Denition and some Properties of Generalized Elementary Functions of a Real Variable
|
|
- Erica Long
- 6 years ago
- Views:
Transcription
1 Denition and some Properties of Generalized Elementary Functions of a Real Variable I. Introduction The term elementary function is very often mentioned in many math classes and in books, e.g. Calculus books. In fact, the very vast majority of the functions that students come across are elementary functions of a real variable. However, there is a lack of a precise mathematical denition of elementary functions. Only a few authors in their textbooks, e.g. Stewart in his Calculus books try to give a description of elementary functions. Unfortunately, these descriptions are not given properly. For example, from the descriptions of elementary functions in Stewart's book, one could conclude that the function sin x, x 5 f(x) = ln x, x > 5 is elementary!! This is simply incorrect. Thus, this note is written to introduce a precise mathematical denition of generalized elementary functions of a real variable, which is a most broader class of functions that includes all the elementary functions. It is not claimed to be an original research article, but rather a note that could serve the students to see a proper mathematical denition of the term Generalized Elementary Function of a Real Variable. After the denition is introduced, it is easy to see that the generalized elementary functions of a real variable possess properties (which all elementary functions also possess) that could greatly simplify the mathematical analysis needed to be done on them. Also, many problems in mathematics deal with generalized elementary functions or even if the functions are non-elementary, very often the studying of these non-elementary functions lead to generalized elementary functions. According to the denition of generalized elementary functions given in this note, there are functions that are usually considered to be non-elementary, but are generalized elementary functions according to the denition. Since the problems are much simplied when the functions involved are generalized elementary functions, it is a good idea to have a precise mathematical denition of the generalized elementary functions. This denition and some more properties of generalized elementary functions are provided in this note. II. Denitions Denition. The following ve functions are referred to as the fundamental elementary functions of a real variable f (x) = c, c R with domain D R (D ) f 2 (x) = x, with domain D R (D ) f 3 (x) = e x, with domain D R (D )
2 f 4 (x) = sin x, with domain D R (D ) f 5 (x) =, with domain D R \ 0} (D ) x Denition 2. For any two functions (of a real variable) f(x) and g(x) with domains D f, D g and ranges R f, R g, respectively, the following operations are called The Fundamental Elementary Operations on Functions:. Addition: x D = D f Dg, f(x) and g(x) are both dened and have values a and b (a R and a R) respectively. Thus, x D there is a unique corresponding real number c = a + b. Hence, we dene a new function O (x) = f(x) + g(x) where the domain of O (x) is D = D f Dg, which is called the sum of f(x) and g(x). 2. Multiplication: In a similar fashion as in, we dene O 2 (x) = f(x) g(x) where the domain of O 2 (x) is D = D f Dg. O 2 (x) is called the product of f(x) and g(x). 3. Composition of Functions: x D f and R f D g, f(x) is dened and has a value f(x) = a, a R. Since x D f, a R f D g, hence a D g. Since a D g, g(a) = g(f(x)) is dened and has value g(a) = g(f(x)) = b. Thus, we dene O 3 (x) to be the composite function of f(x) and g(x) i R f D g O 3 (x) = g(f(x)), x D O3 (x) = D f 4. Inverse Functions: The function g(y) (domain D g and range R g ) is called inverse (function) of f(x) (domain D f and range R f ), i. R g D f 2. D g = R f 3. y D g, f(g(y)) = y We shall also call the operations that could be obtained from the above four fundamental operations an elementary operation. Denition 3. A function F (x) with domain D is called an invertible function if F (x) has a unique inverse function. Usually, this unique inverse function of F (x) is denoted as F (x). Denition 4. A function F (x) with domain D is called a generalized elementary function, if it can be obtained from one and the same set of fundamental elementary functions using a nite number of fundamental elementary operations in one and the same order. Remark. If F (x) is a generalized elementary function, then there exists exactly one formula to calculate the value of F (x) at any point x in its domain. Theorem. Subtraction could be obtained from applying the fundamental elementary operations, i.e. subtraction is an elementary operation. 2
3 Proof. f(x) g(x) = f(x) + ( g(x)) = f(x) + ( )(g(x)). Thus, the theorem follows. Theorem 2. Division could be obtained from applying the fundamental elementary operations, i.e. division is an elementary operation. Proof. Similarly done as in Theorem. Theorem 3. All polynomial functions are generalized elementary functions. Proof. A general polynomial function is dened as f(x) = a n x n + a n x n a x + a 0 where n N, a n 0, a i R, i = 0,,..., n, and D f R. For an arbitrary n N, x n is obtained from n multiplication(s) of x and multiplication is a fundamental elementary operation. Thus x n is an elementary function. Since a i R (i = 0,, 2,..., n), a i is an elementary function. Since multiplication (a fundamental elementary operation) of two fundamental elementary functions is a generalized elementary function, the theorem is proven. Theorem 4. All rational functions are generalized elementary functions. Proof. A rational function could be dened as r(x) = f(x) g(x) where f(x) and g(x) are polynomial functions with domains D f and D g respectively. Then the domain of r(x) is D r = D f Dg \ x g(x) = 0}. The proof could be done in a similar fashion as in Theorem 3. Theorem 5. All algebraic functions are generalized elementary functions. Proof. Roots are inverses of powers (fundamental elementary operations) and rational powers could be dened in terms of roots, the theorem follows. Theorem 6. All trigonometric functions are generalized elementary functions. Proof. First, it will be shown that f(x) = cos(x), x R is a generalized elementary function. Note that cos x sin 2 x, cos x = ± sin 2 x depending on values of x. Thus, it is not a good idea to go this way. However, sin(x + π) = cos x. Since, x R is a fundamental elementary function and π is a fundamental 2 2 elementary function, x + π is an elementary function. Since sin x (x R) is a fundamental elementary 2 function, the composition of sin x with x + π which gives sin(x + π ) = cos x is also an elementary function 2 2 Since cos x is an elementary function, tan x = sin x cos x, cot x =, sec x =, and csc x = are all cos x sin x cos x sin x generalized elementary functions. Thus, all trigonometric functions are generalized elementary functions. It follows that all trigonometric function of generalized elementary functions are also generalized elementary functions. Theorem 7. ln x (x > 0) is a generalized elementary function. Proof. Since ln x (x > 0) is the inverse function of e x (fundamental elementary function), ln x (x > 0) is a generalized elementary function. Theorem 8. All logarithmic function of generalized elementary functions are generalized elementary functions. 3
4 Proof. log u(x) v(x) = ln v(x) ln u(x) = ln v(x) ln u(x) for values of x such that u(x), u(x) > 0, and v(x) > 0. Since ln v(x) is a generalized elementary function (composition of ln x and v(x)) and ln u(x) is also a generalized elementary function (composition of x, ln x and u(x)), ln v(x) ln u(x) = log u(x) v(x) is a generalized elementary function. Theorem 9. Functions of the form u(x) v(x) (u(x) > 0), where u(x) and v(x) are generalized elementary functions, are also generalized elementary functions.. Proof. For all values of x such that u(x) and v(x) are dened and greater than 0, u(x) v(x) = e v(x) ln u(x). Since the operations involved are either the fundamental elementary operations or a combination of them, u(x) v(x) is a generalized elementary function. Denition 5. For a function f(x) dened in a domain D, a point a is called an isolated point of D i. a D 2. ɛ > 0 such that x (a ɛ, a) (a, a + ɛ), x / D. The following statements are theorems that are very often proven in textbooks and thus will only be stated without proofs. All ve, f (x) - f 5 (x), fundamental elementary functions are continuous everywhere in their domains except at the isolated points and are discontinuous at the isolated points. 2. The sum of two continuous functions is also continuous everywhere in its domain except at the isolated points and is discontinuous at the isolated points. 3. The product of two continuous functions is also continuous everywhere in its domain except at the 4. The composition of two continuous functions is also continuous everywhere in its domain except at the 5. An inverse function of a continuous function is also continuous everywhere in its domain except at the Thus, one has the following very important theorem concerning generalized elementary functions Theorem 0. All generalized elementary functions are continuous in their domains, except at the isolated points at which they are discontinuous. Proof. This proof consists of applying various well-known theorems on continuity of functions. Theorem. The function is a generalized elementary function. x = x, x 0 x, x < 0 Proof. x is usually misunderstood as being a non-elementary function, since the function has two separate formulas. However, it is possible to write x with only one formula. That is x = x 2 which has only the elementary operations, square and square root. elementary function. Thus x for x R is a generalized Theorem 2. For a function f(x) dened as f(x) = g(x) h(x), x < a, x > a where g(x) is a generalized elementary function in D g = (, a) and h(x) is a generalized elementary function in D h = (a, + ). The function f(x) with domain D f = R \ a} is a generalized elementary function. 4
5 Proof. Note that and x + x 2x x + x 2x = = 0, x < 0, x > 0, x < 0 0, x > 0 Applying a small shifting in x, the function f(x) could be written as x a x a x a + x a f(x) = g(x) + h(x) 2(x a) 2(x a) with domain D f = R \ a}. Since the formula to calculate the value of f(x) at any point in the domain D f of f(x) consists of only generalized elementary functions and elementary operations, f(x) is a generalized elementary function. Note: The precise denition of the elementary functions (not generalized elementary functions) will be given in another note. I would like to thank my graduate student Tan Nguyen for his contribution to editing/typing of this text. Angel S. Muleshkov, Ph.D. Associate Professor of Mathematics, UNLV, 989 5
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
More informationfunction independent dependent domain range graph of the function The Vertical Line Test
Functions A quantity y is a function of another quantity x if there is some rule (an algebraic equation, a graph, a table, or as an English description) by which a unique value is assigned to y by a corresponding
More informationSection 4.8 Anti Derivative and Indefinite Integrals 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures College of Science MATHS 101: Calculus I (University of Bahrain) 1 / 28 Indefinite Integral Given a function f, if F is a function such that
More informationDRAFT - Math 102 Lecture Note - Dr. Said Algarni
Math02 - Term72 - Guides and Exercises - DRAFT 7 Techniques of Integration A summery for the most important integrals that we have learned so far: 7. Integration by Parts The Product Rule states that if
More informationChapter 6: Rational Expr., Eq., and Functions Lecture notes Math 1010
Section 6.1: Rational Expressions and Functions Definition of a rational expression Let u and v be polynomials. The algebraic expression u v is a rational expression. The domain of this rational expression
More information1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
More informationUniversity Calculus I. Worksheet # 8 Mar b. sin tan e. sin 2 sin 1 5. b. tan. c. sec sin 1 ( x )) cos 1 ( x )) f. csc. c.
MATH 6 WINTER 06 University Calculus I Worksheet # 8 Mar. 06-0 The topic covered by this worksheet is: Derivative of Inverse Functions and the Inverse Trigonometric functions. SamplesolutionstoallproblemswillbeavailableonDL,
More informationChapter 7: Techniques of Integration
Chapter 7: Techniques of Integration MATH 206-01: Calculus II Department of Mathematics University of Louisville last corrected September 14, 2013 1 / 43 Chapter 7: Techniques of Integration 7.1. Integration
More informationMath 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
More informationCALCULUS ASSESSMENT REVIEW
CALCULUS ASSESSMENT REVIEW DEPARTMENT OF MATHEMATICS CHRISTOPHER NEWPORT UNIVERSITY 1. Introduction and Topics The purpose of these notes is to give an idea of what to expect on the Calculus Readiness
More informationMath 180 Written Homework Solutions Assignment #4 Due Tuesday, September 23rd at the beginning of your discussion class.
Math 180 Written Homework Solutions Assignment #4 Due Tuesday, September 23rd at the beginning of your discussion class. Directions. You are welcome to work on the following problems with other MATH 180
More informationf(g(x)) g (x) dx = f(u) du.
1. Techniques of Integration Section 8-IT 1.1. Basic integration formulas. Integration is more difficult than derivation. The derivative of every rational function or trigonometric function is another
More informationEvaluating Limits Analytically. By Tuesday J. Johnson
Evaluating Limits Analytically By Tuesday J. Johnson Suggested Review Topics Algebra skills reviews suggested: Evaluating functions Rationalizing numerators and/or denominators Trigonometric skills reviews
More informationb n x n + b n 1 x n b 1 x + b 0
Math Partial Fractions Stewart 7.4 Integrating basic rational functions. For a function f(x), we have examined several algebraic methods for finding its indefinite integral (antiderivative) F (x) = f(x)
More informationCalculus. 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
More informationIntegration Using Tables and Summary of Techniques
Integration Using Tables and Summary of Techniques Philippe B. Laval KSU Today Philippe B. Laval (KSU) Summary Today 1 / 13 Introduction We wrap up integration techniques by discussing the following topics:
More information1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents. (a) 5
More informationInverse Trig Functions
6.6i Inverse Trigonometric Functions Inverse Sine Function Does g(x) = sin(x) have an inverse? What restriction would we need to make so that at least a piece of this function has an inverse? Given f (x)
More informationMAT01A1: Functions and Mathematical Models
MAT01A1: Functions and Mathematical Models Dr Craig 21 February 2017 Introduction Who: Dr Craig What: Lecturer & course coordinator for MAT01A1 Where: C-Ring 508 acraig@uj.ac.za Web: http://andrewcraigmaths.wordpress.com
More information1.2 A List of Commonly Occurring Functions
Arkansas Tech University MATH 2914: Calculus I Dr. Marcel B. Finan 1.2 A List of Commonly Occurring Functions In this section, we discuss the most common functions occurring in calculus. Linear Functions
More informationCalculus. Weijiu Liu. Department of Mathematics University of Central Arkansas 201 Donaghey Avenue, Conway, AR 72035, USA
Calculus Weijiu Liu Department of Mathematics University of Central Arkansas 201 Donaghey Avenue, Conway, AR 72035, USA 1 Opening Welcome to your Calculus I class! My name is Weijiu Liu. I will guide you
More informationsin cos 1 1 tan sec 1 cot csc Pre-Calculus Mathematics Trigonometric Identities and Equations
Pre-Calculus Mathematics 12 6.1 Trigonometric Identities and Equations Goal: 1. Identify the Fundamental Trigonometric Identities 2. Simplify a Trigonometric Expression 3. Determine the restrictions on
More informationTopics from Algebra and Pre-Calculus. (Key contains solved problems)
Topics from Algebra and Pre-Calculus (Key contains solved problems) Note: The purpose of this packet is to give you a review of basic skills. You are asked not to use the calculator, except on p. (8) and
More informationCalculus : Summer Study Guide Mr. Kevin Braun Bishop Dunne Catholic School. Calculus Summer Math Study Guide
1 Calculus 2018-2019: Summer Study Guide Mr. Kevin Braun (kbraun@bdcs.org) Bishop Dunne Catholic School Name: Calculus Summer Math Study Guide After you have practiced the skills on Khan Academy (list
More informationMATH 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
More informationEven and odd functions
Connexions module: m15279 1 Even and odd functions Sunil Kumar Singh This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License Even and odd functions are
More informationSESSION 6 Trig. Equations and Identities. Math 30-1 R 3. (Revisit, Review and Revive)
SESSION 6 Trig. Equations and Identities Math 30-1 R 3 (Revisit, Review and Revive) 1 P a g e 2 P a g e Mathematics 30-1 Learning Outcomes Specific Outcome 5: Solve, algebraically and graphically, first
More informationFunctions. Remark 1.2 The objective of our course Calculus is to study functions.
Functions 1.1 Functions and their Graphs Definition 1.1 A function f is a rule assigning a number to each of the numbers. The number assigned to the number x via the rule f is usually denoted by f(x).
More informationMTH30 Review Sheet. y = g(x) BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE
BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH0 Review Sheet. Given the functions f and g described by the graphs below: y = f(x) y = g(x) (a)
More informationChapter 8: Techniques of Integration
Chapter 8: Techniques of Integration Section 8.1 Integral Tables and Review a. Important Integrals b. Example c. Integral Tables Section 8.2 Integration by Parts a. Formulas for Integration by Parts b.
More informationUNIT 3: DERIVATIVES STUDY GUIDE
Calculus I UNIT 3: Derivatives REVIEW Name: Date: UNIT 3: DERIVATIVES STUDY GUIDE Section 1: Section 2: Limit Definition (Derivative as the Slope of the Tangent Line) Calculating Rates of Change (Average
More informationR3.6 Solving Linear Inequalities. 3) Solve: 2(x 4) - 3 > 3x ) Solve: 3(x 2) > 7-4x. R8.7 Rational Exponents
Level D Review Packet - MMT This packet briefly reviews the topics covered on the Level D Math Skills Assessment. If you need additional study resources and/or assistance with any of the topics below,
More information3. Use absolute value notation to write an inequality that represents the statement: x is within 3 units of 2 on the real line.
PreCalculus Review Review Questions 1 The following transformations are applied in the given order) to the graph of y = x I Vertical Stretch by a factor of II Horizontal shift to the right by units III
More information7.1. Calculus of inverse functions. Text Section 7.1 Exercise:
Contents 7. Inverse functions 1 7.1. Calculus of inverse functions 2 7.2. Derivatives of exponential function 4 7.3. Logarithmic function 6 7.4. Derivatives of logarithmic functions 7 7.5. Exponential
More informationMth Review Problems for Test 2 Stewart 8e Chapter 3. For Test #2 study these problems, the examples in your notes, and the homework.
For Test # study these problems, the examples in your notes, and the homework. Derivative Rules D [u n ] = nu n 1 du D [ln u] = du u D [log b u] = du u ln b D [e u ] = e u du D [a u ] = a u ln a du D [sin
More informationpage 1 of 14 1 for all x because f 1 = f and1 f = f. The identity for = x for all x because f
page of 4 Entry # Inverses in General The term inverse is used in very different contexts in mathematics. For example, the multiplicative inverse of a number, the inverse of a function, and the inverse
More informationMath 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
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) ±
Final Review for Pre Calculus 009 Semester Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation algebraically. ) v + 5 = 7 - v
More informationDerivative and Integral Rules These are on the inside of the back cover of your text.
Derivative and Integral Rules These are on the inside of the back cover of your text. General Derivative Rule General Integral Rule d dx u(x) r = r u(x) r - 1 u(x) u(x)r u(x) dx = u(x) r1 r1 + C r U -1
More informationSOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS )
SOLVING TRIGONOMETRIC INEQUALITIES (CONCEPT, METHODS, AND STEPS ) Definition. A trig inequality is an inequality in standard form: R(x) > 0 (or < 0) that contains one or a few trig functions of the variable
More informationAP Calculus Summer Homework MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
AP Calculus Summer Homework 2015-2016 Part 2 Name Score MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the distance d(p1, P2) between the points
More informationLecture 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
More information12) y = -2 sin 1 2 x - 2
Review -Test 1 - Unit 1 and - Math 41 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find and simplify the difference quotient f(x + h) - f(x),
More informationCalculus I Sample Exam #01
Calculus I Sample Exam #01 1. Sketch the graph of the function and define the domain and range. 1 a) f( x) 3 b) g( x) x 1 x c) hx ( ) x x 1 5x6 d) jx ( ) x x x 3 6 . Evaluate the following. a) 5 sin 6
More informationAP Calculus Summer Packet
AP Calculus Summer Packet Writing The Equation Of A Line Example: Find the equation of a line that passes through ( 1, 2) and (5, 7). ü Things to remember: Slope formula, point-slope form, slopeintercept
More informationSection 0.2 & 0.3 Worksheet. Types of Functions
MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2
More information8.7 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 informationJUST THE MATHS UNIT NUMBER ORDINARY DIFFERENTIAL EQUATIONS 1 (First order equations (A)) A.J.Hobson
JUST THE MATHS UNIT NUMBER 5. ORDINARY DIFFERENTIAL EQUATIONS (First order equations (A)) by A.J.Hobson 5.. Introduction and definitions 5..2 Exact equations 5..3 The method of separation of the variables
More informationPreliminaries Lectures. Dr. Abdulla Eid. Department of Mathematics MATHS 101: Calculus I
Preliminaries 2 1 2 Lectures Department of Mathematics http://www.abdullaeid.net/maths101 MATHS 101: Calculus I (University of Bahrain) Prelim 1 / 35 Pre Calculus MATHS 101: Calculus MATHS 101 is all about
More informationChapter 8 Indeterminate Forms and Improper Integrals Math Class Notes
Chapter 8 Indeterminate Forms and Improper Integrals Math 1220-004 Class Notes Section 8.1: Indeterminate Forms of Type 0 0 Fact: The it of quotient is equal to the quotient of the its. (book page 68)
More informationPrinciple 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 informationMath 121: Calculus 1 - Fall 2013/2014 Review of Precalculus Concepts
Introduction Math 121: Calculus 1 - Fall 201/2014 Review of Precalculus Concepts Welcome to Math 121 - Calculus 1, Fall 201/2014! This problems in this packet are designed to help you review the topics
More informationHAND IN PART. Prof. Girardi Math 142 Spring Exam 1. NAME: key
HAND IN PART Prof. Girardi Math 4 Spring 4..4 Exam MARK BOX problem points 7 % NAME: key PIN: INSTRUCTIONS The mark box above indicates the problems along with their points. Check that your copy of the
More informationAlgebra 2/Trig AIIT.17 Trig Identities Notes. Name: Date: Block:
Algebra /Trig AIIT.7 Trig Identities Notes Mrs. Grieser Name: Date: Block: Trigonometric Identities When two trig expressions can be proven to be equal to each other, the statement is called a trig identity
More informationCALCULUS II MATH Dr. Hyunju Ban
CALCULUS II MATH 2414 Dr. Hyunju Ban Introduction Syllabus Chapter 5.1 5.4 Chapters To Be Covered: Chap 5: Logarithmic, Exponential, and Other Transcendental Functions (2 week) Chap 7: Applications of
More informationAP Calculus Summer Homework
Class: Date: AP Calculus Summer Homework Show your work. Place a circle around your final answer. 1. Use the properties of logarithms to find the exact value of the expression. Do not use a calculator.
More informationTrigonometric Identities Exam Questions
Trigonometric Identities Exam Questions Name: ANSWERS January 01 January 017 Multiple Choice 1. Simplify the following expression: cos x 1 cot x a. sin x b. cos x c. cot x d. sec x. Identify a non-permissible
More informationFUNCTIONAL EQUATIONS
FUNCTIONAL EQUATIONS ZHIQIN LU 1. What is a functional equation An equation contains an unknown function is called a functional equation. Example 1.1 The following equations can be regarded as functional
More informationAP Calculus AB Summer Math Packet
Name Date Section AP Calculus AB Summer Math Packet This assignment is to be done at you leisure during the summer. It is meant to help you practice mathematical skills necessary to be successful in Calculus
More informationCALCULUS OPTIONAL SUMMER WORK
NAME JUNE 016 CALCULUS OPTIONAL SUMMER WORK PART I - NO CALCULATOR I. COORDINATE GEOMETRY 1) Identify the indicated quantities for -8x + 15y = 0. x-int y-int slope ) A line has a slope of 5/7 and contains
More informationMath 180, Final Exam, Fall 2012 Problem 1 Solution
Math 80, Final Exam, Fall 0 Problem Solution. Find the derivatives of the following functions: (a) ln(ln(x)) (b) x 6 + sin(x) e x (c) tan(x ) + cot(x ) (a) We evaluate the derivative using the Chain Rule.
More informationALGEBRA AND TRIGONOMETRY REVIEW by Dr TEBOU, FIU. A. Fundamental identities Throughout this section, a and b denotes arbitrary real numbers.
ALGEBRA AND TRIGONOMETRY REVIEW by Dr TEBOU, FIU A. Fundamental identities Trougout tis section, a and b denotes arbitrary real numbers. i) Square of a sum: (a+b) =a +ab+b ii) Square of a difference: (a-b)
More informationMath 113 (Calculus 2) Exam 4
Math 3 (Calculus ) Exam 4 November 0 November, 009 Sections 0, 3 7 Name Student ID Section Instructor In some cases a series may be seen to converge or diverge for more than one reason. For such problems
More informationSummer Review for Students Taking Calculus in No calculators allowed. To earn credit: Be sure to show all work in the area provided.
Summer Review for Students Taking Calculus in 2016-2017 No calculators allowed. To earn credit: Be sure to show all work in the area provided. 1 Graph each equation on the axes provided. Include any relevant
More informationTaylor and Maclaurin Series. Approximating functions using Polynomials.
Taylor and Maclaurin Series Approximating functions using Polynomials. Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear
More information1.3 Limits and Continuity
40 CHAPTER 1. RATES OF CHANGE AND THE DERIVATIVE 1.3 Limits and Continuity Despite the fact that we have placed all of the proofs of the theorems from this section in the Technical Matters section, Section
More informationCalculus Summer Math Practice. 1. Find inverse functions Describe in words how you use algebra to determine the inverse function.
1 Calculus 2017-2018: Summer Study Guide Mr. Kevin Braun (kbraun@bdcs.org) Bishop Dunne Catholic School Calculus Summer Math Practice Please see the math department document for instructions on setting
More informationPre-Calculus Exam 2009 University of Houston Math Contest. Name: School: There is no penalty for guessing.
Pre-Calculus Exam 009 University of Houston Math Contest Name: School: Please read the questions carefully and give a clear indication of your answer on each question. There is no penalty for guessing.
More informationMath 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
More informationSection 2.5. Evaluating Limits Algebraically
Section 2.5 Evaluating Limits Algebraically (1) Determinate and Indeterminate Forms (2) Limit Calculation Techniques (A) Direct Substitution (B) Simplification (C) Conjugation (D) The Squeeze Theorem (3)
More informationIn this note we will evaluate the limits of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0. f(x)
L Hôpital s Rule In this note we will evaluate the its of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0 f() Suppose a f() = 0 and a g() = 0. Then a g() the indeterminate
More informationThe function graphed below is continuous everywhere. The function graphed below is NOT continuous everywhere, it is discontinuous at x 2 and
Section 1.4 Continuity A function is a continuous at a point if its graph has no gaps, holes, breaks or jumps at that point. If a function is not continuous at a point, then we say it is discontinuous
More informationMATH 1231 MATHEMATICS 1B CALCULUS. Section 5: - Power Series and Taylor Series.
MATH 1231 MATHEMATICS 1B CALCULUS. Section 5: - Power Series and Taylor Series. The objective of this section is to become familiar with the theory and application of power series and Taylor series. By
More informationMATH 250 TOPIC 13 INTEGRATION. 13B. Constant, Sum, and Difference Rules
Math 5 Integration Topic 3 Page MATH 5 TOPIC 3 INTEGRATION 3A. Integration of Common Functions Practice Problems 3B. Constant, Sum, and Difference Rules Practice Problems 3C. Substitution Practice Problems
More informationFebruary 21 Math 1190 sec. 63 Spring 2017
February 21 Math 1190 sec. 63 Spring 2017 Chapter 2: Derivatives Let s recall the efinitions an erivative rules we have so far: Let s assume that y = f (x) is a function with c in it s omain. The erivative
More informationAP Calculus Summer Prep
AP Calculus Summer Prep Topics from Algebra and Pre-Calculus (Solutions are on the Answer Key on the Last Pages) The purpose of this packet is to give you a review of basic skills. You are asked to have
More informationTaylor and Maclaurin Series. Approximating functions using Polynomials.
Taylor and Maclaurin Series Approximating functions using Polynomials. Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear
More information( ) - 4(x -3) ( ) 3 (2x -3) - (2x +12) ( x -1) 2 x -1) 2 (3x -1) - 2(x -1) Section 1: Algebra Review. Welcome to AP Calculus!
Welcome to AP Calculus! Successful Calculus students must have a strong foundation in algebra and trigonometry. The following packet was designed to help you review your algebra skills in preparation for
More informationMathematics 136 Calculus 2 Everything You Need Or Want To Know About Partial Fractions (and maybe more!) October 19 and 21, 2016
Mathematics 36 Calculus 2 Everything You Need Or Want To Know About Partial Fractions (and maybe more!) October 9 and 2, 206 Every rational function (quotient of polynomials) can be written as a polynomial
More informationExam Review 2 nd Semester 6-1 Operations on Functions
NAME DATE PERIOD Exam Review 2 nd Semester 6-1 Operations on Functions Find (f + g)(x), (f g)(x), (f g)(x), and (x) for each f(x) and g(x). 1. f(x) = 8x 3; g(x) = 4x + 5 2. f(x) = + x 6; g(x) = x 2 If
More informationChapter 6: Messy Integrals
Chapter 6: Messy Integrals Review: Solve the following integrals x 4 sec x tan x 0 0 Find the average value of 3 1 x 3 3 Evaluate 4 3 3 ( x 1), then find the area of ( x 1) 4 Section 6.1: Slope Fields
More information2.1 Limits, Rates of Change and Slopes of Tangent Lines
2.1 Limits, Rates of Change and Slopes of Tangent Lines (1) Average rate of change of y f x over an interval x 0,x 1 : f x 1 f x 0 x 1 x 0 Instantaneous rate of change of f x at x x 0 : f x lim 1 f x 0
More informationMath 480 The Vector Space of Differentiable Functions
Math 480 The Vector Space of Differentiable Functions The vector space of differentiable functions. Let C (R) denote the set of all infinitely differentiable functions f : R R. Then C (R) is a vector space,
More informationContinuity, Intermediate Value Theorem (2.4)
Continuity, Intermediate Value Theorem (2.4) Xiannan Li Kansas State University January 29th, 2017 Intuitive Definition: A function f(x) is continuous at a if you can draw the graph of y = f(x) without
More information±. Then. . x. lim g( x) = lim. cos x 1 sin x. and (ii) lim
MATH 36 L'H ˆ o pital s Rule Si of the indeterminate forms of its may be algebraically determined using L H ˆ o pital's Rule. This rule is only stated for the / and ± /± indeterminate forms, but four other
More informationMath Practice Exam 3 - solutions
Math 181 - Practice Exam 3 - solutions Problem 1 Consider the function h(x) = (9x 2 33x 25)e 3x+1. a) Find h (x). b) Find all values of x where h (x) is zero ( critical values ). c) Using the sign pattern
More informationSANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET
SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET 017-018 Name: 1. This packet is to be handed in on Monday August 8, 017.. All work must be shown on separate paper attached to the packet. 3.
More informationExamples 2: Composite Functions, Piecewise Functions, Partial Fractions
Examples 2: Composite Functions, Piecewise Functions, Partial Fractions September 26, 206 The following are a set of examples to designed to complement a first-year calculus course. objectives are listed
More informationChapter 8B - Trigonometric Functions (the first part)
Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 8B-I! Page 79 Chapter 8B - Trigonometric Functions (the first part) Recall from geometry that if 2 corresponding triangles have 2 angles of
More informationJUST THE MATHS UNIT NUMBER DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) A.J.Hobson
JUST THE MATHS UNIT NUMBER.5 DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) by A.J.Hobson.5. Maclaurin s series.5. Standard series.5.3 Taylor s series.5.4 Exercises.5.5 Answers to exercises
More informationDRAFT - Math 101 Lecture Note - Dr. Said Algarni
2 Limits 2.1 The Tangent Problems The word tangent is derived from the Latin word tangens, which means touching. A tangent line to a curve is a line that touches the curve and a secant line is a line that
More informationMATH1190 CALCULUS 1 - NOTES AND AFTERNOTES
MATH90 CALCULUS - NOTES AND AFTERNOTES DR. JOSIP DERADO. Historical background Newton approach - from physics to calculus. Instantaneous velocity. Leibniz approach - from geometry to calculus Calculus
More informationMath 152 Take Home Test 1
Math 5 Take Home Test Due Monday 5 th October (5 points) The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics I
More information3 Algebraic Methods. we can differentiate both sides implicitly to obtain a differential equation involving x and y:
3 Algebraic Methods b The first appearance of the equation E Mc 2 in Einstein s handwritten notes. So far, the only general class of differential equations that we know how to solve are directly integrable
More informationChapter 1. Functions 1.1. Functions and Their Graphs
1.1 Functions and Their Graphs 1 Chapter 1. Functions 1.1. Functions and Their Graphs Note. We start by assuming that you are familiar with the idea of a set and the set theoretic symbol ( an element of
More informationC3 Revision Questions. (using questions from January 2006, January 2007, January 2008 and January 2009)
C3 Revision Questions (using questions from January 2006, January 2007, January 2008 and January 2009) 1 2 1. f(x) = 1 3 x 2 + 3, x 2. 2 ( x 2) (a) 2 x x 1 Show that f(x) =, x 2. 2 ( x 2) (4) (b) Show
More informationDifferentiation. Table of contents Definition Arithmetics Composite and inverse functions... 5
Differentiation Table of contents. Derivatives................................................. 2.. Definition................................................ 2.2. Arithmetics...............................................
More informationSOLVING TRIGONOMETRIC INEQUALITIES: METHODS, AND STEPS By Nghi H. Nguyen
DEFINITION. SOLVING TRIGONOMETRIC INEQUALITIES: METHODS, AND STEPS By Nghi H. Nguyen A trig inequality is an inequality in standard form: R(x) > 0 (or < 0) that contains one or a few trig functions of
More informationREVIEW: MORE FUNCTIONS AP CALCULUS :: MR. VELAZQUEZ
REVIEW: MORE FUNCTIONS AP CALCULUS :: MR. VELAZQUEZ INVERSE FUNCTIONS Two functions are inverses if they undo each other. In other words, composing one function in the other will result in simply x (the
More informationTechniques of Integration
Chapter 8 Techniques of Integration 8. Trigonometric Integrals Summary (a) Integrals of the form sin m x cos n x. () sin k+ x cos n x = ( cos x) k cos n x (sin x ), then apply the substitution u = cos
More information