Limits and Continuous Functions. 2.2 Introduction to Limits. We first interpret limits loosely. We write. lim f(x) = L
|
|
- Kelly White
- 6 years ago
- Views:
Transcription
1 2 Limits and Continuous Functions 2.2 Introduction to Limits We first interpret limits loosel. We write lim f() = L and sa the limit of f() as approaches c, equals L if we can make the values of f() arbitraril close to L (as close as we like) b taking to be sufficientl close to c (on either side of c) but possibl not equal to c. Eample 1. Evaluate the function e 1 at several points near = 0 and use the result to 2 estimate e 1 lim 0 2 Methods for Finding Limits To calculate limits we ma use 1. a table of values. 2. a graph. 3. algebra. 1
2 2 2 Limits and Continuous Functions Notation. The statement, the limit of f() as approaches c from the left is L is written lim f() = L. Similarl, the limit of f() as approaches c from the right is L is written f() = L. We call these statements the one-sided limits or the left and right hand + lim limits respectivel. Eample 2. Show that lim 0 does not eist. 1 Eample 3. Discuss the eistence of lim 0 2.
3 3 2.2 Introduction to Limits Eample 4. Discuss the eistence of lim 0 sin π. Noneistence of a Limit When a limit fails to eists, then the function ma have one of the following problems: 1. f() approaches a different value from the right side of c than it approaches from the left side. 2. f() increases or decreases without bound as approaches c. 3. f() oscillates between two fied values as approaches c.
4 4 2 Limits and Continuous Functions Definition. Let f be a function defined on an open interval containing c (ecept possibl at c) and let L be a real number. We sa that the limit of f() as approaches c is L and write lim f() = L provided that for each ε > 0 there eists a δ > 0 such that if 0 < c < δ, then f() L < ε.
5 5 2.2 Introduction to Limits Eample 5. Given lim 3 (2 1) = 5, find δ such that (2 1) 5 < 0.01 whenever 0 < 3 < δ. Eample 6. Prove that lim 3 (2 1) = 5.
6 6 2 Limits and Continuous Functions Eample 7. Prove that lim 3 2 = 9.
7 7 2.3 Limit Theorems 2.3 Limit Theorems Theorem 2.1. Let k and c be real numbers and let n be a positive integer. 1. lim k = 2. lim = 3. lim n =
8 8 2 Limits and Continuous Functions Theorem 2.2. Let k and c be real numbers and let n be a positive integer. Suppose that f and g are functions such that lim f() = L and lim g() = M 1. lim [ kf() ] = 2. lim [ f() ± g() ] = [ ] 3. lim f()g() = [ ] f() 4. lim = g() 5. lim [ f() ] n = Eample 1. Evaluate using the properties of limits. lim 1 ( ) Eample 2. Evaluate using the properties of limits. lim 2 5 3
9 9 2.3 Limit Theorems Theorem If p is a polnomial function and c is a real number, then lim p() = 2. If r is a rational function given b r() = p()/q(), where p and q are polnomials, and c is a real number such that q(c) 0, then lim r() = Eample 3. Evaluate the following limit. lim Theorem 2.4. Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid for c > 0 if n is even: lim n = Theorem 2.5. Suppose that f and g are functions such that lim g() = M and lim f() = L M lim f( g() ) = Eample 4. Evaluate the following limit. lim
10 10 2 Limits and Continuous Functions Theorem 2.6. Let c be a real number in the domain of each of the following functions: 1. lim sin = 3. lim tan = 5. lim sec = 7. lim a = 2. lim cos = 4. lim cot = 6. lim csc = 8. lim ln = Eample 5. Evaluate. lim e (5 + ln ) Eample 6. Evaluate. lim 1 ln cos Theorem 2.7. Let c be a real number and let f() = g() for all c in an open interval containing c. If lim g() eists, then also eists and lim f() = lim f() Eample 7. Evaluate. lim
11 Limit Theorems Eample 8. Evaluate. lim Theorem 2.8 (The Squeeze Theorem). If g() f() h() for all in an open interval containing c, ecept possibl at c itself, and if lim g() = eists, then lim f() =
12 12 2 Limits and Continuous Functions Theorem lim sin 1 cos = 2. lim =
13 Limit Theorems Eample 9. Evaluate. lim 0 tan Eample 10. Evaluate. lim 0 sin(5)
14 14 2 Limits and Continuous Functions 2.4 Continuit Definition. A function f is said to be continuous at a point c if the following conditions are met: 1. f(c) 2. lim f() 3. lim f() Furthermore, a function is continuous on an open interval (a, b) if it is continuous at each point in the interval. A function that is continuous on (, ) is said to be continuous everwhere. Note. Another wa to understand continuit is to consider where a function is discontinuous. This is done b looking at the negations of the above definition.
15 2.4 Continuit Note. Discontinuities fall into two categories: 1. Removable. Algebraicall: Graphicall: 2. Nonremovable. Algebraicall: Graphicall: Eample 1. Discuss the continuit of f() = 1 1 Eample 2. Discuss the continuit of g() = Eample 3. Discuss the continuit of h() = { 1 1 log > 1 Eample 4. Discuss the continuit of = cos 15
16 16 2 Limits and Continuous Functions Recall the use of one sided limits. Eample 5. Evaluate. lim Definition. The greatest integer function, denoted b [[]], is the greatest integer n such that n.
17 Continuit Theorem Let f be a function and let c and L be real numbers. Then lim f() = L if and onl if lim f() = L and lim f() = L + Definition. A function f is continuous on the closed interval [a, b] if it is continuous on the open interval (a, b) and f() = f(a) and lim f() = f(b) + lim a The function f is continuous from the right at a and continuous from the left at b. b Eample 6. Discuss the continuit of f() = 9 2. Theorem Let k be a real number and let f and g be continuous at = c. The following functions are also continuous at c
18 18 2 Limits and Continuous Functions Theorem If g is continuous at and f is continuous at g(c), then, ( f g ) () = f ( g() ) is continuous at c. Eample 7. Discuss the continuit of f() = cot. Eample 8. Discuss the continuit of g() = { cos = 0. Eample 9. Discuss the continuit of h() = { cos = 0.
19 Continuit Theorem 2.13 (The Intermediate Value Theorem). If f is continuous on [a, b] and k is an number between f(a) and f(b) (in other words f(a) k f(b) or f(b) k f(a)), then there eists at least one number c in [a, b] such that f(c) = k. Note. The theorem implies that on an great circle around the world, the temperature, pressure, elevation, carbon dioide concentration, or anthing else that varies continuousl, there will alwas eist two antipodal points that share the same value for that variable. Eample 10. Use the Intermediate Value Theorem to show that the polnomial function has a zero in [0, 3].
20 20 2 Limits and Continuous Functions 2.5 Infinite Limits Eample 1. Construct a table of values to determine the behavior of f() = as approaches Definition. Let f be a function that is defined at ever real number in some open interval containing c (ecept possibl at c itself). The statement lim f() = means that for each M > 0, there eists a δ > 0 such that f() > M whenever 0 < c < δ. Definition. If f() approaches infinit (or negative infinit) as approaches c from the right or the left, then the line = c is a vertical asmptote of the graph of f. Theorem Let f and g be continuous on an open interval containing c. If f(c) 0, g(c) = 0, and there eists an open interval containing c such that g() 0 for all c in the interval, then the graph of the function given b h() = f() g() has a vertical asmptote at = c. Eample 2. Find all vertical asmptotes of f() = 1 2
21 Infinite Limits Eample 3. Find all vertical asmptotes of g() = Eample 4. Find all vertical asmptotes of h() = Eample 5. Find all vertical asmptotes of = tan
22 22 2 Limits and Continuous Functions Theorem Let c and L be real numbers and let f and g be functions such that Then 1. lim [ f() ± g() ] = 2. lim [ f()g() ] = 3. lim g() f() = lim Note. The third item gives a special asmptote. f() = and lim g() = L. Eample 6. Suppose that f() = 3, g() = 1 4, and h() = 1. We illustrate item two as follows:
Limits and Their Properties
Chapter 1 Limits and Their Properties Course Number Section 1.1 A Preview of Calculus Objective: In this lesson you learned how calculus compares with precalculus. I. What is Calculus? (Pages 42 44) Calculus
More informationAn Intro to Limits Sketch to graph of 3
Limits and Their Properties A Preview of Calculus Objectives: Understand what calculus is and how it compares with precalculus.understand that the tangent line problem is basic to calculus. Understand
More informationThe main way we switch from pre-calc. to calc. is the use of a limit process. Calculus is a "limit machine".
A Preview of Calculus Limits and Their Properties Objectives: Understand what calculus is and how it compares with precalculus. Understand that the tangent line problem is basic to calculus. Understand
More information2.1 Rates of Change and Limits AP Calculus
. Rates of Change and Limits AP Calculus. RATES OF CHANGE AND LIMITS Limits Limits are what separate Calculus from pre calculus. Using a it is also the foundational principle behind the two most important
More informationSection 1.2: A Catalog of Functions
Section 1.: A Catalog of Functions As we discussed in the last section, in the sciences, we often tr to find an equation which models some given phenomenon in the real world - for eample, temperature as
More information2.5. Infinite Limits and Vertical Asymptotes. Infinite Limits
. Infinite Limits and Vertical Asmptotes. Infinite Limits and Vertical Asmptotes In this section we etend the concept of it to infinite its, which are not its as before, but rather an entirel new use of
More informationCHAPTER 1 Limits and Their Properties
CHAPTER Limits and Their Properties Section. A Preview of Calculus................... 305 Section. Finding Limits Graphically and Numerically....... 305 Section.3 Evaluating Limits Analytically...............
More information1.2 Functions and Their Properties PreCalculus
1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given
More informationCHAPTER 2 Limits and Their Properties
CHAPTER Limits and Their Properties Section. A Preview of Calculus...5 Section. Finding Limits Graphically and Numerically...5 Section. Section. Evaluating Limits Analytically...5 Continuity and One-Sided
More informationIn this note we will evaluate the limits of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0. f(x)
L Hôpital s Rule In this note we will evaluate the its of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0 f() Suppose a f() = 0 and a g() = 0. Then a g() the indeterminate
More information4.3 Mean-Value Theorem and Monotonicity
.3 Mean-Value Theorem and Monotonicit 1. Mean Value Theorem Theorem: Suppose that f is continuous on the interval a, b and differentiable on the interval a, b. Then there eists a number c in a, b such
More information1. d = 1. or Use Only in Pilot Program F Review Exercises 131
or Use Onl in Pilot Program F 0 0 Review Eercises. Limit proof Suppose f is defined for all values of near a, ecept possibl at a. Assume for an integer N 7 0, there is another integer M 7 0 such that f
More informationTHS Step By Step Calculus Chapter 1
Name: Class Period: Throughout this packet there will be blanks you are epected to fill in prior to coming to class. This packet follows your Larson Tetbook. Do NOT throw away! Keep in 3 ring binder until
More informationSEE and DISCUSS the pictures on pages in your text. Key picture:
Math 6 Notes 1.1 A PREVIEW OF CALCULUS There are main problems in calculus: 1. Finding a tangent line to a curve though a point on the curve.. Finding the area under a curve on some interval. SEE and DISCUSS
More information2.1 Rates of Change and Limits AP Calculus
.1 Rates of Change and Limits AP Calculus.1 RATES OF CHANGE AND LIMITS Limits Limits are what separate Calculus from pre calculus. Using a it is also the foundational principle behind the two most important
More information2.5 CONTINUITY. a x. Notice that Definition l implicitly requires three things if f is continuous at a:
SECTION.5 CONTINUITY 9.5 CONTINUITY We noticed in Section.3 that the it of a function as approaches a can often be found simpl b calculating the value of the function at a. Functions with this propert
More information1.2 Functions and Their Properties PreCalculus
1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given
More informationCHAPTER 3 Applications of Differentiation
CHAPTER Applications of Differentiation Section. Etrema on an Interval................... 0 Section. Rolle s Theorem and the Mean Value Theorem...... 0 Section. Increasing and Decreasing Functions and
More informationRolle s Theorem, the Mean Value Theorem, and L Hôpital s Rule
Rolle s Theorem, the Mean Value Theorem, and L Hôpital s Rule 5. Rolle s Theorem In the following problems (a) Verify that the three conditions of Rolle s theorem have been met. (b) Find all values z that
More information2.1 Limits, Rates of Change and Slopes of Tangent Lines
2.1 Limits, Rates of Change and Slopes of Tangent Lines (1) Average rate of change of y f x over an interval x 0,x 1 : f x 1 f x 0 x 1 x 0 Instantaneous rate of change of f x at x x 0 : f x lim 1 f x 0
More informationSummer Review Packet (Limits & Derivatives) 1. Answer the following questions using the graph of ƒ(x) given below.
Name AP Calculus BC Summer Review Packet (Limits & Derivatives) Limits 1. Answer the following questions using the graph of ƒ() given below. (a) Find ƒ(0) (b) Find ƒ() (c) Find f( ) 5 (d) Find f( ) 0 (e)
More information1.6 CONTINUITY OF TRIGONOMETRIC, EXPONENTIAL, AND INVERSE FUNCTIONS
.6 Continuit of Trigonometric, Eponential, and Inverse Functions.6 CONTINUITY OF TRIGONOMETRIC, EXPONENTIAL, AND INVERSE FUNCTIONS In this section we will discuss the continuit properties of trigonometric
More informationProperties of Limits
33460_003qd //04 :3 PM Page 59 SECTION 3 Evaluating Limits Analticall 59 Section 3 Evaluating Limits Analticall Evaluate a it using properties of its Develop and use a strateg for finding its Evaluate
More informationContinuity, End Behavior, and Limits. Unit 1 Lesson 3
Unit Lesson 3 Students will be able to: Interpret ke features of graphs and tables in terms of the quantities, and sketch graphs showing ke features given a verbal description of the relationship. Ke Vocabular:
More informationCHAPTER 3 Applications of Differentiation
CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. 0 Section. Rolle s Theorem and the Mean Value Theorem. 07 Section. Increasing and Decreasing Functions and the First
More informationChapter 1 Limits and Their Properties
Chapter 1 Limits and Their Properties Calculus: Chapter P Section P.2, P.3 Chapter P (briefly) WARM-UP 1. Evaluate: cot 6 2. Find the domain of the function: f( x) 3x 3 2 x 4 g f ( x) f ( x) x 5 3. Find
More information1.3 LIMITS AT INFINITY; END BEHAVIOR OF A FUNCTION
. Limits at Infinit; End Behavior of a Function 89. LIMITS AT INFINITY; END BEHAVIOR OF A FUNCTION Up to now we have been concerned with its that describe the behavior of a function f) as approaches some
More informationFinding Limits Graphically and Numerically. An Introduction to Limits
60_00.qd //0 :05 PM Page 8 8 CHAPTER Limits and Their Properties Section. Finding Limits Graphicall and Numericall Estimate a it using a numerical or graphical approach. Learn different was that a it can
More informationFinding Limits Graphically and Numerically. An Introduction to Limits
8 CHAPTER Limits and Their Properties Section Finding Limits Graphicall and Numericall Estimate a it using a numerical or graphical approach Learn different was that a it can fail to eist Stud and use
More informationUNIVERSIDAD CARLOS III DE MADRID MATHEMATICS II EXERCISES (SOLUTIONS )
UNIVERSIDAD CARLOS III DE MADRID MATHEMATICS II EXERCISES (SOLUTIONS ) CHAPTER : Limits and continuit of functions in R n. -. Sketch the following subsets of R. Sketch their boundar and the interior. Stud
More informationInfinite Limits. Let f be the function given by. f x 3 x 2.
0_005.qd //0 :07 PM Page 8 SECTION.5 Infinite Limits 8, as Section.5, as + f() = f increases and decreases without bound as approaches. Figure.9 Infinite Limits Determine infinite its from the left and
More informationIntegration. 5.1 Antiderivatives and Indefinite Integration. Suppose that f(x) = 5x 4. Can we find a function F (x) whose derivative is f(x)?
5 Integration 5. Antiderivatives and Indefinite Integration Suppose that f() = 5 4. Can we find a function F () whose derivative is f()? Definition. A function F is an antiderivative of f on an interval
More informationLimits and Continuity
Limits and Continuity Philippe B. Laval Kennesaw State University January 2, 2005 Contents Abstract Notes and practice problems on its and continuity. Limits 2. Introduction... 2.2 Theory:... 2.2. GraphicalMethod...
More informationCalculus I Exam 1 Review Fall 2016
Problem 1: Decide whether the following statements are true or false: (a) If f, g are differentiable, then d d x (f g) = f g. (b) If a function is continuous, then it is differentiable. (c) If a function
More informationx c x c This suggests the following definition.
110 Chapter 1 / Limits and Continuit 1.5 CONTINUITY A thrown baseball cannot vanish at some point and reappear someplace else to continue its motion. Thus, we perceive the path of the ball as an unbroken
More informationCHAPTER 3 Applications of Differentiation
CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. Section. Rolle s Theorem and the Mean Value Theorem. 7 Section. Increasing and Decreasing Functions and the First Derivative
More informationA function from a set D to a set R is a rule that assigns a unique element in R to each element in D.
1.2 Functions and Their Properties PreCalculus 1.2 FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1.2 1. Determine whether a set of numbers or a graph is a function 2. Find the domain of a function
More informationAP Calculus AB/IB Math SL2 Unit 1: Limits and Continuity. Name:
AP Calculus AB/IB Math SL Unit : Limits and Continuity Name: Block: Date:. A bungee jumper dives from a tower at time t = 0. Her height h (in feet) at time t (in seconds) is given by the graph below. In
More informationx f x
MATC 00 Class Notes - Sec.. Limits Idea: Look at the behavior of f as gets closer and closer to a specific number. Let f. We want to know the behavior of f when is close to a specific number, sa. Look
More information8 Differential Calculus 1 Introduction
8 Differential Calculus Introduction The ideas that are the basis for calculus have been with us for a ver long time. Between 5 BC and 5 BC, Greek mathematicians were working on problems that would find
More informationChapter 2. Limits and Continuity. 2.1 Rates of change and Tangents to Curves. The average Rate of change of y = f(x) with respect to x over the
Chapter 2 Limits and Continuity 2.1 Rates of change and Tangents to Curves Definition 2.1.1 : interval [x 1, x 2 ] is The average Rate of change of y = f(x) with respect to x over the y x = f(x 2) f(x
More informationCalculus I. 1. Limits and Continuity
2301107 Calculus I 1. Limits and Continuity Outline 1.1. Limits 1.1.1 Motivation:Tangent 1.1.2 Limit of a function 1.1.3 Limit laws 1.1.4 Mathematical definition of a it 1.1.5 Infinite it 1.1. Continuity
More informationContinuity at a Point
Continuity at a Point When we eplored the limit of f() as approaches c, the emphasis was on the function values close to = c rather than what happens to the function at = c. We will now consider the following
More informationSection 3.1. ; X = (0, 1]. (i) f : R R R, f (x, y) = x y
Paul J. Bruillard MATH 0.970 Problem Set 6 An Introduction to Abstract Mathematics R. Bond and W. Keane Section 3.1: 3b,c,e,i, 4bd, 6, 9, 15, 16, 18c,e, 19a, 0, 1b Section 3.: 1f,i, e, 6, 1e,f,h, 13e,
More informationContinuity and One-Sided Limits. By Tuesday J. Johnson
Continuity and One-Sided Limits By Tuesday J. Johnson Suggested Review Topics Algebra skills reviews suggested: Evaluating functions Rationalizing numerators and/or denominators Trigonometric skills reviews
More informationLecture Notes for Math 1000
Lecture Notes for Math 1000 Dr. Xiang-Sheng Wang Memorial University of Newfoundland Office: HH-2016, Phone: 864-4321 Office hours: 13:00-15:00 Wednesday, 12:00-13:00 Friday Email: swang@mun.ca Course
More information1 DL3. Infinite Limits and Limits at Infinity
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 78 Mark Sparks 01 Infinite Limits and Limits at Infinity In our graphical analysis of its, we have already seen both an infinite
More information3.5 Continuity of a Function One Sided Continuity Intermediate Value Theorem... 23
Chapter 3 Limit and Continuity Contents 3. Definition of Limit 3 3.2 Basic Limit Theorems 8 3.3 One sided Limit 4 3.4 Infinite Limit, Limit at infinity and Asymptotes 5 3.4. Infinite Limit and Vertical
More informationDRAFT - Math 101 Lecture Note - Dr. Said Algarni
2 Limits 2.1 The Tangent Problems The word tangent is derived from the Latin word tangens, which means touching. A tangent line to a curve is a line that touches the curve and a secant line is a line that
More informationChapter 2: Functions, Limits and Continuity
Chapter 2: Functions, Limits and Continuity Functions Limits Continuity Chapter 2: Functions, Limits and Continuity 1 Functions Functions are the major tools for describing the real world in mathematical
More informationLimits 4: Continuity
Limits 4: Continuit 55 Limits 4: Continuit Model : Continuit I. II. III. IV. z V. VI. z a VII. VIII. IX. Construct Your Understanding Questions (to do in class). Which is the correct value of f (a) in
More informationPre-Calculus Mathematics Limit Process Calculus
NOTES : LIMITS AND DERIVATIVES Name: Date: Period: Mrs. Nguyen s Initial: LESSON.1 THE TANGENT AND VELOCITY PROBLEMS Pre-Calculus Mathematics Limit Process Calculus The type of it that is used to find
More informationL 8.6 L 10.6 L 13.3 L Psud = u 3-4u 2 + 5u; [1, 2]
. Rates of Change and Tangents to Curves 6 p Q (5, ) (, ) (5, ) (, 65) Slope of PQ p/ t (flies/ da) - 5 5 - - 5 - - 5 5-65 - 5 - L 8.6 L.6 L. L 6. Number of flies 5 B(5, 5) Q(5, ) 5 5 P(, 5) 5 5 A(, )
More informationC) 2 D) 4 E) 6. ? A) 0 B) 1 C) 1 D) The limit does not exist.
. The asymptotes of the graph of the parametric equations = t, y = t t + are A) =, y = B) = only C) =, y = D) = only E) =, y =. What are the coordinates of the inflection point on the graph of y = ( +
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.4 Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at
More informationTrigonometry Outline
Trigonometr Outline Introduction Knowledge of the content of this outline is essential to perform well in calculus. The reader is urged to stud each of the three parts of the outline. Part I contains the
More informationLimits and Continuity of Functions of several Variables
Lesson: Limits and Continuit of Functions of several Variables Lesson Developer: Kapil Kumar Department/College: Assistant Professor, Department of Mathematics, A.R.S.D. College, Universit of Delhi Institute
More informationSolutions to Problem Sheet for Week 6
THE UNIVERSITY OF SYDNEY SCHOOL OF MATHEMATICS AND STATISTICS Solutions to Problem Sheet for Week 6 MATH90: Differential Calculus (Advanced) Semester, 07 Web Page: sydney.edu.au/science/maths/u/ug/jm/math90/
More informationMcKinney High School AP Calculus Summer Packet
McKinne High School AP Calculus Summer Packet (for students entering AP Calculus AB or AP Calculus BC) Name:. This packet is to be handed in to our Calculus teacher the first week of school.. ALL work
More informationLimits and Their Properties
Limits and Their Properties The it of a function is the primar concept that distinguishes calculus from algebra and analtic geometr. The notion of a it is fundamental to the stud of calculus. Thus, it
More informationOctober 27, 2018 MAT186 Week 3 Justin Ko. We use the following notation to describe the limiting behavior of functions.
October 27, 208 MAT86 Week 3 Justin Ko Limits. Intuitive Definitions of Limits We use the following notation to describe the iting behavior of functions.. (Limit of a Function A it is written as f( = L
More informationAdditional Material On Recursive Sequences
Penn State Altoona MATH 141 Additional Material On Recursive Sequences 1. Graphical Analsis Cobweb Diagrams Consider a generic recursive sequence { an+1 = f(a n ), n = 1,, 3,..., = Given initial value.
More informationCHAPTER 3 Applications of Differentiation
CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. 78 Section. Rolle s Theorem and the Mean Value Theorem. 8 Section. Increasing and Decreasing Functions and the First
More informationy »x 2» x 1. Find x if a = be 2x, lna = 7, and ln b = 3 HAL ln 7 HBL 2 HCL 7 HDL 4 HEL e 3
. Find if a = be, lna =, and ln b = HAL ln HBL HCL HDL HEL e a = be and taing the natural log of both sides, we have ln a = ln b + ln e ln a = ln b + = + = B. lim b b b = HAL b HBL b HCL b HDL b HEL b
More informationz-axis SUBMITTED BY: Ms. Harjeet Kaur Associate Professor Department of Mathematics PGGCG 11, Chandigarh y-axis x-axis
z-ais - - SUBMITTED BY: - -ais - - - - - - -ais Ms. Harjeet Kaur Associate Proessor Department o Mathematics PGGCG Chandigarh CONTENTS: Function o two variables: Deinition Domain Geometrical illustration
More informationTRIG REVIEW NOTES. Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents will equal)
TRIG REVIEW NOTES Convert from radians to degrees: multiply by 0 180 Convert from degrees to radians: multiply by 0. 180 Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.4 Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at
More informationDIFFERENTIATION. 3.1 Approximate Value and Error (page 151)
CHAPTER APPLICATIONS OF DIFFERENTIATION.1 Approimate Value and Error (page 151) f '( lim 0 f ( f ( f ( f ( f '( or f ( f ( f '( f ( f ( f '( (.) f ( f '( (.) where f ( f ( f ( Eample.1 (page 15): Find
More informationFixed-Point Iterations (2.2)
Fied-Point Iterations (.). Mean Value Theorem: a. Rolle s Theorem : Suppose that f is continuous on a, b and is differentiable on a, b.iff a f b, then there eists a number c in a, b such that f c. b. Mean
More informationLimits. or Use Only in Pilot Program F The Idea of Limits 2.2 Definitions of Limits 2.3 Techniques for Computing.
Limits or Use Onl in Pilot Program F 03 04. he Idea of Limits. Definitions of Limits.3 echniques for Computing Limits.4 Infinite Limits.5 Limits at Infinit.6 Continuit.7 Precise Definitions of Limits Biologists
More information1969 AP Calculus BC: Section I
969 AP Calculus BC: Section I 9 Minutes No Calculator Note: In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e).. t The asymptotes of the graph of the parametric
More information6.5 Trigonometric Equations
6. Trigonometric Equations In this section, we discuss conditional trigonometric equations, that is, equations involving trigonometric functions that are satisfied only by some values of the variable (or
More informationQUIZ ON CHAPTERS 1 AND 2 - SOLUTIONS REVIEW / LIMITS AND CONTINUITY; MATH 150 SPRING 2017 KUNIYUKI 105 POINTS TOTAL, BUT 100 POINTS = 100%
QUIZ ON CHAPTERS AND 2 - SOLUTIONS REVIEW / LIMITS AND CONTINUITY; MATH 50 SPRING 207 KUNIYUKI 05 POINTS TOTAL, BUT 00 POINTS = 00% ) For a), b), and c) below, bo in the correct answer. (6 points total;
More informationChapter One. Chapter One
Chapter One Chapter One CHAPTER ONE Hughes Hallett et al c 005, John Wile & Sons ConcepTests and Answers and Comments for Section.. Which of the following functions has its domain identical with its range?
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals 8. Basic Integration Rules In this section we will review various integration strategies. Strategies: I. Separate
More informationThe function graphed below is continuous everywhere. The function graphed below is NOT continuous everywhere, it is discontinuous at x 2 and
Section 1.4 Continuity A function is a continuous at a point if its graph has no gaps, holes, breaks or jumps at that point. If a function is not continuous at a point, then we say it is discontinuous
More informationIndeterminate Forms and L Hospital s Rule
APPLICATIONS OF DIFFERENTIATION Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at certain points. INDETERMINATE FORM TYPE
More information10.5 Graphs of the Trigonometric Functions
790 Foundations of Trigonometr 0.5 Graphs of the Trigonometric Functions In this section, we return to our discussion of the circular (trigonometric functions as functions of real numbers and pick up where
More informationCopyright c 2007 Jason Underdown Some rights reserved. quadratic formula. absolute value. properties of absolute values
Copyright & License Formula Copyright c 2007 Jason Underdown Some rights reserved. quadratic formula absolute value properties of absolute values equation of a line in various forms equation of a circle
More information2.3 The Fixed-Point Algorithm
.3 The Fied-Point Algorithm 1. Mean Value Theorem: Theorem Rolle stheorem: Suppose that f is continuous on a, b and is differentiable on a, b. If f a f b, then there eists a number c in a, b such that
More informationThings to remember: x n a 1. x + a 0. x n + a n-1. P(x) = a n. Therefore, lim g(x) = 1. EXERCISE 3-2
lim f() = lim (0.8-0.08) = 0, " "!10!10 lim f() = lim 0 = 0.!10!10 Therefore, lim f() = 0.!10 lim g() = lim (0.8 - "!10!10 0.042-3) = 1, " lim g() = lim 1 = 1.!10!0 Therefore, lim g() = 1.!10 EXERCISE
More informationMATH 1010E University Mathematics Lecture Notes (week 8) Martin Li
MATH 1010E University Mathematics Lecture Notes (week 8) Martin Li 1 L Hospital s Rule Another useful application of mean value theorems is L Hospital s Rule. It helps us to evaluate its of indeterminate
More information7.7. Inverse Trigonometric Functions. Defining the Inverses
7.7 Inverse Trigonometric Functions 57 7.7 Inverse Trigonometric Functions Inverse trigonometric functions arise when we want to calculate angles from side measurements in triangles. The also provide useful
More information3.1 Graphing Quadratic Functions. Quadratic functions are of the form.
3.1 Graphing Quadratic Functions A. Quadratic Functions Completing the Square Quadratic functions are of the form. 3. It is easiest to graph quadratic functions when the are in the form using transformations.
More information4 Inverse function theorem
Tel Aviv Universit, 2013/14 Analsis-III,IV 53 4 Inverse function theorem 4a What is the problem................ 53 4b Simple observations before the theorem..... 54 4c The theorem.....................
More informationFUNCTIONS. Note: Example of a function may be represented diagrammatically. The above example can be written diagrammatically as follows.
FUNCTIONS Def : A relation f from a set A into a set is said to be a function or mapping from A into if for each A there eists a unique such that (, ) f. It is denoted b f : A. Note: Eample of a function
More informationInduction, sequences, limits and continuity
Induction, sequences, limits and continuity Material covered: eclass notes on induction, Chapter 11, Section 1 and Chapter 2, Sections 2.2-2.5 Induction Principle of mathematical induction: Let P(n) be
More informationIn everyday speech, a continuous. Limits and Continuity. Critical Thinking Exercises
062 Chapter Introduction to Calculus Critical Thinking Eercises Make Sense? In Eercises 74 77, determine whether each statement makes sense or does not make sense, and eplain our reasoning. 74. I evaluated
More informationPolynomial and Rational Functions
Name Date Chapter Polnomial and Rational Functions Section.1 Quadratic Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Important Vocabular Define
More informationContinuity. MATH 161 Calculus I. J. Robert Buchanan. Fall Department of Mathematics
Continuity MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Fall 2017 Intuitive Idea A process or an item can be described as continuous if it exists without interruption. The mathematical
More informationSection 4.1 Increasing and Decreasing Functions
Section.1 Increasing and Decreasing Functions The graph of the quadratic function f 1 is a parabola. If we imagine a particle moving along this parabola from left to right, we can see that, while the -coordinates
More informationAP Calculus BC Prerequisite Knowledge
AP Calculus BC Prerequisite Knowledge Please review these ideas over the summer as they come up during our class and we will not be reviewing them during class. Also, I feel free to quiz you at any time
More informationLESSON #42 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART 2 COMMON CORE ALGEBRA II
LESSON #4 - INVERSES OF FUNCTIONS AND FUNCTION NOTATION PART COMMON CORE ALGEBRA II You will recall from unit 1 that in order to find the inverse of a function, ou must switch and and solve for. Also,
More informationKEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1
Chapter Function Transformations. Horizontal and Vertical Translations A translation can move the graph of a function up or down (vertical translation) and right or left (horizontal translation). A translation
More informationCONTINUITY AND DIFFERENTIABILITY
5. Introduction The whole of science is nothing more than a refinement of everyday thinking. ALBERT EINSTEIN This chapter is essentially a continuation of our stu of differentiation of functions in Class
More informationb. Create a graph that gives a more complete representation of f.
or Use Onl in Pilot Program F 96 Chapter Limits 6 7. Steep secant lines a. Given the graph of f in the following figures, find the slope of the secant line that passes through, and h, f h in terms of h,
More informationName DIRECTIONS: PLEASE COMPLET E ON A SEPARATE SHEET OF PAPER. USE THE ANSWER KEY PROVIDED TO CORRECT YOUR WORK. THIS WILL BE COLLECTED!!!
FINAL EXAM REVIEW 0 PRECALCULUS Name DIRECTIONS: PLEASE COMPLET E ON A SEPARATE SHEET OF PAPER. USE THE ANSWER KEY PROVIDED TO CORRECT YOUR WORK. THIS WILL BE COLLECTED!!! State the domain of the rational
More information1.1 Introduction to Limits
Chapter 1 LIMITS 1.1 Introduction to Limits Why Limit? Suppose that an object steadily moves forward, with s(t) denotes the position at time t. The average speed over the interval [1,2] is The average
More informationReview Exercises for Chapter 3. Review Exercises for Chapter r v 0 2. v ft sec. x 1 2 x dx f x x 99.4.
Review Eercises for Chapter 6. r v 0 sin. Let f, 00, d 0.6. v 0 00 ftsec changes from 0 to dr 00 cos d 6 0 d 0 r dr 80 00 6 96 feet 80 cos 0 96 feet 8080 f f fd d f 99. 00 0.6 9.97 00 Using a calculator:
More informationUnit 2 Review. No Calculator Allowed. 1. Find the domain of each function. (1.2)
PreCalculus Unit Review Name: No Calculator Allowed 1. Find the domain of each function. (1.) log7 a) g 9 7 b) hlog7 c) h 97 For questions &, (1.) (a) Find the domain (b) Identif an discontinuities as
More information