INTERMEDIATE VALUE THEOREM
|
|
- Anne Gray
- 5 years ago
- Views:
Transcription
1 INTERMEDIATE VALUE THEOREM Section 1.4B Calculus AP/Dual, Revised 017 7/30/018 1:9 AM 1.4B: Intermediate Value Theorem 1
2 DEFINITION OF CONTINUITY A function is continuous at the point x = c if and only if: 1. f(c) is continuous ) lim f x exists x c 3) lim f x = f(c) x c 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem
3 EXAMPLES OF DISCONTINUOUS 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 3
4 PROOF OF INTERMEDIATE VALUE THEOREM Can you prove that at one time, you were exactly feet tall? f b 5'5" k 5'3" 5' f a If f is continuous on a, b and k is between f(a) and f(b) then there exists a number c between a and b such that f(c) = k 10yrs a c 11yrs b 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 4
5 INTERMEDIATE VALUE THEOREM A. If f(x) is continuous on the closed interval a, b B. f a f b C. If k is between f a and f b then there exists a number c between a and b for f c = k 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 5
6 EXAMPLE 1 Use the IVT to prove that the function f x = x is 7 on the interval between, a c 5 b f x a b f b r A. If is continuous on the closed interval, B. f a C. If k is between f a and f b then there exists a number c between a and b fo f c k c f c Since f x is continuous on,5 and 7 is between f and f 5, then IVT exists where a number between and 5 for 7 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 6
7 EXAMPLE If f x = ln x, prove by the IVT that there is a root on the interval of 1, 3. 1 A. f ( x ) is continuous on the closed interval,3 where lim and lim are real numbers f x f x x0.5 x3 1 1 B. f f 3 ;ln 0;ln C.If k is between f and f (3) then there exists a number " c" between and 3 for f ( c) k 1 Since f x is continuous on,3, then IVT exists 1 where a number "c" between and 3 for f c k 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 7
8 EXAMPLE 3 If f x = x + x 1, prove the IVT holds through the indicated interval of 0, 5. If the IVT applies, find the value of c for f c = 11. What are the extremes? (other words f a and f b )? f 0 x x 1 f f 0 1 f x x x 1 Interval These are the two extremes. : 0,5 f 5 x x 1 f /30/018 1:30 AM 1.4B: Intermediate Value Theorem 8 f 5 9
9 EXAMPLE 3 If f x = x + x 1, prove the IVT holds through the indicated interval of 0, 5. If the IVT applies, find the value of c for f c = 11. f a k f b x x1 0 x x1 f x x 3 x 4 f c 11 4,3 or x 4, x 3 then IVT exists where c 3. 3 is in 0,5 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 9 A. f ( x ) is continuous on the closed interval 0,5 where lim f x 1 and lim f x 9 x0 x5 is f and f and for f c B. f 0 f 5 ; f 0 11 and f 5 11 C.If 11 between 0 5 then there exists a number "c" between Since f x is continuous on 0,5 and 11 is between f 0 and f 5,
10 EXAMPLE 4 If f x = x +x, prove the IVT holds through the indicated interval of 5 x 1, 4 if f c = 6. If the IVT applies, find the value of c for f c = Since f x is continuous on,4 and f c is between f and f 4, then IVT exists where c 3 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 10
11 EXAMPLE 4 (EXTENSION) Would the IVT hold for f x = x +x, through the indicated interval of 3, 7? Explain why. x 1 Since f x has a non-removable discontinuity at x 1 on 3,7 then IVT does not exist where. 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 11
12 EXAMPLE 5 If f x = x 6x + 8, prove the IVT holds through the indicated interval of 0, 3. If the IVT applies, find the value of c for f c = 3. Since f x is continuous on 0,3 and 3 is between f 0 and f 5, then IVT exists where c 1. 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 1
13 If f x = 1 x YOUR TURN, use the Intermediate Value Theorem to prove for c on the interval 5, 7 if f c = Since f x is continuous on,7 and f c is between f and f 4, 1 then IVT exists where f c 4 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 13
14 TO EARN FULL CREDIT: A. The function, f x (or whatever they give) is identified, and stated to be CONTINUOUS. B. Include the function is continuous in a, b where a and b are defined C. State the value of c, if asked to be defined. 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 14
15 PIECEWISE FUNCTIONS A. For a piecewise function to be continuous each function must be continuous on its specified interval and the limit of the endpoints of each interval must be equal. 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 15
16 EXAMPLE 6 What value of k will make the given piecewise function f x continuous x +5x 3 at x = 3 of f x = ቐ, x 3 x 9 k, x = 3 x 5x3 lim x3 x 9 x1 x3 lim x3 x3 x3 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 16? x 1 lim x3 x 3
17 EXAMPLE 6 What value of k will make the given piecewise function f x continuous x +5x 3 at x = 3 of f x = ቐ x 9 k, lim x3 lim x3, x 3 x = 3 x 1 x 3 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem k 7 6?
18 EXAMPLE 6 What value of k will make the given piecewise function f x continuous x +5x 3 at x = 3 of f x = ቐ x 9 k,, x 3 x = 3? k 7 6 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 18
19 IN CONCLUSION A function exists when: 1. Point Exists. Limit Exists 3. Limit = Point 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 19
20 AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON-CALCULATOR) Let f be a continuous function on the closed interval 3, 6. If f 3 = 1 and f 6 = 3, then the Intermediate Value Theorem guarantees that: (A) f c = 4 for at least one c between 3 and 6 9 (B) 1 f(x) 3 for all x between 3 and 6 (C) f(c) = 1 for at least one c between 3 and 6 (D) f(c) = 0 for at least one c between 1 and 3 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 0
21 AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON-CALCULATOR) Let f be a continuous function on the closed interval 3, 6. If f 3 = 1 and f 6 = 3, then the Intermediate Value Theorem guarantees that: Vocabulary Connections and Process Answer and Justifications Continuous IVT Interval: 3, 6 f b f a Point: 3, 1, 6,3 c A) f ' MVT b a B) 1 y 3 Not necessarily true. How do we know? C) Is there a point in the I 3,6 when the y-value is 1? True. Since the range of the points are 1 and 3. D) Is there a point in the I 1,3 when the y-value of the point = 0? Possible. We don't know for sure. C There's a f c in the I 3,6 when the y-value is 1. Since the range of the points are 1 and 3, 1 must exist. 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem 1
22 ASSIGNMENT Worksheet 7/30/018 1:30 AM 1.4B: Intermediate Value Theorem
INTERMEDIATE VALUE THEOREM
INTERMEDIATE VALUE THEOREM Section 1.4B Calculus AP/Dual, Revised 017 viet.dang@humbleisd.net 7/30/018 1:36 AM 1.4B: Intermediate Value Theorem 1 PROOF OF INTERMEDIATE VALUE THEOREM Can you prove that
More informationIntermediate Value Theorem
Stewart Section 2.5 Continuity p. 1/ Intermediate Value Theorem The intermediate value theorem states that, if a function f is continuous on a closed interval [a,b] (that is, an interval that includes
More informationMEAN VALUE THEOREM. Section 3.2 Calculus AP/Dual, Revised /30/2018 1:16 AM 3.2: Mean Value Theorem 1
MEAN VALUE THEOREM Section 3. Calculus AP/Dual, Revised 017 viet.dang@humbleisd.net 7/30/018 1:16 AM 3.: Mean Value Theorem 1 ACTIVITY A. Draw a curve (x) on a separate sheet o paper within a deined closed
More informationStudent Study Session. Theorems
Students should be able to apply and have a geometric understanding of the following: Intermediate Value Theorem Mean Value Theorem for derivatives Extreme Value Theorem Name Formal Statement Restatement
More informationHomework for Section 1.4, Continuity and One sided Limits. Study 1.4, # 1 21, 27, 31, 37 41, 45 53, 61, 69, 87, 91, 93. Class Notes: Prof. G.
GOAL: 1. Understand definition of continuity at a point. 2. Evaluate functions for continuity at a point, and on open and closed intervals 3. Understand the Intermediate Value Theorum (IVT) Homework for
More informationSection 4.2: The Mean Value Theorem
Section 4.2: The Mean Value Theorem Before we continue with the problem of describing graphs using calculus we shall briefly pause to examine some interesting applications of the derivative. In previous
More informationConsequences of Continuity and Differentiability
Consequences of Continuity and Differentiability We have seen how continuity of functions is an important condition for evaluating limits. It is also an important conceptual tool for guaranteeing the existence
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 informationPARTICLE MOTION. Section 3.7A Calculus BC AP/Dual, Revised /30/2018 1:20 AM 3.7A: Particle Motion 1
PARTICLE MOTION Section 3.7A Calculus BC AP/Dual, Revised 2017 viet.dang@humbleisd.net 7/30/2018 1:20 AM 3.7A: Particle Motion 1 WHEN YOU SEE THINK When you see Think Initially t = 0 At rest v t = 0 At
More informationAVERAGE VALUE AND MEAN VALUE THEOREM
AVERAGE VALUE AND MEAN VALUE THEOREM Section 4.4A Calculus AP/Dual, Revised 017 viet.dang@humbleisd.net 7/30/018 3:00 AM 4.4A: Average Value and Mean Value Theorem 1 MATERIALS NEEDED A. Grid Paper B. Compass
More information10/9/10. The line x = a is a vertical asymptote of the graph of a function y = f(x) if either. Definitions and Theorems.
Definitions and Theorems Introduction Unit 2 Limits and Continuity Definition - Vertical Asymptote Definition - Horizontal Asymptote Definition Continuity Unit 3 Derivatives Definition - Derivative Definition
More information1.5 Continuity Continued & IVT
1.5 Continuity Continued & IVT A continuous function A non-continuous function To review the 3-step definition of continuity at a point, A function f ( x) is continuous at a point x = c if lim ( ) = (
More informationContinuity. To handle complicated functions, particularly those for which we have a reasonable formula or formulas, we need a more precise definition.
Continuity Intuitively, a function is continuous if its graph can be traced on paper in one motion without lifting the pencil from the paper. Thus the graph has no tears or holes. To handle complicated
More informationCHAIN RULE: DAY 2 WITH TRIG FUNCTIONS. Section 2.4A Calculus AP/Dual, Revised /30/2018 1:44 AM 2.4A: Chain Rule Day 2 1
CHAIN RULE: DAY WITH TRIG FUNCTIONS Section.4A Calculus AP/Dual, Revised 018 viet.dang@humbleisd.net 7/30/018 1:44 AM.4A: Chain Rule Day 1 THE CHAIN RULE A. d dx f g x = f g x g x B. If f(x) is a differentiable
More informationThe Intermediate Value Theorem If a function f (x) is continuous in the closed interval [ a,b] then [ ]
Lecture 2 5B Evaluating Limits Limits x ---> a The Intermediate Value Theorem If a function f (x) is continuous in the closed interval [ a,b] then [ ] the y values f (x) must take on every value on the
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 informationDEFINITION OF A DERIVATIVE
DEFINITION OF A DERIVATIVE Section 2.1 Calculus AP/Dual, Revised 2017 viet.dang@umbleisd.net 2.1: Definition of a Derivative 1 DEFINITION A. Te derivative of a function allows you to find te SLOPE OF THE
More informationMIDTERM 2. Section: Signature:
MIDTERM 2 Math 3A 11/17/2010 Name: Section: Signature: Read all of the following information before starting the exam: Check your exam to make sure all pages are present. When you use a major theorem (like
More informationLesson 59 Rolle s Theorem and the Mean Value Theorem
Lesson 59 Rolle s Theorem and the Mean Value Theorem HL Math - Calculus After this lesson, you should be able to: Understand and use Rolle s Theorem Understand and use the Mean Value Theorem 1 Rolle s
More informationAP Calculus AB Worksheet - Differentiability
Name AP Calculus AB Worksheet - Differentiability MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The figure shows the graph of a function. At the
More information( ) 4 and 20, find the value. v c is equal to this average CALCULUS WORKSHEET 1 ON PARTICLE MOTION
CALCULUS WORKSHEET 1 ON PARTICLE MOTION Work these on notebook paper. Use your calculator only on part (f) of problems 1. Do not use your calculator on the other problems. Write your justifications in
More informationChapter 2: Limits & Continuity
Name: Date: Period: AP Calc AB Mr. Mellina Chapter 2: Limits & Continuity Sections: v 2.1 Rates of Change of Limits v 2.2 Limits Involving Infinity v 2.3 Continuity v 2.4 Rates of Change and Tangent Lines
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 information5.5 Worksheet - Linearization
AP Calculus 4.5 Worksheet 5.5 Worksheet - Linearization All work must be shown in this course for full credit. Unsupported answers ma receive NO credit. 1. Consider the function = sin. a) Find the equation
More informationMath Section Bekki George: 08/28/18. University of Houston. Bekki George (UH) Math /28/18 1 / 37
Math 1431 Section 14616 Bekki George: bekki@math.uh.edu University of Houston 08/28/18 Bekki George (UH) Math 1431 08/28/18 1 / 37 Office Hours: Tuesdays and Thursdays 12:30-2pm (also available by appointment)
More informationLimits, Continuity, and the Derivative
Unit #2 : Limits, Continuity, and the Derivative Goals: Study and define continuity Review limits Introduce the derivative as the limit of a difference quotient Discuss the derivative as a rate of change
More informationSection 1.4 Tangents and Velocity
Math 132 Tangents and Velocity Section 1.4 Section 1.4 Tangents and Velocity Tangent Lines A tangent line to a curve is a line that just touches the curve. In terms of a circle, the definition is very
More informationLemma 15.1 (Sign preservation Lemma). Suppose that f : E R is continuous at some a R.
15. Intermediate Value Theorem and Classification of discontinuities 15.1. Intermediate Value Theorem. Let us begin by recalling the definition of a function continuous at a point of its domain. Definition.
More informationPARTICLE MOTION: DAY 2
PARTICLE MOTION: DAY 2 Section 3.6A Calculus AP/Dual, Revised 2018 viet.dang@humbleisd.net 7/30/2018 1:24 AM 3.6A: Particle Motion Day 2 1 WHEN YOU SEE THINK When you see Think Initially t = 0 At rest
More informationStudent Study Session Topic: Table Problems
Student Notes Student Study Session Topic: Table Problems The AP Calculus exams include multiple choice and free response questions in which the stem of the question includes a table of numerical information
More informationTheorems Solutions. Multiple Choice Solutions
Solutions We have intentionally included more material than can be covered in most Student Study Sessions to account for groups that are able to answer the questions at a faster rate. Use your own judgment,
More informationAP CALCULUS BC 2008 SCORING GUIDELINES
AP CALCULUS BC 2008 SCORING GUIDELINES Question 4 A particle moves along the x-axis so that its velocity at time t, for 0 t 6, is given by a differentiable function v whose graph is shown above. The velocity
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 informationMath 112 (Calculus I) Final Exam
Name: Student ID: Section: Instructor: Math 112 (Calculus I) Final Exam Dec 18, 7:00 p.m. Instructions: Work on scratch paper will not be graded. For questions 11 to 19, show all your work in the space
More informationIMPLICIT DIFFERENTIATION
IMPLICIT DIFFERENTIATION Section.5 Calculus AP/Dual, Revised 017 viet.dang@humbleisd.net 7/30/018 1:47 AM.5: Implicit Differentiation 1 REVIEW Solve or d of 4 + 3 = 6 4 3 6 4 3 6 4 3 4 ' 3 3 7/30/018 1:47
More informationThe Mean Value Theorem and its Applications
The Mean Value Theorem and its Applications Professor Richard Blecksmith richard@math.niu.edu Dept. of Mathematical Sciences Northern Illinois University http://math.niu.edu/ richard/math229 1. Extreme
More informationMean Value Theorem. MATH 161 Calculus I. J. Robert Buchanan. Summer Department of Mathematics
Mean Value Theorem MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Summer 2018 Background: Corollary to the Intermediate Value Theorem Corollary Suppose f is continuous on the closed interval
More informationTheorems (IVT, EVT, and MVT)
Theorems (IVT, EVT, and MVT) Students should be able to apply and have a geometric understanding of the following: Intermediate Value Theorem Mean Value Theorem for derivatives Extreme Value Theorem Multiple
More informationName Date Period. Worksheet 1.5 Continuity on Intervals & IVT. Show all work. No Calculator (unless stated otherwise) Short Answer
Name Date Period Worksheet 1.5 Continuity on Intervals & IVT Show all work. No Calculator (unless stated otherwise) Short Answer 1. A piece of the graph of (a) Sketch a graph of f ( x). x, x 1 f( x) =
More informationWhat is on today. 1 Linear approximation. MA 123 (Calculus I) Lecture 17: November 2, 2017 Section A2. Professor Jennifer Balakrishnan,
Professor Jennifer Balakrishnan, jbala@bu.edu What is on today 1 Linear approximation 1 1.1 Linear approximation and concavity....................... 2 1.2 Change in y....................................
More informationAP Calculus AB Chapter 1 Limits
AP Calculus AB Chapter Limits SY: 206 207 Mr. Kunihiro . Limits Numerical & Graphical Show all of your work on ANOTHER SHEET of FOLDER PAPER. In Exercises and 2, a stone is tossed vertically into the air
More informationMean Value Theorem. MATH 161 Calculus I. J. Robert Buchanan. Summer Department of Mathematics
Mean Value Theorem MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Summer 2018 Background: Corollary to the Intermediate Value Theorem Corollary Suppose f is continuous on the closed interval
More information1) Find the equations of lines (in point-slope form) passing through (-1,4) having the given characteristics:
AP Calculus AB Summer Worksheet Name 10 This worksheet is due at the beginning of class on the first day of school. It will be graded on accuracy. You must show all work to earn credit. You may work together
More informationMVT and Rolle s Theorem
AP Calculus CHAPTER 4 WORKSHEET APPLICATIONS OF DIFFERENTIATION MVT and Rolle s Teorem Name Seat # Date UNLESS INDICATED, DO NOT USE YOUR CALCULATOR FOR ANY OF THESE QUESTIONS In problems 1 and, state
More informationO.K. But what if the chicken didn t have access to a teleporter.
The intermediate value theorem, and performing algebra on its. This is a dual topic lecture. : The Intermediate value theorem First we should remember what it means to be a continuous function: A function
More information1.10 Continuity Brian E. Veitch
1.10 Continuity Definition 1.5. A function is continuous at x = a if 1. f(a) exists 2. lim x a f(x) exists 3. lim x a f(x) = f(a) If any of these conditions fail, f is discontinuous. Note: From algebra
More informationMath 122 Test 3. April 15, 2014
SI: Math 1 Test 3 April 15, 014 EF: 1 3 4 5 6 7 8 Total Name Directions: 1. No books, notes or 6 year olds with ear infections. You may use a calculator to do routine arithmetic computations. You may not
More informationAB Calc Sect Notes Monday, November 28, 2011
Assignments & Opportunities: I will TRY to have Sketchpad projects back to you next Monday or Tuesday. Tomorrow: p268; 5,22,27,45 & p280; 9 AB Calc Sect 4.3 - Notes Monday, November 28, 2011 Today's Topics
More informationTest 3 Review. y f(a) = f (a)(x a) y = f (a)(x a) + f(a) L(x) = f (a)(x a) + f(a)
MATH 2250 Calculus I Eric Perkerson Test 3 Review Sections Covered: 3.11, 4.1 4.6. Topics Covered: Linearization, Extreme Values, The Mean Value Theorem, Consequences of the Mean Value Theorem, Concavity
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus.1 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The only way to guarantee the eistence of a it is to algebraically prove
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 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 informationAP Calculus AB. Chapter IV Lesson B. Curve Sketching
AP Calculus AB Chapter IV Lesson B Curve Sketching local maxima Absolute maximum F I A B E G C J Absolute H K minimum D local minima Summary of trip along curve critical points occur where the derivative
More informationMA 123 (Calculus I) Lecture 6: September 19, 2016 Section A3. Professor Joana Amorim,
Professor Joana Amorim, jamorim@bu.edu What is on today 1 Continuity 1 1.1 Continuity checklist................................ 2 1.2 Continuity on an interval............................. 3 1.3 Intermediate
More informationMath 117: Honours Calculus I Fall, 2002 List of Theorems. a n k b k. k. Theorem 2.1 (Convergent Bounded) A convergent sequence is bounded.
Math 117: Honours Calculus I Fall, 2002 List of Theorems Theorem 1.1 (Binomial Theorem) For all n N, (a + b) n = n k=0 ( ) n a n k b k. k Theorem 2.1 (Convergent Bounded) A convergent sequence is bounded.
More informationUnit 4 Applications of Derivatives (Part I)
Unit 4 Applications of Derivatives (Part I) What is the same? 1. Same HW policy 2. Same expectations 3. Same level of difficulty but different format 4. Still understanding at a conceptual level but more
More informationSection I Multiple Choice 45 questions. Section II Free Response 6 questions
Section I Multiple Choice 45 questions Each question = 1.2 points, 54 points total Part A: No calculator allowed 30 questions in 60 minutes = 2 minutes per question Part B: Calculator allowed 15 questions
More informationInfinite Limits. Infinite Limits. Infinite Limits. Previously, we discussed the limits of rational functions with the indeterminate form 0/0.
Infinite Limits Return to Table of Contents Infinite Limits Infinite Limits Previously, we discussed the limits of rational functions with the indeterminate form 0/0. Now we will consider rational functions
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 informationAP Calculus AB Winter Break Packet Happy Holidays!
AP Calculus AB Winter Break Packet 04 Happy Holidays! Section I NO CALCULATORS MAY BE USED IN THIS PART OF THE EXAMINATION. Directions: Solve each of the following problems. After examining the form of
More informationAB Calculus: Rates of Change and Tangent Lines
AB Calculus: Rates of Change and Tangent Lines Name: The World Record Basketball Shot A group called How Ridiculous became YouTube famous when they successfully made a basket from the top of Tasmania s
More informationContinuity, Intermediate Value Theorem (2.4)
Continuity, Intermediate Value Theorem (2.4) Xiannan Li Kansas State University January 29th, 2017 Intuitive Definition: A function f(x) is continuous at a if you can draw the graph of y = f(x) without
More informationMAT 145. Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points
MAT 15 Test #2 Name Solution Guide Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points Use te grap of a function sown ere as you respond to questions 1 to 8. 1. lim f (x) 0 2. lim
More informationMath 106: Calculus I, Spring 2018: Midterm Exam II Monday, April Give your name, TA and section number:
Math 106: Calculus I, Spring 2018: Midterm Exam II Monday, April 6 2018 Give your name, TA and section number: Name: TA: Section number: 1. There are 6 questions for a total of 100 points. The value of
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus. Worksheet Day All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. The only way to guarantee the eistence of a it is to algebraically prove it.
More informationDo now as a warm up: Is there some number a, such that this limit exists? If so, find the value of a and find the limit. If not, explain why not.
Do now as a warm up: Is there some number a, such that this limit exists? If so, find the value of a and find the limit. If not, explain why not. 1 Continuity and One Sided Limits To say that a function
More informationMATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions.
MATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions. Continuity Definition. Given a set E R, a function f : E R, and a point c E, the function f is continuous at c if
More informationJustifications on the AP Calculus Exam
Justifications on the AP Calculus Exam Students are expected to demonstrate their knowledge of calculus concepts in 4 ways. 1. Numerically (Tables/Data) 2. Graphically 3. Analytically (Algebraic equations)
More informationMath 1: Calculus with Algebra Midterm 2 Thursday, October 29. Circle your section number: 1 Freund 2 DeFord
Math 1: Calculus with Algebra Midterm 2 Thursday, October 29 Name: Circle your section number: 1 Freund 2 DeFord Please read the following instructions before starting the exam: This exam is closed book,
More information1 Limits and continuity
1 Limits and continuity Question 1. Which of the following its can be evaluated by continuity ( plugging in )? sin(x) (a) x + 1 (d) x 3 x 2 + x 6 (b) e x sin(x) (e) x 2 + x 6 (c) x 2 x 2 + x 6 (f) n (
More informationChapter 5 Integrals. 5.1 Areas and Distances
Chapter 5 Integrals 5.1 Areas and Distances We start with a problem how can we calculate the area under a given function ie, the area between the function and the x-axis? If the curve happens to be something
More informationToday Applications of MVT Find where functions are increasing/decreasing Derivative tests for extrema
Today Applications of MVT Find where functions are increasing/decreasing Derivative tests for extrema Mean Value Theorem (proved by Cauchy in 1823) If f is continuous on [a, b] f(b) differentiable on (a,
More informationAP Calculus Prep Session Handout. Table Problems
AP Calculus Prep Session Handout The AP Calculus Exams include multiple choice and free response questions in which the stem of the question includes a table of numerical information from which the students
More informationRolle s Theorem. The theorem states that if f (a) = f (b), then there is at least one number c between a and b at which f ' (c) = 0.
Rolle s Theorem Rolle's Theorem guarantees that there will be at least one extreme value in the interior of a closed interval, given that certain conditions are satisfied. As with most of the theorems
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 informationAP CALCULUS (AB) Outline Chapter 4 Overview. 2) Recovering a function from its derivatives and a single point;
AP CALCULUS (AB) Outline Chapter 4 Overview NAME Date Objectives of Chapter 4 1) Using the derivative to determine extreme values of a function and the general shape of a function s graph (including where
More information3. (12 points) Find an equation for the line tangent to the graph of f(x) =
April 8, 2015 Name The total number of points available is 168 Throughout this test, show your work Throughout this test, you are expected to use calculus to solve problems Graphing calculator solutions
More informationFundamental Theorem of Calculus
Fundamental Theorem of Calculus MATH 6 Calculus I J. Robert Buchanan Department of Mathematics Summer 208 Remarks The Fundamental Theorem of Calculus (FTC) will make the evaluation of definite integrals
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION We have already investigated some applications of derivatives. However, now that we know the differentiation rules, we are in a better
More informationStudy 5.3 # , 157; 5.2 # 111, 113
Goals: 1. Recognize and understand the Fundamental Theorem of Calculus. 2. Use the Fundamental Theorum of Calculus to evaluate Definite Integrals. 3. Recognize and understand the Mean Value Theorem for
More informationMath Exam 1a. c) lim tan( 3x. 2) Calculate the derivatives of the following. DON'T SIMPLIFY! d) s = t t 3t
Math 111 - Eam 1a 1) Evaluate the following limits: 7 3 1 4 36 a) lim b) lim 5 1 3 6 + 4 c) lim tan( 3 ) + d) lim ( ) 100 1+ h 1 h 0 h ) Calculate the derivatives of the following. DON'T SIMPLIFY! a) y
More informationMATH 409 Advanced Calculus I Lecture 16: Mean value theorem. Taylor s formula.
MATH 409 Advanced Calculus I Lecture 16: Mean value theorem. Taylor s formula. Points of local extremum Let f : E R be a function defined on a set E R. Definition. We say that f attains a local maximum
More informationSection 2.6: Continuity
Section 2.6: Continuity Problem 1 (a) Let f(x) = x 1 x 2 5x. Then f(2) = 1 6 and f(6) = 5, but there is no value of c between 2 6 and 6 for which f(c) = 0. Does this fact violate the Intermediate Value
More informationMath 122 Test 3. April 17, 2018
SI: Math Test 3 April 7, 08 EF: 3 4 5 6 7 8 9 0 Total Name Directions:. No books, notes or April showers. You may use a calculator to do routine arithmetic computations. You may not use your calculator
More informationExam 3 review for Math 1190
Exam 3 review for Math 9 Be sure to be familiar with the following : Extreme Value Theorem Optimization The antiderivative u-substitution as a method for finding antiderivatives Reimann sums (e.g. L 6
More informationNumerical Differentiation & Integration. Numerical Differentiation II
Numerical Differentiation & Integration Numerical Differentiation II Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University
More informationWork the following on notebook paper. You may use your calculator to find
CALCULUS WORKSHEET ON 3.1 Work the following on notebook paper. You may use your calculator to find f values. 1. For each of the labeled points, state whether the function whose graph is shown has an absolute
More informationWithout fully opening the exam, check that you have pages 1 through 11.
Name: Section: Recitation Instructor: INSTRUCTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show all your work on the standard response
More informationMath 117: Honours Calculus I Fall, 2012 List of Theorems. a n k b k. k. Theorem 2.1 (Convergent Bounded): A convergent sequence is bounded.
Math 117: Honours Calculus I Fall, 2012 List of Theorems Theorem 1.1 (Binomial Theorem): For all n N, (a+b) n = n k=0 ( ) n a n k b k. k Theorem 2.1 (Convergent Bounded): A convergent sequence is bounded.
More information4. We accept without proofs that the following functions are differentiable: (e x ) = e x, sin x = cos x, cos x = sin x, log (x) = 1 sin x
4 We accept without proofs that the following functions are differentiable: (e x ) = e x, sin x = cos x, cos x = sin x, log (x) = 1 sin x x, x > 0 Since tan x = cos x, from the quotient rule, tan x = sin
More informationUniversity of Connecticut Department of Mathematics
University of Connecticut Department of Mathematics Math 1131 Sample Exam 2 Fall 2015 Name: Instructor Name: Section: TA Name: Discussion Section: This sample exam is just a guide to prepare for the actual
More informationHigh School AP Calculus AB Curriculum
High School AP Calculus AB Curriculum Course Description: AP Calculus AB is designed for the serious and motivated college-bound student planning to major in math, science or engineering. This course prepares
More informationAP Calculus BC. Chapter 2: Limits and Continuity 2.4: Rates of Change and Tangent Lines
AP Calculus BC Chapter 2: Limits and Continuity 2.4: Rates of Change and Tangent Lines Essential Questions & Why: Essential Questions: What is the difference between average and instantaneous rates of
More informationSection 3.1 Extreme Values
Math 132 Extreme Values Section 3.1 Section 3.1 Extreme Values Example 1: Given the following is the graph of f(x) Where is the maximum (x-value)? What is the maximum (y-value)? Where is the minimum (x-value)?
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus.1 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The only way to guarantee the eistence of a it is to algebraically prove
More informationTaylor and Maclaurin Series. Approximating functions using Polynomials.
Taylor and Maclaurin Series Approximating functions using Polynomials. Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear
More informationSince the two-sided limits exist, so do all one-sided limits. In particular:
SECTION 3.6 IN-SECTION EXERCISES: EXERCISE 1. The Intermediate Value Theorem 1. There are many correct graphs possible. A few are shown below. Since f is continuous on [a, b] and π is between f(a) = 3
More information[ ] with end points at ( a,f(a) ) and b,f(b)
Section 4 2B: Rolle s Theorem and the Mean Value Theorem The intermediate Value Theorem If f(x) is a continuous function on the closed interval a,b [ ] with end points at ( a,f(a) ) and b,f(b) ( )then
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Calculus 1 Instructor: James Lee Practice Exam 3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine from the graph whether the function
More informationThe Mean Value Theorem and the Extended Mean Value Theorem
The Mean Value Theorem and the Extended Mean Value Theorem Willard Miller September 21, 2006 0.1 The MVT Recall the Extreme Value Theorem (EVT) from class: If the function f is defined and continuous on
More information