The Limit Concept. Introduction to Limits. Definition of Limit. Example 1. Example 2. Example 3 4/7/2015
|
|
- Cora Nelson
- 5 years ago
- Views:
Transcription
1 4/7/015 The Limit Concept Introduction to Limits Precalculus 1.1 The notion of a it is a fundamental concept of calculus. We will learn how to evaluate its and how they are used in the two basic problems of calculus: the tangent line problem and the area problem. Eample 1 Definition of Limit You are given 3 inches of wire and are asked to form a rectangle whose area is as large as possible. Determine the dimensions of the rectangle that will produce a maimum area. Eample Use the table to estimate numerically the it: (5 3) f()? Eample 3 Use the table to estimate numerically the it: f()? 1
2 4/7/015 Estimate the it: Eample Eample 5 Find the it of f() as approaches 5. 1, 5 f ( ), 5 Eample 6 Show that the it does not eist, 0 Eample 7 1 Discuss the eistence of the it 0 4 Eample 8 Discuss the eistence of the it 1 cos 0 Limits That Fail to Eist
3 4/7/015 Properties of Limits and Direct Substitution Properties of Limits and Direct Substitution Eample 9 Properties of Limits and Direct Substitution Find each it. 3 a) b) c) d) tan Find each it. Eample 10 a) 5 4 b) Techniques for Evaluating Limits Precalculus 1. 3
4 4/7/015 Eample 1 Eample Find the it: 8 Find the it: 3 4 Eample 3 Eample 4 Find the it: Approimate the it: f()? Eample 5 Approimate the it graphically: 1 cos 0 One-Sided Limits You saw that one way in which a it can fail to eist is when a function approaches a different value from the left side of c than it approaches from the right side of c. This type of behavior can be described more concisely with the concept of a one-sided it. 4
5 4/7/015 Eample 6 Find the it as 0 from the le and the it as 0 from the right for the func on. f ( ) 4 One-Sided Limits For the it of a function to eist as c, it must be true that both one-sided its eist and are equal. Eample 7 Find the it of f() as approaches. 1, f ( ) 1 6, Eample 8 To ship a package overnight, a delivery service charges $9 for the first pound and $1 for each additional pound or portion of a pound. Let represent the weight of the package and let f() represent the shipping cost. Show that the it of f() as approaches 3 does not eist. A Limit from Calculus Eample 9 You will study an important type of it from calculus the it of a difference quotient. f ( h) f ( ) h0 h For the function given by f ( h) f (). h0 h f ( ) 1, find 5
6 4/7/015 Tangent Line to a Graph The Tangent Line Problem Precalculus 1.3 Calculus is a branch of mathematics that studies rates of change of functions. If you go on to take a course in calculus, you will learn that rates of change have many applications in real life. Earlier in the tet, you learned how the slope of a line indicates the rate at which a line rises or falls. For a line, this rate (or slope) is the same at every point on the line. For graphs other than lines, the rate at which the graph rises or falls changes from point to point. Tangent Line to a Graph For instance, in Figure 1.0, the parabola is rising more quickly at the point ( 1, y 1 ) than it is at the point (, y ). At the verte ( 3, y 3 ), the graph levels off, and at the point ( 4, y 4 ), the graph is falling. Tangent Line to a Graph To determine the rate at which a graph rises or falls at a single point, you can find the slope of the tangent line at that point. In simple terms, the tangent line to the graph of a function f at a point P( 1, y 1 ) is the line that best approimates the slope of the graph at the point. Figure 1.0 Tangent Line to a Graph Figure 1.1 shows other eamples of tangent lines. Eample 1 Visually Approimating the Slope of a Graph Use the graph in Figure 1. to approimate the slope of the graph of f () = at the point (1, 1). Figure 1.1 Figure 1. 6
7 4/7/015 Slope and the Limit Process In Eample 1, you approimated the slope of a graph at a point by creating a graph and then eyeballing the tangent line at the point of tangency. A more precise method of approimating tangent lines makes use of a secant line through the point of tangency and a second point on the graph, as shown in Figure 1.4. Slope and the Limit Process If (, f()) is the point of tangency and ( + h, f( + h)) is a second point on the graph of f, the slope of the secant line through the two points is given by Slope of secant line The right side of this equation is called the difference quotient. The denominator h is the change in, and the numerator is the change in y. Figure 1.4 Slope and the Limit Process The beauty of this procedure is that you obtain more and more accurate approimations of the slope of the tangent line by choosing points closer and closer to the point of tangency, as shown in Figure 1.5. Slope and the Limit Process Using the it process, you can find the eact slope of the tangent line at (, f()). As h approaches 0, the secant line approaches the tangent line. From the definition above, you can see that the difference quotient is used frequently in calculus. Using the difference quotient to find the slope of a tangent line to a graph is a major concept of calculus. Figure 1.5 Eample 3 Eample 4 Find the slope of the graph of point (,8). f ( ) 3 at the Find the slope of f ( ) 3 5 when =4. 7
8 4/7/015 Eample 5 Find the formula for the slope of the graph of f ( ). What are the slopes at (-3,7) and (1,-1)? The Derivative of a Function You started with the function f() = + 1 and used the it process to derive another function, m =, that represents the slope of the graph of f at the point (, f()). This derived function is called the derivative of f at. It is denoted by f(), which is read as f prime of. The Derivative of a Function Eample 6 Find the derivative of f ( ) 4 5 Remember that the derivative f() is a formula for the slope of the tangent line to the graph of f at the point (, f()). Eample 7 Find the equation of the tangent line of f ( ) 4 5 at =1. Find f () for f ( ) 1. Then find the slopes of the graph of f at the points (4,3) and (9,4). Where does f ( ) 4 5 tangent line? have a horizontal 8
9 4/7/015 Derivative Shortcuts! Limit Process / Definition of Derivative Vs Shortcut Method If f() = c, then f () = If f() = a + b then f () = If f ( ) a b c then f () = Find the derivative of: f() = 4 f() = f ( ) 3 6 f() = 1/ f ( ) Limits at Infinity and Horizontal Asymptotes Limits at Infinity & Limits of Sequences The graph of f is shown in Figure Precalculus 1.4 Figure 1.30 Limits at Infinity and Horizontal Asymptotes Limits at Infinity and Horizontal Asymptotes From earlier work, you know that is a horizontal asymptote of the graph of this function. Using it notation, this can be written as follows. Horizontal asymptote to the left Horizontal asymptote to the right These its mean that the value of f() gets arbitrarily close to as decreases or increases without bound. 9
10 4/7/015 Limits at Infinity and Horizontal Asymptotes To help evaluate its at infinity, you can use the following definition. Find the it: Eample Eample Limits at Infinity and Horizontal Asymptotes Find the it as approaches for each function. 4 4 a) b) c) Eample 3 You are manufacturing greeting cards that cost $0.65 per card to produce. Your initial investment is $4500, which implies that the total cost of producing cards is given by C( ) Eample 3 cont d The average cost per card is given by C(). Find the average cost per card when a)=5000, b) =50,000, c) =500,000. d) What is the it of C() as approaches? C( ) 10
11 4/7/015 Limits of Sequences The following relationship shows how its of functions of can be used to evaluate the it of a sequence. Eample 4 Find the it of each sequence (assume n begins with 1) a) a 4n 5 4n 5 b) c) n n 1 n 1 4n 5 1n a n a n A sequence that does not converge is said to diverge. Eample 5 Find the it of the sequence whose nth term is a n 5 n( n 1)(n 1) 3 n 6 The Area Problem Precalculus 1.5 Limits of Summations We have used the concept of a it to obtain a formula for the sum S of an infinite geometric series Limits of Summations The following summation formulas and properties are used to evaluate finite and infinite summations. Using it notation, this sum can be written as 11
12 4/7/015 Eample 1 Evaluate the summation. 50 i1 i Eample Simplify the summation. n i n 3 S... n n n n n i1 Eample 3 Find the it of S(n) as n. n i 1 S( n) i1 n n The Area Problem You now have the tools needed to solve the second basic problem of calculus: the area problem. The problem is to find the area of the region R bounded by the graph of a nonnegative, continuous function f, the -ais, and the vertical lines = a and = b, as shown in Figure Figure 1.33 The Area Problem If the region R is a square, a triangle, a trapezoid, or a semicircle, you can find its area by using a geometric formula. For more general regions, however, you must use a different approach one that involves the it of a summation. Eample 4 Use five rectangles of equal width to approimate the area of the region bounded by f ( ) 8, the -ais, and the line =0 and =. The basic strategy is to use a collection of rectangles of equal width that approimates the region R, as illustrated in Eample 4. 1
13 4/7/015 The Area Problem Based on the procedure illustrated in Eample 4, the eact area of a plane region R is given by the it of the sum of n rectangles as n approaches. Eample 5 Find the area of the region bounded by the graph of f ( ), and the -ais between =0 and =1. Eample 6 Find the area of the region bounded by the graph of f ( ) 4, and the -ais between =1 and =3. 13
11.3 The Tangent Line Problem
11.3 The Tangent Line Problem Copyright Cengage Learning. All rights reserved. What You Should Learn Understand the tangent line problem. Use a tangent line to approximate the slope of a graph at a point.
More informationDifferentiation. The Derivative and the Tangent Line Problem 10/9/2014. Copyright Cengage Learning. All rights reserved.
Differentiation Copyright Cengage Learning. All rights reserved. The Derivative and the Tangent Line Problem Copyright Cengage Learning. All rights reserved. 1 Objectives Find the slope of the tangent
More informationCalculus I Review Handout 1.3 Introduction to Calculus - Limits. by Kevin M. Chevalier
Calculus I Review Handout 1.3 Introduction to Calculus - Limits by Kevin M. Chevalier We are now going to dive into Calculus I as we take a look at the it process. While precalculus covered more static
More informationLimits, Continuity, and Asymptotes
LimitsContinuity.nb 1 Limits, Continuity, and Asymptotes Limits Limit evaluation is a basic calculus tool that can be used in many different situations. We will develop a combined numerical, graphical,
More informationSlope of the Tangent Line. Estimating with a Secant Line
Slope of the Tangent Line Given a function f find the slope of the line tangent to the graph of f, that is, to the curve, at the point P(a, f (a)). The graph of a function f and the tangent line at a point
More informationSection 1.2 The Slope of a Tangent
Section 1.2 Te Slope of a Tangent You are familiar wit te concept of a tangent to a curve. Wat geometric interpretation can be given to a tangent to te grap of a function at a point? A tangent is te straigt
More informationPreCalculus Review for Math 400
PreCalculus Review for Math.) Completely factor..) For the function.) For the functions f ( ), evaluate ( ) f. f ( ) and g( ), find and simplify f ( g( )). Then, give the domain of f ( g( ))..) Solve.
More informationPre-Calculus 11: Final Review
Pre-Calculus 11 Name: Block: FORMULAS Sequences and Series Pre-Calculus 11: Final Review Arithmetic: = + 1 = + or = 2 + 1 Geometric: = = or = Infinite geometric: = Trigonometry sin= cos= tan= Sine Law:
More information4.2 and 4.6 filled in notes.notebook. December 08, Integration. Copyright Cengage Learning. All rights reserved.
4 Integration Copyright Cengage Learning. All rights reserved. 1 4.2 Area Copyright Cengage Learning. All rights reserved. 2 Objectives Use sigma notation to write and evaluate a sum. Understand the concept
More informationLimits and Their Properties. Copyright Cengage Learning. All rights reserved.
1 Limits and Their Properties Copyright Cengage Learning. All rights reserved. 1.1 A Preview of Calculus Copyright Cengage Learning. All rights reserved. What Is Calculus? 3 Calculus Calculus is the mathematics
More informationTechnology Assignment: Limits at Infinity
The goal of this technology assignment is to find the location of the horizontal asymptote for your model from Technology Assignment: Rational Model. You will produce a graph similar to the one below.
More information1.2 Functions and Graphs
Section.2 Functions and Graphs 3.2 Functions and Graphs You will be able to use the language, notation, and graphical representation of functions to epress relationships between variable quantities. Function,
More informationThe directional derivative of f x, y in the direction of at x, y u. f x sa y sb f x y (, ) (, ) 0 0 y 0 0
Review: 0, lim D f u 0 0 0 0 u The directional derivative of f, in the direction of at, is denoted b D f, : u a, b must a unit vector u f sa sb f s 0 (, ) (, ) s f (, ) a f (, ) b 0 0 0 0 0 0 D f, f u
More informationAP Calculus BC Course Description
AP Calculus BC Course Description COURSE OUTLINE: The following topics define the AP Calculus BC course as it is taught over three trimesters, each consisting of twelve week grading periods. Limits and
More informationlim x c x 2 x +2. Suppose that, instead of calculating all the values in the above tables, you simply . What do you find? x +2
MA123, Chapter 3: The idea of its (pp. 47-67, Gootman) Chapter Goals: Evaluate its. Evaluate one-sided its. Understand the concepts of continuity and differentiability and their relationship. Assignments:
More informationGraphing Functions. 0, < x < 0 1, 0 x < is defined everywhere on R but has a jump discontinuity at x = 0. h(x) =
Graphing Functions Section. of your tetbook is devoted to reviewing a series of steps that you can use to develop a reasonable graph of a function. Here is my version of a list of things to check. You
More informationApplications of Integration. Copyright Cengage Learning. All rights reserved.
Applications of Integration Copyright Cengage Learning. All rights reserved. Volume: The Disk Method Copyright Cengage Learning. All rights reserved. Objectives Find the volume of a solid of revolution
More informationExample 1: Given below is the graph of the quadratic function f. Use the function and its graph to find the following: Outputs
Quadratic Functions: - functions defined by quadratic epressions (a 2 + b + c) o the degree of a quadratic function is ALWAYS 2 - the most common way to write a quadratic function (and the way we have
More informationUnit 1: Sections Skill Set
MthSc 106 Fall 2011 Calculus of One Variable I : Calculus by Briggs and Cochran Section 1.1: Review of Functions Unit 1: Sections 1.1 3.3 Skill Set Find the domain and range of a function. 14, 17 13, 15,
More informationTABLE OF CONTENTS CHAPTER 1 LIMIT AND CONTINUITY... 26
TABLE OF CONTENTS CHAPTER LIMIT AND CONTINUITY... LECTURE 0- BASIC ALGEBRAIC EXPRESSIONS AND SOLVING EQUATIONS... LECTURE 0- INTRODUCTION TO FUNCTIONS... 9 LECTURE 0- EXPONENTIAL AND LOGARITHMIC FUNCTIONS...
More informationFinding Asymptotes KEY
Unit: 0 Lesson: 0 Discontinuities Rational functions of the form f ( are undefined at values of that make 0. Wherever a rational function is undefined, a break occurs in its graph. Each such break is called
More informationDerivatives and Graphs of Functions
Derivatives and Graphs of Functions September 8, 2014 2.2 Second Derivatives, Concavity, and Graphs In the previous section, we discussed how our derivatives can be used to obtain useful information about
More information3 Applications of Di erentiation
9 Applications of Di erentiation This chapter is intended to illustrate some of the applications of di erentiation. The activities of Section. illustrate the relationship between the values of rst and
More informationMath 14 Lecture Notes Ch. 6.1
6.1 Normal Distribution What is normal? a 10-year old boy that is 4' tall? 5' tall? 6' tall? a 25-year old woman with a shoe size of 5? 7? 9? an adult alligator that weighs 200 pounds? 500 pounds? 800
More informationEXPLORING RATIONAL FUNCTIONS GRAPHICALLY
EXPLORING RATIONAL FUNCTIONS GRAPHICALLY Precalculus Project Objectives: To find patterns in the graphs of rational functions. To construct a rational function using its properties. Required Information:
More informationLimits and Derivatives (Review of Math 249 or 251)
Chapter 3 Limits and Derivatives (Review of Math 249 or 251) 3.1 Overview This is the first of two chapters reviewing material from calculus; its and derivatives are discussed in this chapter, and integrals
More informationHonors Precalculus: Solving equations and inequalities graphically and algebraically. Page 1
Solving equations and inequalities graphically and algebraically 1. Plot points on the Cartesian coordinate plane. P.1 2. Represent data graphically using scatter plots, bar graphs, & line graphs. P.1
More informationSequences and Series. Copyright Cengage Learning. All rights reserved.
Sequences and Series Copyright Cengage Learning. All rights reserved. 12.3 Geometric Sequences Copyright Cengage Learning. All rights reserved. 2 Objectives Geometric Sequences Partial Sums of Geometric
More informationMA 180 Lecture Chapter 7 College Algebra and Calculus by Larson/Hodgkins Limits and Derivatives
MA 180 Lecture Chapter 7 College Algebra and Calculus by Larson/Hodgkins Limits and Derivatives 7.1) Limits An important concept in the study of mathematics is that of a it. It is often one of the harder
More informationDerivatives. Day 8 - Tangents and Linearizations
Derivatives Day 8 - Tangents and Linearizations Learning Objectives Write an equation for the tangent line to a graph Write an equation for the normal line to a graph Find the locations of horizontal and
More informationLines and Their Slopes
8.2 Lines and Their Slopes Linear Equations in Two Variables In the previous chapter we studied linear equations in a single variable. The solution of such an equation is a real number. A linear equation
More information7.5. Systems of Inequalities. The Graph of an Inequality. What you should learn. Why you should learn it
0_0705.qd /5/05 9:5 AM Page 5 Section 7.5 7.5 Sstems of Inequalities 5 Sstems of Inequalities What ou should learn Sketch the graphs of inequalities in two variables. Solve sstems of inequalities. Use
More informationYou should be able to visually approximate the slope of a graph. The slope m of the graph of f at the point x, f x is given by
Section. Te Tangent Line Problem 89 87. r 5 sin, e, 88. r sin sin Parabola 9 9 Hperbola e 9 9 9 89. 7,,,, 5 7 8 5 ortogonal 9. 5, 5,, 5, 5. Not multiples of eac oter; neiter parallel nor ortogonal 9.,,,
More information1.5 LIMITS. The Limit of a Function
60040_005.qd /5/05 :0 PM Page 49 SECTION.5 Limits 49.5 LIMITS Find its of functions graphicall and numericall. Use the properties of its to evaluate its of functions. Use different analtic techniques to
More informationThe following information is for reviewing the material since Exam 3:
Outcomes List for Math 121 Calculus I Fall 2010-2011 General Information: The purpose of this Outcomes List is to give you a concrete summary of the material you should know, and the skills you should
More informationModule 3 Graphing and Optimization
Module 3 Graphing and Optimization One of the most important applications of calculus to real-world problems is in the area of optimization. We will utilize the knowledge gained in the previous chapter,
More informationMath 20C. Lecture Examples.
Math 20C. Lecture Eamples. (8/7/0) Section 4.4. Linear approimations and tangent planes A function z = f(,) of two variables is linear if its graph in z-space is a plane. Equations of planes were found
More informationDomain: The domain of f is all real numbers except those values for which Q(x) =0.
Math 1330 Section.3.3: Rational Functions Definition: A rational function is a function that can be written in the form P() f(), where f and g are polynomials. Q() The domain of the rational function such
More informationAlgebra II Lesson 10-5: Hyperbolas Mrs. Snow, Instructor
Algebra II Lesson 10-5: Hyperbolas Mrs. Snow, Instructor In this section, we will look at the hyperbola. A hyperbola is a set of points P in a plane such that the absolute value of the difference between
More information3.6-Rational Functions & Their Graphs
.6-Rational Functions & Their Graphs What is a Rational Function? A rational function is a function that is the ratio of two polynomial functions. This definition is similar to a rational number which
More informationGraphing f ( x) = ax 2 + c
. Graphing f ( ) = a + c Essential Question How does the value of c affect the graph of f () = a + c? Graphing = a + c Work with a partner. Sketch the graphs of the functions in the same coordinate plane.
More information1-1. Functions. Lesson 1-1. What You ll Learn. Active Vocabulary. Scan Lesson 1-1. Write two things that you already know about functions.
1-1 Functions What You ll Learn Scan Lesson 1- Write two things that ou alread know about functions. Lesson 1-1 Active Vocabular New Vocabular Write the definition net to each term. domain dependent variable
More informationTI-89 Clinic. Let s first set up our calculators so that we re all working in the same mode.
TI-89 Clinic Preliminaries Let s first set up our calculators so that we re all working in the same mode. From the home screen, select F6 new problem. Hit enter to eecute that command. This erases previous
More informationMultivariable Calculus
Multivariable Calculus Chapter 10 Topics in Analytic Geometry (Optional) 1. Inclination of a line p. 5. Circles p. 4 9. Determining Conic Type p. 13. Angle between lines p. 6. Parabolas p. 5 10. Rotation
More informationWhat is the reasonable domain of this volume function? (c) Can there exist a volume of 0? (d) Estimate a maximum volume for the open box.
MA 15800 Lesson 11 Summer 016 E 1: From a rectangular piece of cardboard having dimensions 0 inches by 0 inches, an open bo is to be made by cutting out identical squares of area from each corner and,
More informationSLOPE A MEASURE OF STEEPNESS through 2.1.4
SLOPE A MEASURE OF STEEPNESS 2.1.2 through 2.1.4 Students used the equation = m + b to graph lines and describe patterns in previous courses. Lesson 2.1.1 is a review. When the equation of a line is written
More informationNow each of you should be familiar with inverses from your previous mathematical
5. Inverse Functions TOOTLIFTST: Knowledge of derivatives of basic functions, including power, eponential, logarithmic, trigonometric, and inverse trigonometric functions. Now each of you should be familiar
More informationPRACTICE FINAL - MATH 1210, Spring 2012 CHAPTER 1
PRACTICE FINAL - MATH 2, Spring 22 The Final will have more material from Chapter 4 than other chapters. To study for chapters -3 you should review the old practice eams IN ADDITION TO what appears here.
More information2.1. Definition: If a < b, then f(a) < f(b) for every a and b in that interval. If a < b, then f(a) > f(b) for every a and b in that interval.
1.1 Concepts: 1. f() is INCREASING on an interval: Definition: If a < b, then f(a) < f(b) for every a and b in that interval. A positive slope for the secant line. A positive slope for the tangent line.
More informationThis handout will discuss three kinds of asymptotes: vertical, horizontal, and slant.
CURVE SKETCHING This is a handout that will help you systematically sketch functions on a coordinate plane. This handout also contains definitions of relevant terms needed for curve sketching. ASYMPTOTES:
More informationGraphing Review. Math Tutorial Lab Special Topic
Graphing Review Math Tutorial Lab Special Topic Common Functions and Their Graphs Linear Functions A function f defined b a linear equation of the form = f() = m + b, where m and b are constants, is called
More information3.5 Rational Functions
0 Chapter Polnomial and Rational Functions Rational Functions For a rational function, find the domain and graph the function, identifing all of the asmptotes Solve applied problems involving rational
More informationPrecalculus Notes Unit 1 Day 1
Precalculus Notes Unit Day Rules For Domain: When the domain is not specified, it consists of (all real numbers) for which the corresponding values in the range are also real numbers.. If is in the numerator
More informationMATH 122 Final Exam Review Precalculus Mathematics for Calculus, 7 th ed., Stewart, et al
MATH Final Eam Review Precalculus Mathematics for Calculus, 7 th ed., Stewart, et al.. Mark the point determined by on the unit circle... Sketch a graph of y = sin( ) by hand... Find the amplitude, period,
More informationMastery. PRECALCULUS Student Learning Targets
PRECALCULUS Student Learning Targets Big Idea: Sequences and Series 1. I can describe a sequence as a function where the domain is the set of natural numbers. Connections (Pictures, Vocabulary, Definitions,
More informationUnit 12 Topics in Analytic Geometry - Classwork
Unit 1 Topics in Analytic Geometry - Classwork Back in Unit 7, we delved into the algebra and geometry of lines. We showed that lines can be written in several forms: a) the general form: Ax + By + C =
More informationUsing Characteristics of a Quadratic Function to Describe Its Graph. The graphs of quadratic functions can be described using key characteristics:
Chapter Summar Ke Terms standard form of a quadratic function (.1) factored form of a quadratic function (.1) verte form of a quadratic function (.1) concavit of a parabola (.1) reference points (.) transformation
More information1. (12 points) Find an equation for the line tangent to the graph of f(x) = xe 2x+4 at the point (2, f(2)).
April 13, 2011 Name The problems count as marked The total number of points available is 159 Throughout this test, show your work Use calculus to work the problems Calculator solutions which circumvent
More informationExploring AP Calculus With Colorful Calculator Investigations Deedee Stanfield
Eploring AP Calculus With Colorful Calculator Investigations Deedee Stanfield dstanfield.oh@oford.k12.al.us Eplore Limits, Derivatives, and Integration through hands-on activities that involve color-enhanced
More information(ii) Use Simpson s rule with two strips to find an approximation to Use your answers to parts (i) and (ii) to show that ln 2.
C umerical Methods. June 00 qu. 6 (i) Show by calculation that the equation tan = 0, where is measured in radians, has a root between.0 and.. [] Use the iteration formula n+ = tan + n with a suitable starting
More informationThe Fundamental Theorem of Calculus Using the Rule of Three
The Fundamental Theorem of Calculus Using the Rule of Three A. Approimations with Riemann sums. The area under a curve can be approimated through the use of Riemann (or rectangular) sums: n Area f ( k
More informationDownloaded from
1 Class XI: Math Chapter 13: Limits and Derivatives Chapter Notes Key-Concepts 1. The epected value of the function as dictated by the points to the left of a point defines the left hand it of the function
More information1.1 calculator viewing window find roots in your calculator 1.2 functions find domain and range (from a graph) may need to review interval notation
1.1 calculator viewing window find roots in your calculator 1.2 functions find domain and range (from a graph) may need to review interval notation functions vertical line test function notation evaluate
More informationName w s2q0f1q7r XKkuxt[az usrodfxtdw^atruev hlglucz.s r katldli SrCifgshPtMsw tryems`e_rgviesdr.
Precalculus Name w sq0f1q7r XKkut[az usrodftdw^atruev hlglucz.s r katldli SrCifgshPtMsw tryems`e_rgviesdr. Spring Final Review Solve each triangle. Round our answers to the nearest tenth. 1) ) B A 7 17
More informationAlgebra 2 Semester 2 Final Exam Study Outline Semester 2 Final Exam Study Tips and Information
Algebra 2 Semester 2 Final Exam Study Outline 2013 Semester 2 Final Exam Study Tips and Information The final exam is CUMULATIVE and will include all concepts taught from Chapter 1 through Chapter 13.
More information3.2 Extrema & Function Analysis Name: 1
Precalculus Write our questions and thoughts here! 3.2 Etrema & Function Analsis Name: 1 Absolute ma/min absolutel the. Relative ma/min a point on the function that is. Finding a ma/min means finding the
More informationChapter 1: Limits and Their Properties
1. Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus,
More information7h2 Graphing Basic Rational Functions (Gebrochenrationaler Funktionen)
7h Graphing Basic Rational Functions (Gebrochenrationaler Funktionen) Goals: Graph rational functions Find the zeros and y-intercepts Identify and understand the asymptotes Match graphs to their equations
More informationLimits. f(x) and lim. g(x) g(x)
Limits Limit Laws Suppose c is constant, n is a positive integer, and f() and g() both eist. Then,. [f() + g()] = f() + g() 2. [f() g()] = f() g() [ ] 3. [c f()] = c f() [ ] [ ] 4. [f() g()] = f() g()
More information3.9 Differentials. Tangent Line Approximations. Exploration. Using a Tangent Line Approximation
3.9 Differentials 3 3.9 Differentials Understand the concept of a tangent line approimation. Compare the value of the differential, d, with the actual change in,. Estimate a propagated error using a differential.
More informationMath Precalculus (12H/4H) Review. CHSN Review Project
Math Precalculus (12H/4H) Review CHSN Review Project Contents Functions 3 Polar and Complex Numbers 9 Sequences and Series 15 This review guide was written by Dara Adib. Prateek Pratel checked the Polar
More informationIB SL REVIEW and PRACTICE
IB SL REVIEW and PRACTICE Topic: CALCULUS Here are sample problems that deal with calculus. You ma use the formula sheet for all problems. Chapters 16 in our Tet can help ou review. NO CALCULATOR Problems
More informationMoore Catholic High School Math Department
Moore Catholic High School Math Department Geometry Vocabulary The following is a list of terms and properties which are necessary for success in a Geometry class. You will be tested on these terms during
More informationSupplemental 1.5. Objectives Interval Notation Increasing & Decreasing Functions Average Rate of Change Difference Quotient
Supplemental 1.5 Objectives Interval Notation Increasing & Decreasing Functions Average Rate of Change Difference Quotient Interval Notation Many times in this class we will only want to talk about what
More information9.3 Hyperbolas and Rotation of Conics
9.3 Hyperbolas and Rotation of Conics Copyright Cengage Learning. All rights reserved. What You Should Learn Write equations of hyperbolas in standard form. Find asymptotes of and graph hyperbolas. Use
More informationSECTION 3-4 Rational Functions
20 3 Polnomial and Rational Functions 0. Shipping. A shipping bo is reinforced with steel bands in all three directions (see the figure). A total of 20. feet of steel tape is to be used, with 6 inches
More informationx=2 26. y 3x Use calculus to find the area of the triangle with the given vertices. y sin x cos 2x dx 31. y sx 2 x dx
4 CHAPTER 6 APPLICATIONS OF INTEGRATION 6. EXERCISES 4 Find the area of the shaded region.. =5-. (4, 4) =. 4. = - = (_, ) = -4 =œ + = + =.,. sin,. cos, sin,, 4. cos, cos, 5., 6., 7.,, 4, 8., 8, 4 4, =_
More informationUnit 2 Functions Analyzing Graphs of Functions (Unit 2.2)
Unit 2 Functions Analzing Graphs of Functions (Unit 2.2) William (Bill) Finch Mathematics Department Denton High School Introduction Domain/Range Vert Line Zeros Incr/Decr Min/Ma Avg Rate Change Odd/Even
More information7.8. Approximate Integration
7.8. Approimate Integration Recall that when we originally defined the definite integral of a function interval, that we did so with the limit of a Riemann sum: on an where is the number of intervals that
More informationExam 1 Review. MATH Intuitive Calculus Fall Name:. Show your reasoning. Use standard notation correctly.
MATH 11012 Intuitive Calculus Fall 2012 Name:. Exam 1 Review Show your reasoning. Use standard notation correctly. 1. Consider the function f depicted below. y 1 1 x (a) Find each of the following (or
More informationCalculus Course Overview
Description: Walk in the footsteps of Newton and Leibnitz! An interactive text and graphing software combine with the exciting on-line course delivery to make Calculus an adventure. This course includes
More informationGraphs of Other Trig Functions
Graph y = tan. y 0 0 6 3 3 3 5 6 3 3 1 Graphs of Other Trig Functions.58 3 1.7 undefined 3 3 3 1.7-1 0.58 3 CHAT Pre-Calculus 3 The Domain is all real numbers ecept multiples of. (We say the domain is
More informationSLOPE A MEASURE OF STEEPNESS through 7.1.5
SLOPE A MEASURE OF STEEPNESS 7.1. through 7.1.5 Students have used the equation = m + b throughout this course to graph lines and describe patterns. When the equation is written in -form, the m is the
More informationLesson/Unit Plan Name: Comparing Linear and Quadratic Functions. Timeframe: 50 minutes + up to 60 minute assessment/extension activity
Grade Level/Course: Algebra 1 Lesson/Unit Plan Name: Comparing Linear and Quadratic Functions Rationale/Lesson Abstract: This lesson will enable students to compare the properties of linear and quadratic
More informationTheorems & Postulates Math Fundamentals Reference Sheet Page 1
Math Fundamentals Reference Sheet Page 1 30-60 -90 Triangle In a 30-60 -90 triangle, the length of the hypotenuse is two times the length of the shorter leg, and the length of the longer leg is the length
More information1.2 Visualizing and Graphing Data
6360_ch01pp001-075.qd 10/16/08 4:8 PM Page 1 1 CHAPTER 1 Introduction to Functions and Graphs 9. Volume of a Cone The volume V of a cone is given b V = 1 3 pr h, where r is its radius and h is its height.
More informationConic Sections. College Algebra
Conic Sections College Algebra Conic Sections A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines
More informationLevel 3 will generally. Level 2 may demonstrate limited ability to: Same as Level 2 Same as Level 2 identify models or
identify models or representations of multidigit division apply the distributive property to solve multi-digit division problems divide multi-digit whole numbers fluently using the standard algorithm Same
More informationCalculus Limits Images in this handout were obtained from the My Math Lab Briggs online e-book.
Calculus Limits Images in this handout were obtained from the My Math Lab Briggs online e-book. A it is the value a function approaches as the input value gets closer to a specified quantity. Limits are
More information15. PARAMETRIZED CURVES AND GEOMETRY
15. PARAMETRIZED CURVES AND GEOMETRY Parametric or parametrized curves are based on introducing a parameter which increases as we imagine travelling along the curve. Any graph can be recast as a parametrized
More informationInclination of a Line
0_00.qd 78 /8/05 Chapter 0 8:5 AM Page 78 Topics in Analtic Geometr 0. Lines What ou should learn Find the inclination of a line. Find the angle between two lines. Find the distance between a point and
More informationMth Test 3 Review Stewart 8e Chapter 4. For Test #3 study these problems, the examples in your notes, and the homework.
For Test #3 study these problems, the eamples in your notes, and the homework. I. Absolute Etrema A function, continuous on a closed interval, always has an absolute maimum and absolute minimum. They occur
More informationProblem 1: The relationship of height, in cm. and basketball players, names is a relation:
Chapter - Functions and Graphs Chapter.1 - Functions, Relations and Ordered Pairs Relations A relation is a set of ordered pairs. Domain of a relation is the set consisting of all the first elements of
More informationSection 10.1 Polar Coordinates
Section 10.1 Polar Coordinates Up until now, we have always graphed using the rectangular coordinate system (also called the Cartesian coordinate system). In this section we will learn about another system,
More informationDerivatives 3: The Derivative as a Function
Derivatives : The Derivative as a Function 77 Derivatives : The Derivative as a Function Model : Graph of a Function 9 8 7 6 5 g() - - - 5 6 7 8 9 0 5 6 7 8 9 0 5 - - -5-6 -7 Construct Your Understanding
More informationAccelerated Pre-Calculus Unit 1 Task 1: Our Only Focus: Circles & Parabolas Review
Accelerated Pre-Calculus Unit 1 Task 1: Our Only Focus: Circles & Parabolas Review Name: Date: Period: For most students, you last learned about conic sections in Analytic Geometry, which was a while ago.
More informationUnit 4 Trigonometry. Study Notes 1 Right Triangle Trigonometry (Section 8.1)
Unit 4 Trigonometr Stud Notes 1 Right Triangle Trigonometr (Section 8.1) Objective: Evaluate trigonometric functions of acute angles. Use a calculator to evaluate trigonometric functions. Use trigonometric
More informationCK-12 PreCalculus Concepts 1
Chapter Functions and Graphs Answer Ke. Functions Families. - - - - - - - -. - - - - - - - - CK- PreCalculus Concepts Chapter Functions and Graphs Answer Ke. - - - - - - - -. - - - - - - - - 5. - - - -
More informationMath 104, Spring 2010 Course Log
Math 104, Spring 2010 Course Log Date: 1/11 Sections: 1.3, 1.4 Log: Lines in the plane. The point-slope and slope-intercept formulas. Functions. Domain and range. Compositions of functions. Inverse functions.
More informationWalt Whitman High School SUMMER REVIEW PACKET. For students entering AP CALCULUS BC
Walt Whitman High School SUMMER REVIEW PACKET For students entering AP CALCULUS BC Name: 1. This packet is to be handed in to your Calculus teacher on the first day of the school year.. All work must be
More information