Math Section Bekki George: 08/28/18. University of Houston. Bekki George (UH) Math /28/18 1 / 37
|
|
|
- Scot Kennedy
- 5 years ago
- Views:
Transcription
1 Math 1431 Section Bekki George: University of Houston 08/28/18 Bekki George (UH) Math /28/18 1 / 37
2 Office Hours: Tuesdays and Thursdays 12:30-2pm (also available by appointment) Office: 218C PGH Course webpage: Bekki George (UH) Math /28/18 2 / 37
3 Questions Bekki George (UH) Math /28/18 3 / 37
4 Continuity A function f is said to be continuous at a point c if 1 f(c) is defined. 2 lim x c f(x) exists. 3 lim x c f(x) = f(c). Bekki George (UH) Math /28/18 4 / 37
5 Can you give an example of a function where step 1 fails but step 2 doesn t fail? Bekki George (UH) Math /28/18 5 / 37
6 Can you give an example of a function where step 2 fails but step 1 doesn t fail? Bekki George (UH) Math /28/18 6 / 37
7 Can you give an example of a function where step 3 fails but steps 1 and 2 don t fail? Bekki George (UH) Math /28/18 7 / 37
8 Types of discontinuity at a point 1 Removable: 2 Non-Removable - Jump: Bekki George (UH) Math /28/18 8 / 37
9 Types of discontinuity at a point 3 Non-Removable - Infinite: Bekki George (UH) Math /28/18 9 / 37
10 If functions f and g are continuous at the point x = c, then 1 f + g is continuous at c 2 f g is continuous at c 3 αf is continuous at c for each real number α 4 f g is continuous at c 5 f g is continuous at c provided g(c) 0 Lastly, - If g is continuous at c and f is continuous at g(c), then the composition f g is continuous at c. Bekki George (UH) Math /28/18 10 / 37
11 Where are polynomials continuous? Where are rational functions continuous? Bekki George (UH) Math /28/18 11 / 37
12 There is also One-Sided Continuity A function is continuous from the left at c if lim f(x) = f(c) x c and it is continuous from the right at c if lim f(x) = f(c) x c+ Bekki George (UH) Math /28/18 12 / 37
13 Examples: Discuss the continuity for each function. 1 f(x) = x + 2 x 2 x 6 Bekki George (UH) Math /28/18 13 / 37
14 2 f(x) = x2 + 2x x 2 4 Bekki George (UH) Math /28/18 14 / 37
15 3 f(x) = x + 5 x Bekki George (UH) Math /28/18 15 / 37
16 4 f(x) = x + 5 x 2 + 5x Bekki George (UH) Math /28/18 16 / 37
17 5 f(x) = x 3 Bekki George (UH) Math /28/18 17 / 37
18 6 f(x) = x 1 x 2 + 4x 5 Bekki George (UH) Math /28/18 18 / 37
19 7 f(x) = { x 3 x < 1 x x 1 Bekki George (UH) Math /28/18 19 / 37
20 8 f(x) = { 6 x 2 6 x > 2 Bekki George (UH) Math /28/18 20 / 37
21 x + 2 x < 2 4 x 2 2 x < 2 9 g(x) = 1 x = 2 x 2 x > 2 Bekki George (UH) Math /28/18 21 / 37
22 10 Find c so that h(x) is continuous. h(x) = { 2x 3 x < 2 cx x 2 x 2 Bekki George (UH) Math /28/18 22 / 37
23 1 Determine if the following function is continuous at the point where x = 3. 2x x < 3 g(x) = 27 x = 3 x 3 x > 3 Bekki George (UH) Math /28/18 23 / 37
24 x 2 x < 1 3 x = 1 2 Discuss the continuity of f(x) = 2 x 1 < x 1 1 x > 1 x 2 Bekki George (UH) Math /28/18 24 / 37
25 3 Find A and B so that f(x) is continuous. 2x 2 1 x < 2 f(x) = A x = 2 Bx 3 x > 2 Bekki George (UH) Math /28/18 25 / 37
26 Section The Intermediate Value Theorem A very important result of continuity is the Intermediate Value Theorem. If f(x) is continuous on the closed interval [a, b] and K is a value between f(a) and f(b), then there is at least one value c in (a, b) such that f(c) = K. Bekki George (UH) Math /28/18 26 / 37
27 Section The Intermediate Value Theorem Examples: Use the intermediate value theorem to show that there is a solution to the given equation in the indicated interval. 1 x 2 4x + 3 = 0 on the interval [2, 4] 2 x 3 6x 2 x + 2 = 0 on the interval [0, 3] Bekki George (UH) Math /28/18 27 / 37
28 Section The Intermediate Value Theorem 3 2 tan(x) x = 1 on the interval [ 0, π 4 ] 4 Show there is a value of x between 1 and 3 so that 3x 3 + 2x 4 = 7 Bekki George (UH) Math /28/18 28 / 37
29 Section The Intermediate Value Theorem 5 Does the Intermediate Value Theorem guarantee a solution to 0 = x 2 + 6x + 10 on the interval [ 1, 3]? Bekki George (UH) Math /28/18 29 / 37
30 Section The Intermediate Value Theorem 6 Does the Intermediate Value Theorem guarantee a solution to f(x) = 0 for f(x) = 2 sin(x) 8 cos(x) 3x 2 on the interval [ 0, π 2 ]? Bekki George (UH) Math /28/18 30 / 37
31 Section The Intermediate Value Theorem 7 Verify that the IVT applies to this function on the indicated interval and find the value of c guaranteed by the theorem. f(x) = x 2 3x + 1 on the interval [0, 6], f(c) = 5. Bekki George (UH) Math /28/18 31 / 37
32 Section The Intermediate Value Theorem The Intermediate Value Theorem also helps us solve polynomial and rational inequalities. Examples: 1 (x + 2) 2 (3x 2)(x 1) 3 0 Bekki George (UH) Math /28/18 32 / 37
33 Section The Intermediate Value Theorem 2 2x 8x 2 (x + 1) 2 0 Bekki George (UH) Math /28/18 33 / 37
34 Section The Intermediate Value Theorem 3 1 x x + 2 < 0 Bekki George (UH) Math /28/18 34 / 37
35 Section The Intermediate Value Theorem 4 4 x x Bekki George (UH) Math /28/18 35 / 37
36 Section The Intermediate Value Theorem Why did we just work these problems? Bekki George (UH) Math /28/18 36 / 37
37 Section The Intermediate Value Theorem Why did we just work these problems? These inequalities are able to be solved because of the Intermediate Value Theorem (IVT). The IVT basically states that if f(x) is continuous from x = a to x = b, then you must pass through all points (x = c ) plotted along the graph of f(x). Bekki George (UH) Math /28/18 36 / 37
38 Section The Intermediate Value Theorem Why did we just work these problems? These inequalities are able to be solved because of the Intermediate Value Theorem (IVT). The IVT basically states that if f(x) is continuous from x = a to x = b, then you must pass through all points (x = c ) plotted along the graph of f(x). Note: Functions with complex roots do not meet the requirements of the IVT. Why?? Bekki George (UH) Math /28/18 36 / 37
39 To Do Read Sections 1.4 and 1.5. Finish hw 1 Work on quizzes 1, 2 and 3. It helps to do the work problems right after the lesson. Post questions to the discussion board on CASA. Bekki George (UH) Math /28/18 37 / 37
Math Section Bekki George: 02/25/19. University of Houston. Bekki George (UH) Math /25/19 1 / 19
Math 1431 Section 12200 Bekki George: rageorge@central.uh.edu University of Houston 02/25/19 Bekki George (UH) Math 1431 02/25/19 1 / 19 Office Hours: Mondays 1-2pm, Tuesdays 2:45-3:30pm (also available
Math Section Bekki George: 01/16/19. University of Houston. Bekki George (UH) Math /16/19 1 / 31
Math 1431 Section 12200 Bekki George: bekki@math.uh.edu University of Houston 01/16/19 Bekki George (UH) Math 1431 01/16/19 1 / 31 Office Hours: Mondays 1-2pm, Tuesdays 2:45-3:30pm (also available by appointment)
The 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
Continuity, 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
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
Homework 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
Lecture 4. Section 2.5 The Pinching Theorem Section 2.6 Two Basic Properties of Continuity. Jiwen He. Department of Mathematics, University of Houston
Review Pinching Theorem Two Basic Properties Lecture 4 Section 2.5 The Pinching Theorem Section 2.6 Two Basic Properties of Continuity Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu
Continuity 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
Continuity. 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
O.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
Chapter 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
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
2.4 The Precise Definition of a Limit
2.4 The Precise Definition of a Limit Reminders/Remarks: x 4 < 3 means that the distance between x and 4 is less than 3. In other words, x lies strictly between 1 and 7. So, x a < δ means that the distance
Continuity. 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
Math 1431 Final Exam Review
Math 1431 Final Exam Review Comprehensive exam. I recommend you study all past reviews and practice exams as well. Know all rules/formulas. Make a reservation for the final exam. If you miss it, go back
MATH 409 Advanced Calculus I Lecture 11: More on continuous functions.
MATH 409 Advanced Calculus I Lecture 11: More on 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 for any ε > 0 there
Calculus 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
Chapter 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
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
2.2 The Limit of a Function
2.2 The Limit of a Function Introductory Example: Consider the function f(x) = x is near 0. x f(x) x f(x) 1 3.7320508 1 4.236068 0.5 3.8708287 0.5 4.1213203 0.1 3.9748418 0.1 4.0248457 0.05 3.9874607 0.05
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
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
MATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Fall 2018, WEEK 3 JoungDong Kim Week 3 Section 2.3, 2.5, 2.6, Calculating Limits Using the Limit Laws, Continuity, Limits at Infinity; Horizontal Asymptotes. Section
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.
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
Infinite Limits. By Tuesday J. Johnson
Infinite Limits By Tuesday J. Johnson Suggested Review Topics Algebra skills reviews suggested: Evaluating functions Graphing functions Working with inequalities Working with absolute values Trigonometric
1.5 Inverse Trigonometric Functions
1.5 Inverse Trigonometric Functions Remember that only one-to-one functions have inverses. So, in order to find the inverse functions for sine, cosine, and tangent, we must restrict their domains to intervals
Math 106 Calculus 1 Topics for first exam
Chapter 2: Limits and Continuity Rates of change and its: Math 06 Calculus Topics for first exam Limit of a function f at a point a = the value the function should take at the point = the value that the
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
MATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Spring 2018, WEEK 3 JoungDong Kim Week 3 Section 2.5, 2.6, 2.7, Continuity, Limits at Infinity; Horizontal Asymptotes, Derivatives and Rates of Change. Section 2.5 Continuity
1 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 (
CH 2: Limits and Derivatives
2 The tangent and velocity problems CH 2: Limits and Derivatives the tangent line to a curve at a point P, is the line that has the same slope as the curve at that point P, ie the slope of the tangent
Math 261 Calculus I. Test 1 Study Guide. Name. Decide whether the limit exists. If it exists, find its value. 1) lim x 1. f(x) 2) lim x -1/2 f(x)
Math 261 Calculus I Test 1 Study Guide Name Decide whether the it exists. If it exists, find its value. 1) x 1 f(x) 2) x -1/2 f(x) Complete the table and use the result to find the indicated it. 3) If
1.3 Limits and Continuity
.3 Limits and Continuity.3. Limits Problem 8. What will happen to the functional values of as x gets closer and closer to 2? f(x) = Solution. We can evaluate f(x) using x values nearer and nearer to 2
Chapter 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
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
MAT137 Calculus! Lecture 5
MAT137 Calculus! Lecture 5 Today: 2.5 The Pinching Theorem; 2.5 Trigonometric Limits. 2.6 Two Basic Theorems. 3.1 The Derivative Next: 3.2-3.6 DIfferentiation Rules Deadline to notify us if you have a
Chapter 1 Functions and Limits
Contents Chapter 1 Functions and Limits Motivation to Chapter 1 2 4 Tangent and Velocity Problems 3 4.1 VIDEO - Secant Lines, Average Rate of Change, and Applications......................... 3 4.2 VIDEO
DRAFT - 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
Math 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,
Lecture 3 (Limits and Derivatives)
Lecture 3 (Limits and Derivatives) Continuity In the previous lecture we saw that very often the limit of a function as is just. When this is the case we say that is continuous at a. Definition: A function
Introduction to Limits
MATH 136 Introduction to Limits Given a function y = f (x), we wish to describe the behavior of the function as the variable x approaches a particular value a. We should be as specific as possible in describing
Induction, 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
Solution to Review Problems for Midterm #1
Solution to Review Problems for Midterm # Midterm I: Wednesday, September in class Topics:.,.3 and.-.6 (ecept.3) Office hours before the eam: Monday - and 4-6 p.m., Tuesday - pm and 4-6 pm at UH 080B)
Math 12 Final Exam Review 1
Math 12 Final Exam Review 1 Part One Calculators are NOT PERMITTED for this part of the exam. 1. a) The sine of angle θ is 1 What are the 2 possible values of θ in the domain 0 θ 2π? 2 b) Draw these angles
Chapter 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
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
Calculus I. George Voutsadakis 1. LSSU Math 151. Lake Superior State University. 1 Mathematics and Computer Science
Calculus I George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 151 George Voutsadakis (LSSU) Calculus I November 2014 1 / 67 Outline 1 Limits Limits, Rates
2.1 The Tangent and Velocity Problems
2.1 The Tangent and Velocity Problems Tangents What is a tangent? Tangent lines and Secant lines Estimating slopes from discrete data: Example: 1. A tank holds 1000 gallons of water, which drains from
x y More precisely, this equation means that given any ε > 0, there exists some δ > 0 such that
Chapter 2 Limits and continuity 21 The definition of a it Definition 21 (ε-δ definition) Let f be a function and y R a fixed number Take x to be a point which approaches y without being equal to y If there
Consequences 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
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.
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
Math 473: Practice Problems for Test 1, Fall 2011, SOLUTIONS
Math 473: Practice Problems for Test 1, Fall 011, SOLUTIONS Show your work: 1. (a) Compute the Taylor polynomials P n (x) for f(x) = sin x and x 0 = 0. Solution: Compute f(x) = sin x, f (x) = cos x, f
Topic 2 Limits and Continuity c and d) Continuity Handout Notes Assigned Problems: Intro book pg 73, 1-3 and 6-8
c&d. Continuity Handout. Page 1 of 5 Topic Limits and Continuity c and d) Continuity Handout Notes Assigned Problems: Intro book pg 73, 1-3 and 6-8 Recall Limits and Function Values: We have already studied
Test 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
Limit Laws- Theorem 2.2.1
Limit Laws- Theorem 2.2.1 If L, M, c, and k are real numbers and lim x c f (x) =L and lim g(x) =M, then x c 1 Sum Rule: lim x c (f (x)+g(x)) = L + M 2 Difference Rule: lim x c (f (x) g(x)) = L M 3 Constant
Chapter 1: Limits and Continuity
Chapter 1: Limits and Continuity Winter 2015 Department of Mathematics Hong Kong Baptist University 1/69 1.1 Examples where limits arise Calculus has two basic procedures: differentiation and integration.
THE LIMIT PROCESS (AN INTUITIVE INTRODUCTION)
The Limit Process THE LIMIT PROCESS (AN INTUITIVE INTRODUCTION) We could begin by saying that limits are important in calculus, but that would be a major understatement. Without limits, calculus would
Calculus (Math 1A) Lecture 6
Calculus (Math 1A) Lecture 6 Vivek Shende September 5, 2017 Hello and welcome to class! Hello and welcome to class! Last time Hello and welcome to class! Last time We introduced limits, and discussed slopes
MA 137: Calculus I for the Life Sciences
MA 137: Calculus I for the Life Sciences David Murrugarra Department of Mathematics, University of Kentucky http://www.ms.uky.edu/~ma137/ Spring 2018 David Murrugarra (University of Kentucky) MA 137: Lecture
Section 3.7. Rolle s Theorem and the Mean Value Theorem
Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate of change and the average rate of change of
Math 1120 Calculus, sections 3 and 10 Test 1
October 3, 206 Name The problems count as marked The total number of points available is 7 Throughout this test, show your work This is an amalgamation of the tests from sections 3 and 0 (0 points) Find
Limits, 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
Calculus 221 worksheet
Calculus 221 worksheet Graphing A function has a global maximum at some a in its domain if f(x) f(a) for all other x in the domain of f. Global maxima are sometimes also called absolute maxima. A function
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 6 B) 14 C) 10 D) Does not exist
Assn 3.1-3.3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit, if it exists. 1) Find: lim x -1 6x + 5 5x - 6 A) -11 B) - 1 11 C)
Section 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
(e) 2 (f) 2. (c) + (d). Limits at Infinity. 2.5) 9-14,25-34,41-43,46-47,56-57, (c) (d) 2
Math 150A. Final Review Answers, Spring 2018. Limits. 2.2) 7-10, 21-24, 28-1, 6-8, 4-44. 1. Find the values, or state they do not exist. (a) (b) 1 (c) DNE (d) 1 (e) 2 (f) 2 (g) 2 (h) 4 2. lim f(x) = 2,
MAT137 Calculus! Lecture 9
MAT137 Calculus! Lecture 9 Today we will study: Limits at infinity. L Hôpital s Rule. Mean Value Theorem. (11.5,11.6, 4.1) PS3 is due this Friday June 16. Next class: Applications of the Mean Value Theorem.
Calculus. Central role in much of modern science Physics, especially kinematics and electrodynamics Economics, engineering, medicine, chemistry, etc.
Calculus Calculus - the study of change, as related to functions Formally co-developed around the 1660 s by Newton and Leibniz Two main branches - differential and integral Central role in much of modern
MS 2001: Test 1 B Solutions
MS 2001: Test 1 B Solutions Name: Student Number: Answer all questions. Marks may be lost if necessary work is not clearly shown. Remarks by me in italics and would not be required in a test - J.P. Question
Families of Functions, Taylor Polynomials, l Hopital s
Unit #6 : Rule Families of Functions, Taylor Polynomials, l Hopital s Goals: To use first and second derivative information to describe functions. To be able to find general properties of families of functions.
Math 103: L Hopital s Rule
Math 103: L Hopital s Rule Ryan Blair University of Pennsylvania Thursday November 3, 2011 Ryan Blair (U Penn) Math 103: L Hopital s Rule Thursday November 3, 2011 1 / 6 Outline 1 L Hospital s Rule 2 Review
Math Section SR MW 1-2:30pm. Bekki George: University of Houston. Sections
Math 3339 Section 21155 - SR 117 - MW 1-2:30pm Bekki George: bekki@math.uh.edu University of Houston Sections 2.1-2.3 Bekki George (UH) Math 3339 Sections 2.1-2.3 1 / 28 Office Hours: Mondays 11am - 12:30pm,
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
11.10a Taylor and Maclaurin Series
11.10a 1 11.10a Taylor and Maclaurin Series Let y = f(x) be a differentiable function at x = a. In first semester calculus we saw that (1) f(x) f(a)+f (a)(x a), for all x near a The right-hand side of
Math Section TTH 5:30-7:00pm SR 116. James West 620 PGH
Math 1431 Section 15241 TTH 5:30-7:00pm SR 116 James West jdwest@math.uh.edu 620 PGH Office Hours: 2:30 4:30pm TTH in the CASA Tutoring Center or by appointment Class webpage: http://math.uh.edu/~jdwest/teaching/fa14/1431/calendar.html
Calculus 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
14 Increasing and decreasing functions
14 Increasing and decreasing functions 14.1 Sketching derivatives READING Read Section 3.2 of Rogawski Reading Recall, f (a) is the gradient of the tangent line of f(x) at x = a. We can use this fact to
Section 4.2 The Mean Value Theorem
Section 4.2 The Mean Value Theorem Ruipeng Shen October 2nd Ruipeng Shen MATH 1ZA3 October 2nd 1 / 11 Rolle s Theorem Theorem (Rolle s Theorem) Let f (x) be a function that satisfies: 1. f is continuous
MATH CALCULUS I 1.5: Continuity
MATH 12002 - CALCULUS I 1.5: Continuity Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 12 Definition of Continuity Intuitively,
Calculus The Mean Value Theorem October 22, 2018
Calculus The Mean Value Theorem October, 018 Definitions Let c be a number in the domain D of a function f. Then f(c) is the (a) absolute maximum value of f on D, i.e. f(c) = max, if f(c) for all x in
Lesson 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
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
Mathematic 108, Fall 2015: Solutions to assignment #7
Mathematic 08, Fall 05: Solutions to assignment #7 Problem # Suppose f is a function with f continuous on the open interval I and so that f has a local maximum at both x = a and x = b for a, b I with a
Unit 3: HW3.5 Sum and Product
Unit 3: HW3.5 Sum and Product Without solving, find the sum and product of the roots of each equation. 1. x 2 8x + 7 = 0 2. 2x + 5 = x 2 3. -7x + 4 = -3x 2 4. -10x 2 = 5x - 2 5. 5x 2 2x 3 4 6. 1 3 x2 3x
Iowa State University. Instructor: Alex Roitershtein Summer Homework #5. Solutions
Math 50 Iowa State University Introduction to Real Analysis Department of Mathematics Instructor: Alex Roitershtein Summer 205 Homework #5 Solutions. Let α and c be real numbers, c > 0, and f is defined
THS 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
Math 221 Notes on Rolle s Theorem, The Mean Value Theorem, l Hôpital s rule, and the Taylor-Maclaurin formula. 1. Two theorems
Math 221 Notes on Rolle s Theorem, The Mean Value Theorem, l Hôpital s rule, and the Taylor-Maclaurin formula 1. Two theorems Rolle s Theorem. If a function y = f(x) is differentiable for a x b and if
Respect your friends! Do not distract anyone by chatting with people around you Be considerate of others in class.
Math 1431 Dr. Melahat Almus almus@math.uh.edu http://www.math.uh.edu/~almus Visit CASA regularly for announcements and course material! If you e-mail me, please mention your course (1431) in the subject
Advanced Calculus Math 127B, Winter 2005 Solutions: Final. nx2 1 + n 2 x, g n(x) = n2 x
. Define f n, g n : [, ] R by f n (x) = Advanced Calculus Math 27B, Winter 25 Solutions: Final nx2 + n 2 x, g n(x) = n2 x 2 + n 2 x. 2 Show that the sequences (f n ), (g n ) converge pointwise on [, ],
Section 8.7. Taylor and MacLaurin Series. (1) Definitions, (2) Common Maclaurin Series, (3) Taylor Polynomials, (4) Applications.
Section 8.7 Taylor and MacLaurin Series (1) Definitions, (2) Common Maclaurin Series, (3) Taylor Polynomials, (4) Applications. MATH 126 (Section 8.7) Taylor and MacLaurin Series The University of Kansas
McGill University Math 354: Honors Analysis 3
Practice problems McGill University Math 354: Honors Analysis 3 not for credit Problem 1. Determine whether the family of F = {f n } functions f n (x) = x n is uniformly equicontinuous. 1st Solution: The
Last week we looked at limits generally, and at finding limits using substitution.
Math 1314 ONLINE Week 4 Notes Lesson 4 Limits (continued) Last week we looked at limits generally, and at finding limits using substitution. Indeterminate Forms What do you do when substitution gives you
AP 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
Math 320: Real Analysis MWF 1pm, Campion Hall 302 Homework 7 Solutions Please write neatly, and in complete sentences when possible.
Math 320: Real Analysis MWF 1pm, Campion Hall 302 Homework 7 Solutions Please write neatly, and in complete sentences when possible. Do the following problems from the book: 4.2.1, 4.2.3, 4.2.6, 4.2.8,
Chapter 8: Taylor s theorem and L Hospital s rule
Chapter 8: Taylor s theorem and L Hospital s rule Theorem: [Inverse Mapping Theorem] Suppose that a < b and f : [a, b] R. Given that f (x) > 0 for all x (a, b) then f 1 is differentiable on (f(a), f(b))
Math 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.
This Week. Professor Christopher Hoffman Math 124
This Week Sections 2.1-2.3,2.5,2.6 First homework due Tuesday night at 11:30 p.m. Average and instantaneous velocity worksheet Tuesday available at http://www.math.washington.edu/ m124/ (under week 2)
MAT01B1: the Mean Value Theorem
MAT01B1: the Mean Value Theorem Dr Craig 15 August 2018 My details: acraig@uj.ac.za Consulting hours: Monday 14h40 15h25 Thursday 11h20 12h55 Friday (this week): 11h20 12h30 Office C-Ring 508 https://andrewcraigmaths.wordpress.com/
Math 221 Exam II Tuesday Mar 23 5:30-7:00 PM Answers
Math 221 Exam II Tuesday Mar 23 5:30-7:00 PM Answers I. (25 points.) Find. Note: The book sometimes writes D xy for. (a) y = (x 2 x + 1) 7 Answer: Let u = x 2 x + 1. Then y = (x 2 x + 1) 7 = u 7 so = d
Limits and Continuity
Limits and Continuity MATH 151 Calculus for Management J. Robert Buchanan Department of Mathematics Fall 2018 Objectives After this lesson we will be able to: Determine the left-hand and right-hand limits