# 36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?

Save this PDF as:

Size: px
Start display at page:

Download "36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?"

## Transcription

1 36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this is fairly accurate and explicit, it is not precise enough if one wants to prove results about continuous functions. A much more precise de nition is needed. This is the topic of this section. Figure 1.17: At which points is f not continuous? Theory: De nition 62 (Continuity) A function f is said to be continuous at x = a if the three conditions below are satis ed: 1. a is in the domain of f (i.e. f (a) exists ) 2. lim f (x) exists 3. lim f (x) = f (a) De nition 63 (Continuity from the left) A function f is said to be continuous from the left at x = a if the three conditions below are satis ed: 1. a is in the domain of f (i.e. f (a) exists ) 2. lim f (x) exists

2 1.3. CONTINUITY lim f (x) = f (a) De nition 64 (Continuity from the right) A function f is said to be continuous from the right at x = a if the three conditions below are satis ed: 1. a is in the domain of f (i.e. f (a) exists ) 2. lim f (x) exists + 3. lim f (x) = f (a) + Remark 65 To be continuous from the left or the right at x = a, a function has to be de ned there. De nition 66 (Continuity on an interval) A function f is said to be continuous on an interval I if f is continuous at every point of the interval. If a function is not continuous at a point x = a, we say that f is discontinuous at x = a. When looking at the graph of a function, one can tell if the function is continuous because the graph will have no breaks or holes. One should be able to draw such a graph without lifting the pen from the paper. Looking at the graph on gure 1.17, one can see that f is not continuous at x = 3 (not de ned), x = 0 (break), x = 2 and x = 6. It is continuous from the right at x = 0. Example 67 Looking at gure 1.17, we can see the following: 1. f is discontinuous at 3 because it is not de ned there. Since it is not de ned at 3, f cannot be continuous from either the left or the right. 2. f is discontinuous at 0 because the limit does not exists at 0. However, it is continuous from the right because f (0) = 3 and lim = 3. x7! f is discontinuous at x = 2 because f (2) = 1 but lim f (x) = 1. x7!2 4. f is discontinuous at x = 6 because it is not de ned there. 5. f is c0ntinuous on the interval ( 1; 0) [ (0; 2) [ (2; 6) [ (6; 1). We now look at theorems which will help decide where a function is continuous. Given a function, to study its continuity, one has to understand its pattern. If the function is one of the speci c functions studied, then one simply uses our knowledge of that speci c function. However, if the function is a combination of speci c functions, then not only the continuity of each speci c function has to be studied, we also need to see if the way the functions are combined preserves continuity. The rst theorem gives us the continuity of the speci c functions we are likely to encounter in di erential calculus.

3 38 CHAPTER 1. LIMITS AND CONTINUITY Theorem 68 The following is true, regarding continuity of some speci c functions: 1. Any polynomial function is continuous everywhere, that is on ( 1; 1) : 2. Any rational function is continuous everywhere it is de ned. 3. If n is a positive even integer, then np x is continuous on [0; 1) : 4. If n is a positive odd integer, then np x is continuous on ( 1; 1) : 5. The trigonometric functions and their inversesare continuous wherever they are de ned. 6. The natural logarithm functions and the exponential functions are continuous wherever they are de ned. The next theorems are related to continuity of the various operations on functions. Theorem 69 Assume f and g are continuous at a and c is a constant. Then, the following functions are also continuous at a: 1. f + g 2. f g 3. cf 4. fg 5. f g if g(a) 6= 0 In general, functions are continuous almost everywhere. The only places to watch are places where the function will not be de ned, or places where the de nition of the function changes (breaking points for a piecewise function). Thus, we should watch for the following: Fractions: watch for points where the denominator is 0. Be careful that functions which might not appear to be a fraction may be a fraction. For example, tan x = sin x cos x. Piecewise functions: When studying continuity of piecewise functions, one should rst study the continuity of each piece by using the theorems above. Then, one must also check the continuity at each of the breaking points. Functions containing absolute values: Since an absolute value can be written as a piecewise function, they should be treated like a piecewise function.

4 1.3. CONTINUITY Examples We look at several examples to illustrate the de nition of continuity as well as the various theorems stated. Questions regarding continuity usually fall in two categories. They are: Determine if a function is continuous at a given speci c point. Determine if a function is continuous on a given interval. A variation of this question is to determine the set (that is all the values of x) on which a function is continuous. Whenever possible, try to use the theorems. But in situations where the theorems do not apply, you must use the de nition of continuity. 1. Is f (x) = x2 + 5 continuous at x = 2? x 1 f is a rational function, so it is continuous where it is de ned. Since it is de ned at x = 2, it is continuous there. 2. Is f (x) = x2 + 5 continuous at x = 1? x 1 f is not de ned at x = 1, hence it is not continuous at Find where f (x) = x2 + 5 is continuous. x 1 f is a rational function, so it is continuous where it is defnined that is when x 6= 1. We can write the answer in di erent ways: English sentence: f is continuous for all real numbers except x = 1. Set notation: f is continuous on R n f1g. This means the same thing. Interval notation: f is continuous on ( 1; 1) [ (1; 1). 4. Let f (x) = x + 1 if x 2 x 2 if x < 2. Is f continuous at x = 2, x = 3? Continuity at x = 2 Since f is a piecewise function and 2 is a breaking point, we need to investigate the continuity there. We check the three conditions which make a function continuous. First, f is de ned at 2. Next, we see that lim f (x) = 0 and lim f (x) = 3. Thus, lim f (x) does not x!2 x!2 + x!2 exist. f is not continuous at 2. Continuity at 3. When x is close to 3, f (x) = x + 1, which is a polynomial, hence it is continuous.

5 40 CHAPTER 1. LIMITS AND CONTINUITY 5. Find where ln (x 5) is continuous. The natural logarithm function is continuous where it is de ned. ln (x 5) is de ned when x 5 > 0 So, ln (x 5) is continuous on (5; 1). x > 5 6. Find where f (x) = x 1 is continuous. x + 2 Since f is a rational function, it is continuous where it is de ned that is for all reals except x = 2. x + 1 if x > 2 7. Find where f (x) = is continuous. x 2 if x < 2 To study continuity of a piecewise function, one has to study continuity of each branch as well as continuity at the breaking point. When x > 2, f (x) = x+1 is a polynomial, so it is continuous. When x < 2, f (x) = x 2 is also a polynomial, so it is continuous. At x = 2, f is not de ned, so it is not continuous. Thus, f is continuous everywhere, except at Find where f (x) = sin x is continuous. ln x Since f is a fraction, it will be continuous at points where: its numerator is continuous. its denominator is continuous. denominator is not 0 So, we check each condition. sin x is continuous where it is de ned that is on ( 1; 1) ln x is continuous where it is de ned that is on (0; 1) ln x = 0 when x = 1 In conclusion, f is continuous on (0; 1) [ (1; 1) 9. Find where f (x) = ln (5 x) + p x is continuous. f is the sum of two function. f will be continuous where both functions are continuous. p x is continuous on [0; 1) ln (5 x) is continuous where it is de ned that is when 5 x > 0 or 5 > x or x < 5, that is on ( 1; 5) In conclusion, f is continuous on [0; 5)

6 1.3. CONTINUITY Continuity and Limits From the de nition of continuity, we see that if a function is continuous at a point, then to nd the limit at that point, we simply plug in the point. Indeed, if f is continuous at a then lim f (x) = f (a). We can use this knowledge to nd the limit of functions for which we do not have rules yet. This includes the trigonometric functions, the exponential functions, the logarithm functions. We illustrate this idea with some examples. Example 70 Find lim x!0 e x. Since e x is continuous for every real number it is continuous at 0, therefore lim x!0 ex = e 0 = 1 Example 71 Find lim sin x x! 6 Since sin x is always continuous, it is continuous at 6, therefore lim sin x = sin 6 x! 6 = Intermediate Value Theorem We nish this section with one of the important properties of continuous functions: the intermediate value theorem. Theorem 72 Suppose f is continuous in the closed interval [a; b] and let N be a number strictly between f(a) and f(b). Then, there exists a number c in (a; b) such that f(c) = N. The theorem does not tell us how many solutions exist. There might be more than one. It does not tell us either how to nd the solutions when they exist. Though, we will see in the examples that using the theorem, we can nd these solutions with as much accuracy as needed. Figure 1.18 illustratres this theorem with N = 60. This theorem is often used to show an equation has a solution in an interval. Example 73 Show that the equation x 2 3 = 0 has a solution between 1 and 2. Let f (x) = x 2 3. Clearly, f is continuous since it is a polynomial. f (1) = 2, f (2) = 1. So, 0 is between f (1) and f (2). By the intermediate value theorem, there exists a number c between 1 and 2 such that f (c) = 0.

7 42 CHAPTER 1. LIMITS AND CONTINUITY Figure 1.18: f (c) = Sample problems: 1. Do # 3, 5, 11, 13, 15, 17, 25, 29 on pages 126, < x if x < 0 2. Is f (x) = x 2 if 0 < x 2 continuous at 0, 1, 2. : 8 x if x > 2 (answer: not continuous at 0 and 2, continuous at 1) Explain. 3. Find where f(x) = 2x + 1 x 2 is continuous. (answer: Continuous for + x 6 all reals except at x = 2 and x = 3) 4. Find where ln (2 x) is continuous. (answer: continuous on ( 1; 2)) 5. Use the intermediate value theorem to show that x 5 2x 4 x 3 = 0 has a solution in (2; 3). 6. Use the intermediate value theorem to show that there is a number c such that c 2 = 3: This proves the existence of p 3.

### Representation of functions as power series

Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

### Continuity. DEFINITION 1: A function f is continuous at a number a if. lim

Continuity DEFINITION : A function f is continuous at a number a if f(x) = f(a) REMARK: It follows from the definition that f is continuous at a if and only if. f(a) is defined. 2. f(x) and +f(x) exist.

### LIMITS AND CONTINUITY

LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from

### Lecture 5 : Continuous Functions Definition 1 We say the function f is continuous at a number a if

Lecture 5 : Continuous Functions Definition We say the function f is continuous at a number a if f(x) = f(a). (i.e. we can make the value of f(x) as close as we like to f(a) by taking x sufficiently close

### 4.6 Null Space, Column Space, Row Space

NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear

### Homework # 3 Solutions

Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8

### Derivatives: rules and applications (Stewart Ch. 3/4) The derivative f (x) of the function f(x):

Derivatives: rules and applications (Stewart Ch. 3/4) The derivative f (x) of the function f(x): f f(x + h) f(x) (x) = lim h 0 h (for all x for which f is differentiable/ the limit exists) Property:if

### The Mean Value Theorem

The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers

### Math 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) =

Math 50 Eam # Practice Problem Solutions. Find the vertical asymptotes (if any) of the functions g() = + 4, h() = 4. Answer: The only number not in the domain of g is = 0, so the only place where g could

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### x a x 2 (1 + x 2 ) n.

Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number

### Math 113 HW #9 Solutions

Math 3 HW #9 Solutions 4. 50. Find the absolute maximum and absolute minimum values of on the interval [, 4]. f(x) = x 3 6x 2 + 9x + 2 Answer: First, we find the critical points of f. To do so, take the

### TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

### TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### MA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity

MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x

### Math 181 Spring 2007 HW 1 Corrected

Math 181 Spring 2007 HW 1 Corrected February 1, 2007 Sec. 1.1 # 2 The graphs of f and g are given (see the graph in the book). (a) State the values of f( 4) and g(3). Find 4 on the x-axis (horizontal axis)

### FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.

FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is

### APPLICATIONS OF DIFFERENTIATION

4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,

### Partial Fractions Decomposition

Partial Fractions Decomposition Dr. Philippe B. Laval Kennesaw State University August 6, 008 Abstract This handout describes partial fractions decomposition and how it can be used when integrating rational

### Engineering Mathematics II

PSUT Engineering Mathematics II Fourier Series and Transforms Dr. Mohammad Sababheh 4/14/2009 11.1 Fourier Series 2 Fourier Series and Transforms Contents 11.1 Fourier Series... 3 Periodic Functions...

### Pre-Calculus Review Lesson 1 Polynomials and Rational Functions

If a and b are real numbers and a < b, then Pre-Calculus Review Lesson 1 Polynomials and Rational Functions For any real number c, a + c < b + c. For any real numbers c and d, if c < d, then a + c < b

### 55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim

Slant Asymptotes If lim x [f(x) (ax + b)] = 0 or lim x [f(x) (ax + b)] = 0, then the line y = ax + b is a slant asymptote to the graph y = f(x). If lim x f(x) (ax + b) = 0, this means that the graph of

### MTH4100 Calculus I. Lecture notes for Week 8. Thomas Calculus, Sections 4.1 to 4.4. Rainer Klages

MTH4100 Calculus I Lecture notes for Week 8 Thomas Calculus, Sections 4.1 to 4.4 Rainer Klages School of Mathematical Sciences Queen Mary University of London Autumn 2009 Theorem 1 (First Derivative Theorem

### Learning Objectives for Math 165

Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given

### Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

### correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:

Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that

### Fourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks

Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results

### Inverse Functions and Logarithms

Section 3. Inverse Functions and Logarithms 1 Kiryl Tsishchanka Inverse Functions and Logarithms DEFINITION: A function f is called a one-to-one function if it never takes on the same value twice; that

### An Introduction to Separation of Variables with Fourier Series Math 391w, Spring 2010 Tim McCrossen Professor Haessig

An Introduction to Separation of Variables with Fourier Series Math 391w, Spring 2010 Tim McCrossen Professor Haessig Abstract: This paper aims to give students who have not yet taken a course in partial

### 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.

CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if

### CHAPTER 2 FOURIER SERIES

CHAPTER 2 FOURIER SERIES PERIODIC FUNCTIONS A function is said to have a period T if for all x,, where T is a positive constant. The least value of T>0 is called the period of. EXAMPLES We know that =

### Functions: Piecewise, Even and Odd.

Functions: Piecewise, Even and Odd. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway September 21-22, 2015 Tutorials, Online Homework.

### Calculus I Notes. MATH /Richards/

Calculus I Notes MATH 52-154/Richards/10.1.07 The Derivative The derivative of a function f at p is denoted by f (p) and is informally defined by f (p)= the slope of the f-graph at p. Of course, the above

### Items related to expected use of graphing technology appear in bold italics.

- 1 - Items related to expected use of graphing technology appear in bold italics. Investigating the Graphs of Polynomial Functions determine, through investigation, using graphing calculators or graphing

### 6.8 Taylor and Maclaurin s Series

6.8. TAYLOR AND MACLAURIN S SERIES 357 6.8 Taylor and Maclaurin s Series 6.8.1 Introduction The previous section showed us how to find the series representation of some functions by using the series representation

### Notes on Chapter 1, Section 2 Arithmetic and Divisibility

Notes on Chapter 1, Section 2 Arithmetic and Divisibility August 16, 2006 1 Arithmetic Properties of the Integers Recall that the set of integers is the set Z = f0; 1; 1; 2; 2; 3; 3; : : :g. The integers

### Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)

SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f

### Lectures 5-6: Taylor Series

Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,

### Review for Calculus Rational Functions, Logarithms & Exponentials

Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for

### Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs

Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s

### Preparing for the Accuplacer College-Level Math Test

Preparing for the Accuplacer College-Level Math Test Students at GCC are required to take a placement test before their first math class. It is recommended that students review prior to taking this test.

### 100. In general, we can define this as if b x = a then x = log b

Exponents and Logarithms Review 1. Solving exponential equations: Solve : a)8 x = 4! x! 3 b)3 x+1 + 9 x = 18 c)3x 3 = 1 3. Recall: Terminology of Logarithms If 10 x = 100 then of course, x =. However,

### Sept 20, 2011 MATH 140: Calculus I Tutorial 2. ln(x 2 1) = 3 x 2 1 = e 3 x = e 3 + 1

Sept, MATH 4: Calculus I Tutorial Solving Quadratics, Dividing Polynomials Problem Solve for x: ln(x ) =. ln(x ) = x = e x = e + Problem Solve for x: e x e x + =. Let y = e x. Then we have a quadratic

### Fourier Series Chapter 3 of Coleman

Fourier Series Chapter 3 of Coleman Dr. Doreen De eon Math 18, Spring 14 1 Introduction Section 3.1 of Coleman The Fourier series takes its name from Joseph Fourier (1768-183), who made important contributions

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions

Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions

### So here s the next version of Homework Help!!!

HOMEWORK HELP FOR MATH 52 So here s the next version of Homework Help!!! I am going to assume that no one had any great difficulties with the problems assigned this quarter from 4.3 and 4.4. However, if

### LIMITS AND CONTINUITY The following tables show values of f(x, y) and g(x, y), correct to three decimal places, for points (x, y) near the origin.

LIMITS AND Let s compare the behavior of the functions 14. Limits and Continuity In this section, we will learn about: Limits and continuity of various types of functions. sin( ) f ( x, y) and g( x, y)

### SECTION 1.5: PIECEWISE-DEFINED FUNCTIONS; LIMITS AND CONTINUITY IN CALCULUS

(Section.5: Piecewise-Defined Functions; Limits and Continuity in Calculus).5. SECTION.5: PIECEWISE-DEFINED FUNCTIONS; LIMITS AND CONTINUITY IN CALCULUS LEARNING OBJECTIVES Know how to evaluate and graph

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### Contents. 6 Graph Sketching 87. 6.1 Increasing Functions and Decreasing Functions... 87. 6.2 Intervals Monotonically Increasing or Decreasing...

Contents 6 Graph Sketching 87 6.1 Increasing Functions and Decreasing Functions.......................... 87 6.2 Intervals Monotonically Increasing or Decreasing....................... 88 6.3 Etrema Maima

### 5.1 Derivatives and Graphs

5.1 Derivatives and Graphs What does f say about f? If f (x) > 0 on an interval, then f is INCREASING on that interval. If f (x) < 0 on an interval, then f is DECREASING on that interval. A function has

### 2.1 Increasing, Decreasing, and Piecewise Functions; Applications

2.1 Increasing, Decreasing, and Piecewise Functions; Applications Graph functions, looking for intervals on which the function is increasing, decreasing, or constant, and estimate relative maxima and minima.

### Rolle s Theorem. q( x) = 1

Lecture 1 :The Mean Value Theorem We know that constant functions have derivative zero. Is it possible for a more complicated function to have derivative zero? In this section we will answer this question

### Section 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations

Difference Equations to Differential Equations 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

### MTH 233 Calculus I. Part 1: Introduction, Limits and Continuity

MTH 233 Calculus I Tan, Single Variable Calculus: Early Transcendentals, 1st ed. Part 1: Introduction, Limits and Continuity 0 Preliminaries It is assumed that students are comfortable with the material

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### Functions of 2 Variables

Functions of 2 Variables Functions and Graphs In the last chapter, we extended di erential calculus to vector-valued functions. In this chapter, we extend calculus primarily to functions of two variables,

### Review Sheet for Third Midterm Mathematics 1300, Calculus 1

Review Sheet for Third Midterm Mathematics 1300, Calculus 1 1. For f(x) = x 3 3x 2 on 1 x 3, find the critical points of f, the inflection points, the values of f at all these points and the endpoints,

### Real Roots of Univariate Polynomials with Real Coefficients

Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials

### CHAPTER 2. Inequalities

CHAPTER 2 Inequalities In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1. A thorough mastery of this section is essential

### Lecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x)

ecture VI Abstract Before learning to solve partial differential equations, it is necessary to know how to approximate arbitrary functions by infinite series, using special families of functions This process

### 5 Indefinite integral

5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse

### Maths 361 Fourier Series Notes 2

Today s topics: Even and odd functions Real trigonometric Fourier series Section 1. : Odd and even functions Consider a function f : [, ] R. Maths 361 Fourier Series Notes f is odd if f( x) = f(x) for

### 1.7 Graphs of Functions

64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

### Partial Derivatives. @x f (x; y) = @ x f (x; y) @x x2 y + @ @x y2 and then we evaluate the derivative as if y is a constant.

Partial Derivatives Partial Derivatives Just as derivatives can be used to eplore the properties of functions of 1 variable, so also derivatives can be used to eplore functions of 2 variables. In this

### c 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint.

Lecture 2b: Utility c 2008 Je rey A. Miron Outline: 1. Introduction 2. Utility: A De nition 3. Monotonic Transformations 4. Cardinal Utility 5. Constructing a Utility Function 6. Examples of Utility Functions

### 1. Then f has a relative maximum at x = c if f(c) f(x) for all values of x in some

Section 3.1: First Derivative Test Definition. Let f be a function with domain D. 1. Then f has a relative maximum at x = c if f(c) f(x) for all values of x in some open interval containing c. The number

### Taylor and Maclaurin Series

Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

### M344 - ADVANCED ENGINEERING MATHEMATICS Lecture 9: Orthogonal Functions and Trigonometric Fourier Series

M344 - ADVANCED ENGINEERING MATHEMATICS ecture 9: Orthogonal Functions and Trigonometric Fourier Series Before learning to solve partial differential equations, it is necessary to know how to approximate

### 2 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

### Blue Pelican Calculus First Semester

Blue Pelican Calculus First Semester Teacher Version 1.01 Copyright 2011-2013 by Charles E. Cook; Refugio, Tx Edited by Jacob Cobb (All rights reserved) Calculus AP Syllabus (First Semester) Unit 1: Function

### The Derivative and the Tangent Line Problem. The Tangent Line Problem

The Derivative and the Tangent Line Problem Calculus grew out of four major problems that European mathematicians were working on during the seventeenth century. 1. The tangent line problem 2. The velocity

### Georgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1

Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### Limits. Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as:

Limits Limits: Graphical Solutions Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as: The limits are defined as the value that the function approaches as it goes

### MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

### Basic Integration Formulas and the Substitution Rule

Basic Integration Formulas and the Substitution Rule The second fundamental theorem of integral calculus Recall from the last lecture the second fundamental theorem of integral calculus. Theorem Let f(x)

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

### The Derivative. Philippe B. Laval Kennesaw State University

The Derivative Philippe B. Laval Kennesaw State University Abstract This handout is a summary of the material students should know regarding the definition and computation of the derivative 1 Definition

### Algebra 2 Final Exam Review Ch 7-14

Algebra 2 Final Exam Review Ch 7-14 Short Answer 1. Simplify the radical expression. Use absolute value symbols if needed. 2. Simplify. Assume that all variables are positive. 3. Multiply and simplify.

### 3. Exponential and Logarithmic functions

3. ial and s ial and ic... 3.1. Here are a few examples to remind the reader of the definitions and laws for expressions involving exponents: 2 3 = 2 2 2 = 8, 2 0 = 1, 2 1 = 1 2, 2 3 = 1 2 3 = 1 8, 9 1/2

### CHAPTER 10 Limit Formulas

CHAPTER 10 Limit Formulas 10.1 Definition of Limit LIMIT OF A FUNCTION (INFORMAL DEFINITION) The notation D L is read the it of f(x)asx approaches c is L and means that the functional values f(x) can be

### 106 Chapter 5 Curve Sketching. If f(x) has a local extremum at x = a and. THEOREM 5.1.1 Fermat s Theorem f is differentiable at a, then f (a) = 0.

5 Curve Sketching Whether we are interested in a function as a purely mathematical object or in connection with some application to the real world, it is often useful to know what the graph of the function

### The degree of a polynomial function is equal to the highest exponent found on the independent variables.

DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

### Fourier Series. Some Properties of Functions. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Fourier Series Today 1 / 19

Fourier Series Some Properties of Functions Philippe B. Laval KSU Today Philippe B. Laval (KSU) Fourier Series Today 1 / 19 Introduction We review some results about functions which play an important role

### k=1 k2, and therefore f(m + 1) = f(m) + (m + 1) 2 =

Math 104: Introduction to Analysis SOLUTIONS Alexander Givental HOMEWORK 1 1.1. Prove that 1 2 +2 2 + +n 2 = 1 n(n+1)(2n+1) for all n N. 6 Put f(n) = n(n + 1)(2n + 1)/6. Then f(1) = 1, i.e the theorem

### Quadratic Equations and Inequalities

MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose

### I. Pointwise convergence

MATH 40 - NOTES Sequences of functions Pointwise and Uniform Convergence Fall 2005 Previously, we have studied sequences of real numbers. Now we discuss the topic of sequences of real valued functions.

### D f = (2, ) (x + 1)(x 3) (b) g(x) = x 1 solution: We need the thing inside the root to be greater than or equal to 0. So we set up a sign table.

. Find the domains of the following functions: (a) f(x) = ln(x ) We need x > 0, or x >. Thus D f = (, ) (x + )(x 3) (b) g(x) = x We need the thing inside the root to be greater than or equal to 0. So we

### f(x) = g(x), if x A h(x), if x B.

1. Piecewise Functions By Bryan Carrillo, University of California, Riverside We can create more complicated functions by considering Piece-wise functions. Definition: Piecewise-function. A piecewise-function

### 3.3 Real Zeros of Polynomials

3.3 Real Zeros of Polynomials 69 3.3 Real Zeros of Polynomials In Section 3., we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section

### Fourier series. Jan Philip Solovej. English summary of notes for Analysis 1. May 8, 2012

Fourier series Jan Philip Solovej English summary of notes for Analysis 1 May 8, 2012 1 JPS, Fourier series 2 Contents 1 Introduction 2 2 Fourier series 3 2.1 Periodic functions, trigonometric polynomials

### PRE-CALCULUS GRADE 12

PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

### DIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents

DIFFERENTIABILITY OF COMPLEX FUNCTIONS Contents 1. Limit definition of a derivative 1 2. Holomorphic functions, the Cauchy-Riemann equations 3 3. Differentiability of real functions 5 4. A sufficient condition

### MATH Area Between Curves

MATH - Area Between Curves Philippe Laval September, 8 Abstract This handout discusses techniques used to nd the area of regions which lie between two curves. Area Between Curves. Theor Given two functions

### LIES MY CALCULATOR AND COMPUTER TOLD ME

LIES MY CALCULATOR AND COMPUTER TOLD ME See Section Appendix.4 G for a discussion of graphing calculators and computers with graphing software. A wide variety of pocket-size calculating devices are currently

### Asymptotes. Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if

Section 2.1: Vertical and Horizontal Asymptotes Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if, lim x a f(x) =, lim x a x a x a f(x) =, or. + + Definition.

### Section 2.7 One-to-One Functions and Their Inverses

Section. One-to-One Functions and Their Inverses One-to-One Functions HORIZONTAL LINE TEST: A function is one-to-one if and only if no horizontal line intersects its graph more than once. EXAMPLES: 1.