We say that the function f obtains a maximum value provided that there. We say that the function f obtains a minimum value provided that there

Size: px
Start display at page:

Download "We say that the function f obtains a maximum value provided that there. We say that the function f obtains a minimum value provided that there"

Transcription

1 Math 311 W08 Day 10 Section 3.2 Extreme Value Theorem (It s EXTREME!) 1. Definition: For a function f: D R we define the image of the function to be the set f(d) = {y y = f(x) for some x in D} We say that the function f obtains a maximum value provided that there exists x 0 in D such that f(x) f(x 0 ) for all x in D. In this case, x 0 is called the maximizer of f. We say that the function f obtains a minimum value provided that there exists x 0 in D such that f(x 0 ) f(x) for all x in D. In this case, x 0 is called the minimizer of f. So, a function has a maximum value if f(d) has a maximum element and a minimum value if f(d) has a minimum element. 2. Examples: a. The function f: (0, 1) R given by f(x) = x does not have a maximum or minimum value. b. The function f: [0, 1] R given by f(x) = x has a maximum (1) and minimum (0) value. c. The function f: (0, 1) R given by f(x) = 5 does have a maximum and minimum value (both 5). d. The function f: [0, 1] R given by f(0) = 7 and f(x) = 1/x otherwise, does not have a maximum value, but does have a minimum value (1).

2 3. We will prove that a continuous function defined on a closed bounded interval always has a maximum value. To do this, we will first show that the image of the function is bounded above, and then show that the supremum of the image is actually a function value (in the image). 4. Lemming The image of a continuous function on a closed bounded interval f: [a, b] R is bounded above. In other words, there is a real number M such that f(x) M for all x in [a, b]. Proof. Suppose not. Then for each natural number n, there is an x n in [a, b] such that f(x n ) > n. By the Sequential Compactness Theorem, {x n } has a subsequence, { x nk }, that converges to a point in [a, b], call it x 0. Then the image of this subsequence, f( x nk ) must converge to f(x 0 ) since f is continuous. But f( x nk )> n k > k for each natural number k. Thus the sequence f( x nk ) is convergent AND unbounded. I don t think so! Thus the image of the function must be bounded above.

3 5. EXTREME Value Theorem: A continuous function on a closed bounded interval ( f: [a, b] R ) obtains both a maximum and minimum value. Proof: By the previous Lemming, we know that f([a, b]) is bounded above (and it is clearly nonempty) so it has a supremum, c. Now for each natural number n, the number c 1/n is not an upper bound. Thus there is a number in f([a, b]), call it f(x n ) such that c 1/n < f(x n ) c. Clearly the sequence f(x n ) converges to c. And{ x n } is a sequence in [a, b], so by the Sequential Compactness Theorem, it must have a subsequence, x nk { }, that converges to an element in [a, b], call it x 0. Then since f is continuous, the image of this subsequence, f( x nk ), must converge to must converge to f(x 0 ). But since f( x nk ) is a subsequence of a sequence that converges to c, f( x nk ) must converge to c. Thus c = f(x 0 ). So the supremum is a function value WOO HOO! Now note that since f is continuous, so is f. Since f has a maximum (by above) f has a minimum at the same point.

4 3.3 The Intermediate Value Theorem 1. The Nested Interval Theorem (Theorem 2.29). For each natural number n, let a n and b n be numbers such that a n < b n, and consider the sequence of intervals I n = [a n, b n ]. If I n+1 I n and lim n!" [b n # a n ] = 0, then there is exactly one point that belongs to each interval I n for all n and both of the sequences { a n } and { b n } converge to this point. Proof: The sequence { a n } of left endpoints is an increasing sequence that is clearly bounded above by b 1. So it converges (by MCT) to its sup (call it a). Similarly the sequence { b n } of right endpoints is an decreasing sequence that is clearly bounded below by a 1. So it converges to its inf (call it b). Then by the difference property, lim n!" [b n # a n ] = b a (so b = a since this limit is assumed to be 0). Thus both sequences of endpoints converge to the same point. Since this point is the sup of the left endpoints, it is bigger than or equal to all the left endpoints. Since it is the inf of the right endpoints, it is smaller than or equal to all the right endpoints. Thus it is in each interval I n.

5 2. The Intermediate Value Theorem. Suppose f: [a, b] R is continuous and let c be a number that is strictly between f(a) and f(b). Then there exists a point x 0 in the open interval (a, b) at which f(x 0 ) = c. Proof Outline: Assume that f(a) < c < f(b). Let a 1 = a and b 1 = b. Split the interval [a 1, b 1 ]in half. Plug the midpoint, m 1, into the function. If the result is bigger than c, define [a 2, b 2 ] to be [a 1, m 1 ] and if the result is smaller than c, define [a 2, b 2 ] to be [m 1, b 1 ]. (If it is ever equal to c then you are done!) Then do it again and again and again You have made a sequence of nested intervals whose lengths are converging to zero. Thus by the nested interval theorem there is a point, x 0, to which both sequences of endpoints converge. Plugging these sequences into the function, we get two image sequences that converge to f(x 0 ). But since f(a n ) < c, for each index n, we know that f(x 0 ) c, and similarly, f(b n ) c. Thus f(x 0 ) = c. Extra Credit Opportunity! There is a gap (something left unproven) in the proof in the book (and I did not patch it in this outline). I ll add 3 points to your Exam 2 score if you can find it.

6 Class Work for Wednesday 1. Why is every polynomial a continuous function? 2. Prove that f(x) = 2x 3 + 3x 7 has a maximum, minimum, and root in the interval [-5, 10]. 3. Use the sequential definition of continuity to prove that f(x) = 3x 7 is continuous. 4. Use the ε-δ to prove that f(x) = 3x 7 is continuous. 5. Suppose that S is a set of real numbers that is not sequentially compact. Prove that either 1) there is an unbounded sequence of numbers in S, or 2) there is a sequence of numbers in S that converges to a point not in S. 6. Let a and b be real numbers with a < b. a. Find a continuous function on (a, b) with an image that is unbounded below. b. Find a continuous function on (a, b) that has a bounded (from below) image but does not have a minimum value. c. Find a continuous function on (a, b) that has both a maximum and a minimum value.

MATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions.

MATH 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 information

Solutions Final Exam May. 14, 2014

Solutions Final Exam May. 14, 2014 Solutions Final Exam May. 14, 2014 1. Determine whether the following statements are true or false. Justify your answer (i.e., prove the claim, derive a contradiction or give a counter-example). (a) (10

More information

MT804 Analysis Homework II

MT804 Analysis Homework II MT804 Analysis Homework II Eudoxus October 6, 2008 p. 135 4.5.1, 4.5.2 p. 136 4.5.3 part a only) p. 140 4.6.1 Exercise 4.5.1 Use the Intermediate Value Theorem to prove that every polynomial of with real

More information

Econ Lecture 7. Outline. 1. Connected Sets 2. Correspondences 3. Continuity for Correspondences

Econ Lecture 7. Outline. 1. Connected Sets 2. Correspondences 3. Continuity for Correspondences Econ 204 2011 Lecture 7 Outline 1. Connected Sets 2. Correspondences 3. Continuity for Correspondences 1 Connected Sets Definition 1. Two sets A, B in a metric space are separated if Ā B = A B = A set

More information

Homework for MATH 4603 (Advanced Calculus I) Fall Homework 13: Due on Tuesday 15 December. Homework 12: Due on Tuesday 8 December

Homework for MATH 4603 (Advanced Calculus I) Fall Homework 13: Due on Tuesday 15 December. Homework 12: Due on Tuesday 8 December Homework for MATH 4603 (Advanced Calculus I) Fall 2015 Homework 13: Due on Tuesday 15 December 49. Let D R, f : D R and S D. Let a S (acc S). Assume that f is differentiable at a. Let g := f S. Show that

More information

Walker Ray Econ 204 Problem Set 3 Suggested Solutions August 6, 2015

Walker Ray Econ 204 Problem Set 3 Suggested Solutions August 6, 2015 Problem 1. Take any mapping f from a metric space X into a metric space Y. Prove that f is continuous if and only if f(a) f(a). (Hint: use the closed set characterization of continuity). I make use of

More information

Econ Slides from Lecture 1

Econ Slides from Lecture 1 Econ 205 Sobel Econ 205 - Slides from Lecture 1 Joel Sobel August 23, 2010 Warning I can t start without assuming that something is common knowledge. You can find basic definitions of Sets and Set Operations

More information

Math 104 Section 2 Midterm 2 November 1, 2013

Math 104 Section 2 Midterm 2 November 1, 2013 Math 104 Section 2 Midterm 2 November 1, 2013 Name: Complete the following problems. In order to receive full credit, please provide rigorous proofs and show all of your work and justify your answers.

More information

Economics 204 Summer/Fall 2017 Lecture 7 Tuesday July 25, 2017

Economics 204 Summer/Fall 2017 Lecture 7 Tuesday July 25, 2017 Economics 204 Summer/Fall 2017 Lecture 7 Tuesday July 25, 2017 Section 2.9. Connected Sets Definition 1 Two sets A, B in a metric space are separated if Ā B = A B = A set in a metric space is connected

More information

1 Lecture 25: Extreme values

1 Lecture 25: Extreme values 1 Lecture 25: Extreme values 1.1 Outline Absolute maximum and minimum. Existence on closed, bounded intervals. Local extrema, critical points, Fermat s theorem Extreme values on a closed interval Rolle

More information

Math 341 Summer 2016 Midterm Exam 2 Solutions. 1. Complete the definitions of the following words or phrases:

Math 341 Summer 2016 Midterm Exam 2 Solutions. 1. Complete the definitions of the following words or phrases: Math 34 Summer 06 Midterm Exam Solutions. Complete the definitions of the following words or phrases: (a) A sequence (a n ) is called a Cauchy sequence if and only if for every ɛ > 0, there exists and

More information

Math 421, Homework #9 Solutions

Math 421, Homework #9 Solutions Math 41, Homework #9 Solutions (1) (a) A set E R n is said to be path connected if for any pair of points x E and y E there exists a continuous function γ : [0, 1] R n satisfying γ(0) = x, γ(1) = y, and

More information

Numerical Differentiation & Integration. Numerical Differentiation II

Numerical 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 information

Math 131 Exam 1 October 4, :00-9:00 p.m.

Math 131 Exam 1 October 4, :00-9:00 p.m. Name (Last, First) My Solutions ID # Signature Lecturer Section (01, 02, 03, etc.) university of massachusetts amherst department of mathematics and statistics Math 131 Exam 1 October 4, 2017 7:00-9:00

More information

8.7 Taylor s Inequality Math 2300 Section 005 Calculus II. f(x) = ln(1 + x) f(0) = 0

8.7 Taylor s Inequality Math 2300 Section 005 Calculus II. f(x) = ln(1 + x) f(0) = 0 8.7 Taylor s Inequality Math 00 Section 005 Calculus II Name: ANSWER KEY Taylor s Inequality: If f (n+) is continuous and f (n+) < M between the center a and some point x, then f(x) T n (x) M x a n+ (n

More information

Proof. We indicate by α, β (finite or not) the end-points of I and call

Proof. We indicate by α, β (finite or not) the end-points of I and call C.6 Continuous functions Pag. 111 Proof of Corollary 4.25 Corollary 4.25 Let f be continuous on the interval I and suppose it admits non-zero its (finite or infinite) that are different in sign for x tending

More information

Continuity. Chapter 4

Continuity. Chapter 4 Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of

More information

MATH 409 Advanced Calculus I Lecture 7: Monotone sequences. The Bolzano-Weierstrass theorem.

MATH 409 Advanced Calculus I Lecture 7: Monotone sequences. The Bolzano-Weierstrass theorem. MATH 409 Advanced Calculus I Lecture 7: Monotone sequences. The Bolzano-Weierstrass theorem. Limit of a sequence Definition. Sequence {x n } of real numbers is said to converge to a real number a if for

More information

Caculus 221. Possible questions for Exam II. March 19, 2002

Caculus 221. Possible questions for Exam II. March 19, 2002 Caculus 221 Possible questions for Exam II March 19, 2002 These notes cover the recent material in a style more like the lecture than the book. The proofs in the book are in section 1-11. At the end there

More information

Continuity. Chapter 4

Continuity. Chapter 4 Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of

More information

2.4 The Extreme Value Theorem and Some of its Consequences

2.4 The Extreme Value Theorem and Some of its Consequences 2.4 The Extreme Value Theorem and Some of its Consequences The Extreme Value Theorem deals with the question of when we can be sure that for a given function f, (1) the values f (x) don t get too big or

More information

Solutions Final Exam May. 14, 2014

Solutions Final Exam May. 14, 2014 Solutions Final Exam May. 14, 2014 1. (a) (10 points) State the formal definition of a Cauchy sequence of real numbers. A sequence, {a n } n N, of real numbers, is Cauchy if and only if for every ɛ > 0,

More information

1. For each statement, either state that it is True or else Give a Counterexample: (a) If a < b and c < d then a c < b d.

1. For each statement, either state that it is True or else Give a Counterexample: (a) If a < b and c < d then a c < b d. Name: Instructions. Show all work in the space provided. Indicate clearly if you continue on the back side, and write your name at the top of the scratch sheet if you will turn it in for grading. No books

More information

1. Is the set {f a,b (x) = ax + b a Q and b Q} of all linear functions with rational coefficients countable or uncountable?

1. Is the set {f a,b (x) = ax + b a Q and b Q} of all linear functions with rational coefficients countable or uncountable? Name: Instructions. Show all work in the space provided. Indicate clearly if you continue on the back side, and write your name at the top of the scratch sheet if you will turn it in for grading. No books

More information

Math 312 Spring08 Day 13

Math 312 Spring08 Day 13 HW 3: Due Monday May 19 th. 11. 1 # 2, 3, 6, 10 11. 2 # 1, 3, 6 Math 312 Spring08 Day 13 11.1 Continuous Functions and Mappings. 1. Definition. Let A be a subset of R n i. A mapping F: A R n is said to

More information

CHAPTER 3: CONTINUITY ON R 3.1 TWO SIDED LIMITS

CHAPTER 3: CONTINUITY ON R 3.1 TWO SIDED LIMITS CHAPTER 3: CONTINUITY ON R 3.1 TWO SIDED LIMITS DEFINITION. Let a R and let I be an open interval contains a, and let f be a real function defined everywhere except possibly at a. Then f(x) is said to

More information

Math Practice Final - solutions

Math Practice Final - solutions Math 151 - Practice Final - solutions 2 1-2 -1 0 1 2 3 Problem 1 Indicate the following from looking at the graph of f(x) above. All answers are small integers, ±, or DNE for does not exist. a) lim x 1

More information

Notes on Complex Analysis

Notes on Complex Analysis Michael Papadimitrakis Notes on Complex Analysis Department of Mathematics University of Crete Contents The complex plane.. The complex plane...................................2 Argument and polar representation.........................

More information

MA 1124 Solutions 14 th May 2012

MA 1124 Solutions 14 th May 2012 MA 1124 Solutions 14 th May 2012 1 (a) Use True/False Tables to prove (i) P = Q Q = P The definition of P = Q is given by P Q P = Q T T T T F F F T T F F T So Q P Q = P F F T T F F F T T T T T Since the

More information

1.10 Continuity Brian E. Veitch

1.10 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 information

Maximum and Minimum Values section 4.1

Maximum and Minimum Values section 4.1 Maximum and Minimum Values section 4.1 Definition. Consider a function f on its domain D. (i) We say that f has absolute maximum at a point x 0 D if for all x D we have f(x) f(x 0 ). (ii) We say that f

More information

Algebra 2 CP Semester 1 PRACTICE Exam

Algebra 2 CP Semester 1 PRACTICE Exam Algebra 2 CP Semester 1 PRACTICE Exam NAME DATE HR You may use a calculator. Please show all work directly on this test. You may write on the test. GOOD LUCK! THIS IS JUST PRACTICE GIVE YOURSELF 45 MINUTES

More information

MATH 409 Advanced Calculus I Lecture 9: Limit supremum and infimum. Limits of functions.

MATH 409 Advanced Calculus I Lecture 9: Limit supremum and infimum. Limits of functions. MATH 409 Advanced Calculus I Lecture 9: Limit supremum and infimum. Limits of functions. Limit points Definition. A limit point of a sequence {x n } is the limit of any convergent subsequence of {x n }.

More information

(a) For an accumulation point a of S, the number l is the limit of f(x) as x approaches a, or lim x a f(x) = l, iff

(a) For an accumulation point a of S, the number l is the limit of f(x) as x approaches a, or lim x a f(x) = l, iff Chapter 4: Functional Limits and Continuity Definition. Let S R and f : S R. (a) For an accumulation point a of S, the number l is the limit of f(x) as x approaches a, or lim x a f(x) = l, iff ε > 0, δ

More information

REAL VARIABLES: PROBLEM SET 1. = x limsup E k

REAL VARIABLES: PROBLEM SET 1. = x limsup E k REAL VARIABLES: PROBLEM SET 1 BEN ELDER 1. Problem 1.1a First let s prove that limsup E k consists of those points which belong to infinitely many E k. From equation 1.1: limsup E k = E k For limsup E

More information

Exam 2 extra practice problems

Exam 2 extra practice problems Exam 2 extra practice problems (1) If (X, d) is connected and f : X R is a continuous function such that f(x) = 1 for all x X, show that f must be constant. Solution: Since f(x) = 1 for every x X, either

More information

Iowa State University. Instructor: Alex Roitershtein Summer Homework #1. Solutions

Iowa State University. Instructor: Alex Roitershtein Summer Homework #1. Solutions Math 501 Iowa State University Introduction to Real Analysis Department of Mathematics Instructor: Alex Roitershtein Summer 015 EXERCISES FROM CHAPTER 1 Homework #1 Solutions The following version of the

More information

Scalar multiplication and addition of sequences 9

Scalar multiplication and addition of sequences 9 8 Sequences 1.2.7. Proposition. Every subsequence of a convergent sequence (a n ) n N converges to lim n a n. Proof. If (a nk ) k N is a subsequence of (a n ) n N, then n k k for every k. Hence if ε >

More information

This exam contains 5 pages (including this cover page) and 4 questions. The total number of points is 100. Grade Table

This exam contains 5 pages (including this cover page) and 4 questions. The total number of points is 100. Grade Table MAT25-2 Summer Session 2 207 Practice Final August 24th, 207 Time Limit: Hour 40 Minutes Name: Instructor: Nathaniel Gallup This exam contains 5 pages (including this cover page) and 4 questions. The total

More information

MATH 131A: REAL ANALYSIS (BIG IDEAS)

MATH 131A: REAL ANALYSIS (BIG IDEAS) MATH 131A: REAL ANALYSIS (BIG IDEAS) Theorem 1 (The Triangle Inequality). For all x, y R we have x + y x + y. Proposition 2 (The Archimedean property). For each x R there exists an n N such that n > x.

More information

Intermediate Value Theorem

Intermediate 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 information

Chapter 3 Continuous Functions

Chapter 3 Continuous Functions Continuity is a very important concept in analysis. The tool that we shall use to study continuity will be sequences. There are important results concerning the subsets of the real numbers and the continuity

More information

Math 328 Course Notes

Math 328 Course Notes Math 328 Course Notes Ian Robertson March 3, 2006 3 Properties of C[0, 1]: Sup-norm and Completeness In this chapter we are going to examine the vector space of all continuous functions defined on the

More information

d(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N

d(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N Problem 1. Let f : A R R have the property that for every x A, there exists ɛ > 0 such that f(t) > ɛ if t (x ɛ, x + ɛ) A. If the set A is compact, prove there exists c > 0 such that f(x) > c for all x

More information

Section 3.3 Maximum and Minimum Values

Section 3.3 Maximum and Minimum Values Section 3.3 Maximum and Minimum Values Definition For a function f defined on a set S of real numbers and a number c in S. A) f(c) is called the absolute maximum of f on S if f(c) f(x) for all x in S.

More information

A LITTLE REAL ANALYSIS AND TOPOLOGY

A LITTLE REAL ANALYSIS AND TOPOLOGY A LITTLE REAL ANALYSIS AND TOPOLOGY 1. NOTATION Before we begin some notational definitions are useful. (1) Z = {, 3, 2, 1, 0, 1, 2, 3, }is the set of integers. (2) Q = { a b : aεz, bεz {0}} is the set

More information

Math 10850, Honors Calculus 1

Math 10850, Honors Calculus 1 Math 0850, Honors Calculus Homework 0 Solutions General and specific notes on the homework All the notes from all previous homework still apply! Also, please read my emails from September 6, 3 and 27 with

More information

V. Graph Sketching and Max-Min Problems

V. Graph Sketching and Max-Min Problems V. Graph Sketching and Max-Min Problems The signs of the first and second derivatives of a function tell us something about the shape of its graph. In this chapter we learn how to find that information.

More information

Logical Connectives and Quantifiers

Logical Connectives and Quantifiers Chapter 1 Logical Connectives and Quantifiers 1.1 Logical Connectives 1.2 Quantifiers 1.3 Techniques of Proof: I 1.4 Techniques of Proof: II Theorem 1. Let f be a continuous function. If 1 f(x)dx 0, then

More information

Problem Set 2: Solutions Math 201A: Fall 2016

Problem Set 2: Solutions Math 201A: Fall 2016 Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that

More information

Section 4.2: The Mean Value Theorem

Section 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 information

REVIEW OF ESSENTIAL MATH 346 TOPICS

REVIEW OF ESSENTIAL MATH 346 TOPICS REVIEW OF ESSENTIAL MATH 346 TOPICS 1. AXIOMATIC STRUCTURE OF R Doğan Çömez The real number system is a complete ordered field, i.e., it is a set R which is endowed with addition and multiplication operations

More information

Sample Problems for the Second Midterm Exam

Sample Problems for the Second Midterm Exam Math 3220 1. Treibergs σιι Sample Problems for the Second Midterm Exam Name: Problems With Solutions September 28. 2007 Questions 1 10 appeared in my Fall 2000 and Fall 2001 Math 3220 exams. (1) Let E

More information

Analysis Part 1. 1 Chapter Q1(q) 1.2 Q1(r) Book: Measure and Integral by Wheeden and Zygmund

Analysis Part 1. 1 Chapter Q1(q) 1.2 Q1(r) Book: Measure and Integral by Wheeden and Zygmund Analysis Part 1 www.mathtuition88.com Book: Measure and Integral by Wheeden and Zygmund 1 Chapter 1 1.1 Q1(q) Let {T x k } be a sequence of points of T E. Since E is compact, {x k } has a subsequence {x

More information

University of Toronto MAT137Y1 Calculus! Test 2 1 December 2017 Time: 110 minutes

University of Toronto MAT137Y1 Calculus! Test 2 1 December 2017 Time: 110 minutes University of Toronto MAT137Y1 Calculus! Test 2 1 December 2017 Time: 110 minutes Please complete this cover page with ALL CAPITAL LETTERS. Last name......................................................................................

More information

MATH 104 : Final Exam

MATH 104 : Final Exam MATH 104 : Final Exam 10 May, 2017 Name: You have 3 hours to answer the questions. You are allowed one page (front and back) worth of notes. The page should not be larger than a standard US letter size.

More information

Lemma 15.1 (Sign preservation Lemma). Suppose that f : E R is continuous at some a R.

Lemma 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 information

MTH 132 Solutions to Exam 2 Apr. 13th 2015

MTH 132 Solutions to Exam 2 Apr. 13th 2015 MTH 13 Solutions to Exam Apr. 13th 015 Name: Section: Instructor: READ THE FOLLOWING INSTRUCTIONS. Do not open your exam until told to do so. No calculators, cell phones or any other electronic devices

More information

Principle of Mathematical Induction

Principle of Mathematical Induction Advanced Calculus I. Math 451, Fall 2016, Prof. Vershynin Principle of Mathematical Induction 1. Prove that 1 + 2 + + n = 1 n(n + 1) for all n N. 2 2. Prove that 1 2 + 2 2 + + n 2 = 1 n(n + 1)(2n + 1)

More information

We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero

We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero Chapter Limits of Sequences Calculus Student: lim s n = 0 means the s n are getting closer and closer to zero but never gets there. Instructor: ARGHHHHH! Exercise. Think of a better response for the instructor.

More information

Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall. .1 Limits of Sequences. CHAPTER.1.0. a) True. If converges, then there is an M > 0 such that M. Choose by Archimedes an N N such that N > M/ε. Then n N implies /n M/n M/N < ε. b) False. = n does not converge,

More information

Solutions to Homework 9

Solutions to Homework 9 Solutions to Homework 9 Read the proof of proposition 1.7 on p. 271 (section 7.1). Write a more detailed proof. In particular, state the defintion of uniformly continuous and explain the comment whose

More information

EC9A0: Pre-sessional Advanced Mathematics Course. Lecture Notes: Unconstrained Optimisation By Pablo F. Beker 1

EC9A0: Pre-sessional Advanced Mathematics Course. Lecture Notes: Unconstrained Optimisation By Pablo F. Beker 1 EC9A0: Pre-sessional Advanced Mathematics Course Lecture Notes: Unconstrained Optimisation By Pablo F. Beker 1 1 Infimum and Supremum Definition 1. Fix a set Y R. A number α R is an upper bound of Y if

More information

Math 163 (23) - Midterm Test 1

Math 163 (23) - Midterm Test 1 Name: Id #: Math 63 (23) - Midterm Test Spring Quarter 208 Friday April 20, 09:30am - 0:20am Instructions: Prob. Points Score possible 26 2 4 3 0 TOTAL 50 Read each problem carefully. Write legibly. Show

More information

5. Some theorems on continuous functions

5. Some theorems on continuous functions 5. Some theorems on continuous functions The results of section 3 were largely concerned with continuity of functions at a single point (usually called x 0 ). In this section, we present some consequences

More information

Econ Lecture 2. Outline

Econ Lecture 2. Outline Econ 204 2010 Lecture 2 Outline 1. Cardinality (cont.) 2. Algebraic Structures: Fields and Vector Spaces 3. Axioms for R 4. Sup, Inf, and the Supremum Property 5. Intermediate Value Theorem 1 Cardinality

More information

Functions. Chapter Continuous Functions

Functions. Chapter Continuous Functions Chapter 3 Functions 3.1 Continuous Functions A function f is determined by the domain of f: dom(f) R, the set on which f is defined, and the rule specifying the value f(x) of f at each x dom(f). If f is

More information

Part 2 Continuous functions and their properties

Part 2 Continuous functions and their properties Part 2 Continuous functions and their properties 2.1 Definition Definition A function f is continuous at a R if, and only if, that is lim f (x) = f (a), x a ε > 0, δ > 0, x, x a < δ f (x) f (a) < ε. Notice

More information

Exam 2 Solutions. x 1 x. x 4 The generating function for the problem is the fourth power of this, (1 x). 4

Exam 2 Solutions. x 1 x. x 4 The generating function for the problem is the fourth power of this, (1 x). 4 Math 5366 Fall 015 Exam Solutions 1. (0 points) Find the appropriate generating function (in closed form) for each of the following problems. Do not find the coefficient of x n. (a) In how many ways can

More information

Math 165 Final Exam worksheet solutions

Math 165 Final Exam worksheet solutions C Roettger, Fall 17 Math 165 Final Exam worksheet solutions Problem 1 Use the Fundamental Theorem of Calculus to compute f(4), where x f(t) dt = x cos(πx). Solution. From the FTC, the derivative of the

More information

Math 172 HW 1 Solutions

Math 172 HW 1 Solutions Math 172 HW 1 Solutions Joey Zou April 15, 2017 Problem 1: Prove that the Cantor set C constructed in the text is totally disconnected and perfect. In other words, given two distinct points x, y C, there

More information

MATH 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. 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 information

8 Further theory of function limits and continuity

8 Further theory of function limits and continuity 8 Further theory of function limits and continuity 8.1 Algebra of limits and sandwich theorem for real-valued function limits The following results give versions of the algebra of limits and sandwich theorem

More information

Take-Home Final Examination of Math 55a (January 17 to January 23, 2004)

Take-Home Final Examination of Math 55a (January 17 to January 23, 2004) Take-Home Final Eamination of Math 55a January 17 to January 3, 004) N.B. For problems which are similar to those on the homework assignments, complete self-contained solutions are required and homework

More information

Infinite Sequences of Real Numbers (AKA ordered lists) DEF. An infinite sequence of real numbers is a function f : N R.

Infinite Sequences of Real Numbers (AKA ordered lists) DEF. An infinite sequence of real numbers is a function f : N R. Infinite Sequences of Real Numbers (AKA ordered lists) DEF. An infinite sequence of real numbers is a function f : N R. Usually (infinite) sequences are written as lists, such as, x n n 1, x n, x 1,x 2,x

More information

Analysis Qualifying Exam

Analysis Qualifying Exam Analysis Qualifying Exam Spring 2017 Problem 1: Let f be differentiable on R. Suppose that there exists M > 0 such that f(k) M for each integer k, and f (x) M for all x R. Show that f is bounded, i.e.,

More information

1 Which sets have volume 0?

1 Which sets have volume 0? Math 540 Spring 0 Notes #0 More on integration Which sets have volume 0? The theorem at the end of the last section makes this an important question. (Measure theory would supersede it, however.) Theorem

More information

a 2n = . On the other hand, the subsequence a 2n+1 =

a 2n = . On the other hand, the subsequence a 2n+1 = Math 316, Intro to Analysis subsequences. This is another note pack which should last us two days. Recall one of our arguments about why a n = ( 1) n diverges. Consider the subsequence a n = It converges

More information

LECTURE 6. CONTINUOUS FUNCTIONS AND BASIC TOPOLOGICAL NOTIONS

LECTURE 6. CONTINUOUS FUNCTIONS AND BASIC TOPOLOGICAL NOTIONS ANALYSIS FOR HIGH SCHOOL TEACHERS LECTURE 6. CONTINUOUS FUNCTIONS AND BASIC TOPOLOGICAL NOTIONS ROTHSCHILD CAESARIA COURSE, 2011/2 1. The idea of approximation revisited When discussing the notion of the

More information

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9 MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended

More information

Section 1.4 Tangents and Velocity

Section 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 information

Consequences of the Completeness Property

Consequences of the Completeness Property Consequences of the Completeness Property Philippe B. Laval KSU Today Philippe B. Laval (KSU) Consequences of the Completeness Property Today 1 / 10 Introduction In this section, we use the fact that R

More information

MATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous:

MATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous: MATH 51H Section 4 October 16, 2015 1 Continuity Recall what it means for a function between metric spaces to be continuous: Definition. Let (X, d X ), (Y, d Y ) be metric spaces. A function f : X Y is

More information

Mon 3 Nov Tuesday 4 Nov: Quiz 8 ( ) Friday 7 Nov: Exam 2!!! Today: 4.5 Wednesday: REVIEW. In class Covers

Mon 3 Nov Tuesday 4 Nov: Quiz 8 ( ) Friday 7 Nov: Exam 2!!! Today: 4.5 Wednesday: REVIEW. In class Covers Mon 3 Nov 2014 Tuesday 4 Nov: Quiz 8 (4.2-4.4) Friday 7 Nov: Exam 2!!! In class Covers 3.9-4.5 Today: 4.5 Wednesday: REVIEW Linear Approximation and Differentials In section 4.5, you see the pictures on

More information

Exercises from other sources REAL NUMBERS 2,...,

Exercises from other sources REAL NUMBERS 2,..., Exercises from other sources REAL NUMBERS 1. Find the supremum and infimum of the following sets: a) {1, b) c) 12, 13, 14, }, { 1 3, 4 9, 13 27, 40 } 81,, { 2, 2 + 2, 2 + 2 + } 2,..., d) {n N : n 2 < 10},

More information

Describing the Real Numbers

Describing the Real Numbers Describing the Real Numbers Anthony Várilly Math 25a, Fall 2001 1 Introduction The goal of these notes is to uniquely describe the real numbers by taking certain statements as axioms. This exercise might

More information

MA 123 (Calculus I) Lecture 6: September 19, 2016 Section A3. Professor Joana Amorim,

MA 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 information

MATH20101 Real Analysis, Exam Solutions and Feedback. 2013\14

MATH20101 Real Analysis, Exam Solutions and Feedback. 2013\14 MATH200 Real Analysis, Exam Solutions and Feedback. 203\4 A. i. Prove by verifying the appropriate definition that ( 2x 3 + x 2 + 5 ) = 7. x 2 ii. By using the Rules for its evaluate a) b) x 2 x + x 2

More information

First In-Class Exam Solutions Math 410, Professor David Levermore Monday, 1 October 2018

First In-Class Exam Solutions Math 410, Professor David Levermore Monday, 1 October 2018 First In-Class Exam Solutions Math 40, Professor David Levermore Monday, October 208. [0] Let {b k } k N be a sequence in R and let A be a subset of R. Write the negations of the following assertions.

More information

Section I Multiple Choice 45 questions. Section II Free Response 6 questions

Section 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 information

MATH202 Introduction to Analysis (2007 Fall and 2008 Spring) Tutorial Note #7

MATH202 Introduction to Analysis (2007 Fall and 2008 Spring) Tutorial Note #7 MATH202 Introduction to Analysis (2007 Fall and 2008 Spring) Tutorial Note #7 Real Number Summary of terminology and theorems: Definition: (Supremum & infimum) A supremum (or least upper bound) of a non-empty

More information

1 Definition of the Riemann integral

1 Definition of the Riemann integral MAT337H1, Introduction to Real Analysis: notes on Riemann integration 1 Definition of the Riemann integral Definition 1.1. Let [a, b] R be a closed interval. A partition P of [a, b] is a finite set of

More information

What to remember about metric spaces

What to remember about metric spaces Division of the Humanities and Social Sciences What to remember about metric spaces KC Border These notes are (I hope) a gentle introduction to the topological concepts used in economic theory. If the

More information

Quick Tour of the Topology of R. Steven Hurder, Dave Marker, & John Wood 1

Quick Tour of the Topology of R. Steven Hurder, Dave Marker, & John Wood 1 Quick Tour of the Topology of R Steven Hurder, Dave Marker, & John Wood 1 1 Department of Mathematics, University of Illinois at Chicago April 17, 2003 Preface i Chapter 1. The Topology of R 1 1. Open

More information

Section 3.2 : Sequences

Section 3.2 : Sequences Section 3.2 : Sequences Note: Chapter 11 of Stewart s Calculus is a good reference for this chapter of our lecture notes. Definition 52 A sequence is an infinite ordered list a 1, a 2, a 3,... The items

More information

Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi

Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.

More information

Exam 3 review for Math 1190

Exam 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 information

MATH 243E Test #3 Solutions

MATH 243E Test #3 Solutions MATH 4E Test # Solutions () Find a recurrence relation for the number of bit strings of length n that contain a pair of consecutive 0s. You do not need to solve this recurrence relation. (Hint: Consider

More information

G1CMIN Measure and Integration

G1CMIN Measure and Integration G1CMIN Measure and Integration 2003-4 Prof. J.K. Langley May 13, 2004 1 Introduction Books: W. Rudin, Real and Complex Analysis ; H.L. Royden, Real Analysis (QA331). Lecturer: Prof. J.K. Langley (jkl@maths,

More information

INTERMEDIATE VALUE THEOREM

INTERMEDIATE VALUE THEOREM INTERMEDIATE VALUE THEOREM Section 1.4B Calculus AP/Dual, Revised 017 viet.dang@humbleisd.net 7/30/018 1:9 AM 1.4B: Intermediate Value Theorem 1 DEFINITION OF CONTINUITY A function is continuous at the

More information