b) use four rectangles to estimate e area of e region, c) express e area of e region using e definite integral, and d) find e exact area of e region described The region bounded by e graph of e equations Y = X and Y = X.. The region bounded by e graph of e equations Y = X and Y = X The region bounded by e graph of e equations Y = X and Y = X. 4. One section of e region bounded by e graph of e equations Y=sin(X) and Y=cos(X). 1/ 5. The region bounded by e graph of Y = X and e graph of Y = X. 6. The region bounded by e graph of e equations Y = X and Y = - X. In each of e following problems a) sketch a graph of e region described, b) use four horizontal rectangles to estimate e area of e region, c) express e area of e region using e definite integral wi y as e controlling variable, and d) find e exact area of e region described. 7. The region bounded by e graph of e equations,y = X, Y = -1 and Y =. 8. The region bounded by e graph of e equations and Y = X, Y = - and Y =. 9. The region bounded by e graph of e equations Y = X and. 10. The region bounded by e graph of e equations X=sin(Y), X=cos(Y), Y=0 and Y=. 11. The region bounded by e graph of and e graph of. 1. The region bounded by e graph of e equations Y= 1/x, Y= 1/x, and e line X=. Chapter V.G 6/15/14 5

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

AP CALCULUS AB 2008 SCORING GUIDELINES

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

More information

(b)using the left hand end points of the subintervals ( lower sums ) we get the aprroximation

(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 information

Student Performance Q&A:

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

More information

Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20 Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

More information

AP Calculus AB 2011 Scoring Guidelines

AP Calculus AB 2011 Scoring Guidelines AP Calculus AB Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 9, the

More information

Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

More information

Investigating Area Under a Curve

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

More information

AP Calculus AB 2004 Free-Response Questions

AP Calculus AB 2004 Free-Response Questions AP Calculus AB 2004 Free-Response Questions The materials included in these files are intended for noncommercial use by AP teachers for course and exam preparation; permission for any other use must be

More information

Analyzing Piecewise Functions

Analyzing Piecewise Functions Connecting Geometry to Advanced Placement* Mathematics A Resource and Strategy Guide Updated: 04/9/09 Analyzing Piecewise Functions Objective: Students will analyze attributes of a piecewise function including

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

Objectives. Materials

Objectives. Materials Activity 4 Objectives Understand what a slope field represents in terms of Create a slope field for a given differential equation Materials TI-84 Plus / TI-83 Plus Graph paper Introduction One of the ways

More information

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions. Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

More information

2.1 Increasing, Decreasing, and Piecewise Functions; Applications

2.1 Increasing, Decreasing, and Piecewise Functions; Applications 2.1 Increasing, Decreasing, and Piecewise Functions; Applications Graph functions, looking for intervals on which the function is increasing, decreasing, or constant, and estimate relative maxima and minima.

More information

Understanding Basic Calculus

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

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

List the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated

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

More information

1 Functions, Graphs and Limits

1 Functions, Graphs and Limits 1 Functions, Graphs and Limits 1.1 The Cartesian Plane In this course we will be dealing a lot with the Cartesian plane (also called the xy-plane), so this section should serve as a review of it and its

More information

Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass

Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of

More information

Calculus AB 2014 Scoring Guidelines

Calculus AB 2014 Scoring Guidelines P Calculus B 014 Scoring Guidelines 014 The College Board. College Board, dvanced Placement Program, P, P Central, and the acorn logo are registered trademarks of the College Board. P Central is the official

More information

AP Calculus AB 2011 Free-Response Questions

AP Calculus AB 2011 Free-Response Questions AP Calculus AB 11 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in

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

7.6 Approximation Errors and Simpson's Rule

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

More information

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

More information

Graphing Linear Equations

Graphing Linear Equations Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope

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

MATH 132: CALCULUS II SYLLABUS

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

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

3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS. Copyright Cengage Learning. All rights reserved.

3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS. Copyright Cengage Learning. All rights reserved. 3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Copyright Cengage Learning. All rights reserved. What You Should Learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic functions.

More information

Section 3.2 Polynomial Functions and Their Graphs

Section 3.2 Polynomial Functions and Their Graphs Section 3.2 Polynomial Functions and Their Graphs EXAMPLES: P(x) = 3, Q(x) = 4x 7, R(x) = x 2 +x, S(x) = 2x 3 6x 2 10 QUESTION: Which of the following are polynomial functions? (a) f(x) = x 3 +2x+4 (b)

More information

AP Calculus BC 2010 Free-Response Questions

AP Calculus BC 2010 Free-Response Questions AP Calculus BC 2010 Free-Response Questions The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded

More information

MODERN APPLICATIONS OF PYTHAGORAS S THEOREM

MODERN APPLICATIONS OF PYTHAGORAS S THEOREM UNIT SIX MODERN APPLICATIONS OF PYTHAGORAS S THEOREM Coordinate Systems 124 Distance Formula 127 Midpoint Formula 131 SUMMARY 134 Exercises 135 UNIT SIX: 124 COORDINATE GEOMETRY Geometry, as presented

More information

Worksheet 1. What You Need to Know About Motion Along the x-axis (Part 1)

Worksheet 1. What You Need to Know About Motion Along the x-axis (Part 1) Worksheet 1. What You Need to Know About Motion Along the x-axis (Part 1) In discussing motion, there are three closely related concepts that you need to keep straight. These are: If x(t) represents the

More information

AP Calculus AB 2010 Free-Response Questions

AP Calculus AB 2010 Free-Response Questions AP Calculus AB 2010 Free-Response Questions The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded

More information

Graphical Integration Exercises Part Four: Reverse Graphical Integration

Graphical Integration Exercises Part Four: Reverse Graphical Integration D-4603 1 Graphical Integration Exercises Part Four: Reverse Graphical Integration Prepared for the MIT System Dynamics in Education Project Under the Supervision of Dr. Jay W. Forrester by Laughton Stanley

More information

DERIVATIVES AS MATRICES; CHAIN RULE

DERIVATIVES AS MATRICES; CHAIN RULE DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Real-valued Functions Let s first consider functions f : R 2 R. Recall that if the partial derivatives of f exist at the point (x 0, y 0 ), then we

More information

Section 1.1 Linear Equations: Slope and Equations of Lines

Section 1.1 Linear Equations: Slope and Equations of Lines Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

More information

The degree of a polynomial function is equal to the highest exponent found on the independent variables.

The degree of a polynomial function is equal to the highest exponent found on the independent variables. DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

More information

AP Calculus AB 2012 Free-Response Questions

AP Calculus AB 2012 Free-Response Questions AP Calculus AB 1 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in

More information

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

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

More information

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

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

More information

Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only

Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials

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

AP Calculus AB 2006 Scoring Guidelines

AP Calculus AB 2006 Scoring Guidelines AP Calculus AB 006 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college

More information

Logo Symmetry Learning Task. Unit 5

Logo 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 information

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:

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

More information

CHAPTER 1 Linear Equations

CHAPTER 1 Linear Equations CHAPTER 1 Linear Equations 1.1. Lines The rectangular coordinate system is also called the Cartesian plane. It is formed by two real number lines, the horizontal axis or x-axis, and the vertical axis or

More information

AP Calculus BC 2006 Free-Response Questions

AP Calculus BC 2006 Free-Response Questions AP Calculus BC 2006 Free-Response Questions The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to

More information

Functions. MATH 160, Precalculus. J. Robert Buchanan. Fall 2011. Department of Mathematics. J. Robert Buchanan Functions

Functions. MATH 160, Precalculus. J. Robert Buchanan. Fall 2011. Department of Mathematics. J. Robert Buchanan Functions Functions MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: determine whether relations between variables are functions, use function

More information

Review of Fundamental Mathematics

Review of Fundamental Mathematics Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

More information

Arrangements And Duality

Arrangements And Duality Arrangements And Duality 3.1 Introduction 3 Point configurations are tbe most basic structure we study in computational geometry. But what about configurations of more complicated shapes? For example,

More information

AP Calculus AB 2004 Scoring Guidelines

AP Calculus AB 2004 Scoring Guidelines AP Calculus AB 4 Scoring Guidelines The materials included in these files are intended for noncommercial use by AP teachers for course and eam preparation; permission for any other use must be sought from

More information

Lecture 3: Derivatives and extremes of functions

Lecture 3: Derivatives and extremes of functions Lecture 3: Derivatives and extremes of functions Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2011 Lejla Batina Version: spring 2011 Wiskunde 1 1 / 16

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

Method To Solve Linear, Polynomial, or Absolute Value Inequalities:

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

More information

AP Calculus BC 2013 Free-Response Questions

AP Calculus BC 2013 Free-Response Questions AP Calculus BC 013 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in

More information

-2- Reason: This is harder. I'll give an argument in an Addendum to this handout.

-2- Reason: This is harder. I'll give an argument in an Addendum to this handout. LINES Slope The slope of a nonvertical line in a coordinate plane is defined as follows: Let P 1 (x 1, y 1 ) and P 2 (x 2, y 2 ) be any two points on the line. Then slope of the line = y 2 y 1 change in

More information

1.7 Graphs of Functions

1.7 Graphs of Functions 64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

More information

1.6 A LIBRARY OF PARENT FUNCTIONS. Copyright Cengage Learning. All rights reserved.

1.6 A LIBRARY OF PARENT FUNCTIONS. Copyright Cengage Learning. All rights reserved. 1.6 A LIBRARY OF PARENT FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal

More information

Statistics Revision Sheet Question 6 of Paper 2

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

More information

Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities

Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned

More information

Graphing Quadratic Functions

Graphing Quadratic Functions Problem 1 The Parabola Examine the data in L 1 and L to the right. Let L 1 be the x- value and L be the y-values for a graph. 1. How are the x and y-values related? What pattern do you see? To enter the

More information

x), etc. In general, we have

x), etc. In general, we have BASIC CALCULUS REFRESHER. Introduction. Ismor Fischer, Ph.D. Dept. of Statistics UW-Madison This is a very condensed and simplified version of basic calculus, which is a prerequisite for many courses in

More information

7.1 Graphs of Quadratic Functions in Vertex Form

7.1 Graphs of Quadratic Functions in Vertex Form 7.1 Graphs of Quadratic Functions in Vertex Form Quadratic Function in Vertex Form A quadratic function in vertex form is a function that can be written in the form f (x) = a(x! h) 2 + k where a is called

More information

AP Calculus AB 2007 Scoring Guidelines Form B

AP Calculus AB 2007 Scoring Guidelines Form B AP Calculus AB 7 Scoring Guidelines Form B The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to

More information

2.5 Transformations of Functions

2.5 Transformations of Functions 2.5 Transformations of Functions Section 2.5 Notes Page 1 We will first look at the major graphs you should know how to sketch: Square Root Function Absolute Value Function Identity Function Domain: [

More information

Definition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point.

Definition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point. 6.1 Vectors in the Plane PreCalculus 6.1 VECTORS IN THE PLANE Learning Targets: 1. Find the component form and the magnitude of a vector.. Perform addition and scalar multiplication of two vectors. 3.

More information

4.3 Lagrange Approximation

4.3 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 information

AP Calculus AB 2003 Scoring Guidelines Form B

AP 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 information

GRAPHING IN POLAR COORDINATES SYMMETRY

GRAPHING IN POLAR COORDINATES SYMMETRY GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry - y-axis,

More information

Graphing calculators Transparencies (optional)

Graphing calculators Transparencies (optional) What if it is in pieces? Piecewise Functions and an Intuitive Idea of Continuity Teacher Version Lesson Objective: Length of Activity: Students will: Recognize piecewise functions and the notation used

More information

Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

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

Don't Forget the Differential Equations: Finishing 2005 BC4

Don't Forget the Differential Equations: Finishing 2005 BC4 connect to college success Don't Forget the Differential Equations: Finishing 005 BC4 Steve Greenfield available on apcentral.collegeboard.com connect to college success www.collegeboard.com The College

More information

Vocabulary Words and Definitions for Algebra

Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

More information

Unit 7: Radical Functions & Rational Exponents

Unit 7: Radical Functions & Rational Exponents Date Period Unit 7: Radical Functions & Rational Exponents DAY 0 TOPIC Roots and Radical Expressions Multiplying and Dividing Radical Expressions Binomial Radical Expressions Rational Exponents 4 Solving

More information

Solve 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 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 information

1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved.

1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved. 1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs

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

Notes for EER #4 Graph transformations (vertical & horizontal shifts, vertical stretching & compression, and reflections) of basic functions.

Notes for EER #4 Graph transformations (vertical & horizontal shifts, vertical stretching & compression, and reflections) of basic functions. Notes for EER #4 Graph transformations (vertical & horizontal shifts, vertical stretching & compression, and reflections) of basic functions. Basic Functions In several sections you will be applying shifts

More information

Solving Quadratic Equations

Solving Quadratic Equations 9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

More information

AP Calculus AB 2010 Free-Response Questions Form B

AP Calculus AB 2010 Free-Response Questions Form B AP Calculus AB 2010 Free-Response Questions Form B The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity.

More information

Introduction to Quadratic Functions

Introduction to Quadratic Functions Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2

More information

AP Calculus AB 2005 Scoring Guidelines Form B

AP Calculus AB 2005 Scoring Guidelines Form B AP Calculus AB 5 coring Guidelines Form B The College Board: Connecting tudents to College uccess The College Board is a not-for-profit membership association whose mission is to connect students to college

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

COMMON CORE STATE STANDARDS FOR MATHEMATICS 3-5 DOMAIN PROGRESSIONS

COMMON CORE STATE STANDARDS FOR MATHEMATICS 3-5 DOMAIN PROGRESSIONS COMMON CORE STATE STANDARDS FOR MATHEMATICS 3-5 DOMAIN PROGRESSIONS Compiled by Dewey Gottlieb, Hawaii Department of Education June 2010 Operations and Algebraic Thinking Represent and solve problems involving

More information

Math Placement Test Practice Problems

Math Placement Test Practice Problems Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211

More information

Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

More information

AP Calculus AB 2005 Free-Response Questions

AP Calculus AB 2005 Free-Response Questions AP Calculus AB 25 Free-Response Questions The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to

More information

Answer Key for California State Standards: Algebra I

Answer 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 information

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions. Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

More information

If Σ is an oriented surface bounded by a curve C, then the orientation of Σ induces an orientation for C, based on the Right-Hand-Rule.

If Σ is an oriented surface bounded by a curve C, then the orientation of Σ induces an orientation for C, based on the Right-Hand-Rule. Oriented Surfaces and Flux Integrals Let be a surface that has a tangent plane at each of its nonboundary points. At such a point on the surface two unit normal vectors exist, and they have opposite directions.

More information

5.1 Derivatives and Graphs

5.1 Derivatives and Graphs 5.1 Derivatives and Graphs What does f say about f? If f (x) > 0 on an interval, then f is INCREASING on that interval. If f (x) < 0 on an interval, then f is DECREASING on that interval. A function has

More information

AP Calculus AB 2003 Scoring Guidelines

AP 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 information

56 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 224 points.

56 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 224 points. 6.1.1 Review: Semester Review Study Sheet Geometry Core Sem 2 (S2495808) Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which

More information