Single Variable Calculus
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Inverse of Exponential Functions Are Logarithmic Functions
Math Instructional Framework Full Name Math III Unit 3 Lesson 2 Time Frame Unit Name Logarithmic Functions as Inverses of Exponential Functions Learning Task/Topics/ Task 2: How long Does It Take? Themes Task 3: The Population of Exponentia Task 4: Modeling Natural Phenomena on Earth Culminating Task: Traveling to Exponentia Standards and Elements MM3A2. Students will explore logarithmic functions as inverses of exponential functions. c. Define logarithmic functions as inverses of exponential functions. Lesson Essential Questions How can you graph the inverse of an exponential function? Activator PROBLEM 2.Task 3: The Population of Exponentia (Problem 1 could be completed prior) Work Session Inverse of Exponential Functions are Logarithmic Functions A Graph the inverse of exponential functions. B Graph logarithmic functions. See Notes Below. VOCABULARY Asymptote: A line or curve that describes the end behavior of the graph. A graph never crosses a vertical asymptote but it may cross a horizontal or oblique asymptote. Common logarithm: A logarithm with a base of 10. A common logarithm is the power, a, such that 10a = b. The common logarithm of x is written log x. For example, log 100 = 2 because 102 = 100. Exponential functions: A function of the form y = a·bx where a > 0 and either 0 < b < 1 or b > 1. Logarithmic functions: A function of the form y = logbx, with b 1 and b and x both positive. A logarithmic x function is the inverse of an exponential function. The inverse of y = b is y = logbx. Logarithm: The logarithm base b of a number x, logbx, is the power to which b must be raised to equal x. -
An Appreciation of Euler's Formula
Rose-Hulman Undergraduate Mathematics Journal Volume 18 Issue 1 Article 17 An Appreciation of Euler's Formula Caleb Larson North Dakota State University Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Larson, Caleb (2017) "An Appreciation of Euler's Formula," Rose-Hulman Undergraduate Mathematics Journal: Vol. 18 : Iss. 1 , Article 17. Available at: https://scholar.rose-hulman.edu/rhumj/vol18/iss1/17 Rose- Hulman Undergraduate Mathematics Journal an appreciation of euler's formula Caleb Larson a Volume 18, No. 1, Spring 2017 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 [email protected] a scholar.rose-hulman.edu/rhumj North Dakota State University Rose-Hulman Undergraduate Mathematics Journal Volume 18, No. 1, Spring 2017 an appreciation of euler's formula Caleb Larson Abstract. For many mathematicians, a certain characteristic about an area of mathematics will lure him/her to study that area further. That characteristic might be an interesting conclusion, an intricate implication, or an appreciation of the impact that the area has upon mathematics. The particular area that we will be exploring is Euler's Formula, eix = cos x + i sin x, and as a result, Euler's Identity, eiπ + 1 = 0. Throughout this paper, we will develop an appreciation for Euler's Formula as it combines the seemingly unrelated exponential functions, imaginary numbers, and trigonometric functions into a single formula. To appreciate and further understand Euler's Formula, we will give attention to the individual aspects of the formula, and develop the necessary tools to prove it. -
Analysis of Functions of a Single Variable a Detailed Development
ANALYSIS OF FUNCTIONS OF A SINGLE VARIABLE A DETAILED DEVELOPMENT LAWRENCE W. BAGGETT University of Colorado OCTOBER 29, 2006 2 For Christy My Light i PREFACE I have written this book primarily for serious and talented mathematics scholars , seniors or first-year graduate students, who by the time they finish their schooling should have had the opportunity to study in some detail the great discoveries of our subject. What did we know and how and when did we know it? I hope this book is useful toward that goal, especially when it comes to the great achievements of that part of mathematics known as analysis. I have tried to write a complete and thorough account of the elementary theories of functions of a single real variable and functions of a single complex variable. Separating these two subjects does not at all jive with their development historically, and to me it seems unnecessary and potentially confusing to do so. On the other hand, functions of several variables seems to me to be a very different kettle of fish, so I have decided to limit this book by concentrating on one variable at a time. Everyone is taught (told) in school that the area of a circle is given by the formula A = πr2: We are also told that the product of two negatives is a positive, that you cant trisect an angle, and that the square root of 2 is irrational. Students of natural sciences learn that eiπ = 1 and that sin2 + cos2 = 1: More sophisticated students are taught the Fundamental− Theorem of calculus and the Fundamental Theorem of Algebra. -
Variables in Mathematics Education
Variables in Mathematics Education Susanna S. Epp DePaul University, Department of Mathematical Sciences, Chicago, IL 60614, USA http://www.springer.com/lncs Abstract. This paper suggests that consistently referring to variables as placeholders is an effective countermeasure for addressing a number of the difficulties students’ encounter in learning mathematics. The sug- gestion is supported by examples discussing ways in which variables are used to express unknown quantities, define functions and express other universal statements, and serve as generic elements in mathematical dis- course. In addition, making greater use of the term “dummy variable” and phrasing statements both with and without variables may help stu- dents avoid mistakes that result from misinterpreting the scope of a bound variable. Keywords: variable, bound variable, mathematics education, placeholder. 1 Introduction Variables are of critical importance in mathematics. For instance, Felix Klein wrote in 1908 that “one may well declare that real mathematics begins with operations with letters,”[3] and Alfred Tarski wrote in 1941 that “the invention of variables constitutes a turning point in the history of mathematics.”[5] In 1911, A. N. Whitehead expressly linked the concepts of variables and quantification to their expressions in informal English when he wrote: “The ideas of ‘any’ and ‘some’ are introduced to algebra by the use of letters. it was not till within the last few years that it has been realized how fundamental any and some are to the very nature of mathematics.”[6] There is a question, however, about how to describe the use of variables in mathematics instruction and even what word to use for them. -
A Quick Algebra Review
A Quick Algebra Review 1. Simplifying Expressions 2. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Exponents 9. Quadratics 10. Rationals 11. Radicals Simplifying Expressions An expression is a mathematical “phrase.” Expressions contain numbers and variables, but not an equal sign. An equation has an “equal” sign. For example: Expression: Equation: 5 + 3 5 + 3 = 8 x + 3 x + 3 = 8 (x + 4)(x – 2) (x + 4)(x – 2) = 10 x² + 5x + 6 x² + 5x + 6 = 0 x – 8 x – 8 > 3 When we simplify an expression, we work until there are as few terms as possible. This process makes the expression easier to use, (that’s why it’s called “simplify”). The first thing we want to do when simplifying an expression is to combine like terms. For example: There are many terms to look at! Let’s start with x². There Simplify: are no other terms with x² in them, so we move on. 10x x² + 10x – 6 – 5x + 4 and 5x are like terms, so we add their coefficients = x² + 5x – 6 + 4 together. 10 + (-5) = 5, so we write 5x. -6 and 4 are also = x² + 5x – 2 like terms, so we can combine them to get -2. Isn’t the simplified expression much nicer? Now you try: x² + 5x + 3x² + x³ - 5 + 3 [You should get x³ + 4x² + 5x – 2] Order of Operations PEMDAS – Please Excuse My Dear Aunt Sally, remember that from Algebra class? It tells the order in which we can complete operations when solving an equation. -
Number Theory
“mcs-ftl” — 2010/9/8 — 0:40 — page 81 — #87 4 Number Theory Number theory is the study of the integers. Why anyone would want to study the integers is not immediately obvious. First of all, what’s to know? There’s 0, there’s 1, 2, 3, and so on, and, oh yeah, -1, -2, . Which one don’t you understand? Sec- ond, what practical value is there in it? The mathematician G. H. Hardy expressed pleasure in its impracticality when he wrote: [Number theorists] may be justified in rejoicing that there is one sci- ence, at any rate, and that their own, whose very remoteness from or- dinary human activities should keep it gentle and clean. Hardy was specially concerned that number theory not be used in warfare; he was a pacifist. You may applaud his sentiments, but he got it wrong: Number Theory underlies modern cryptography, which is what makes secure online communication possible. Secure communication is of course crucial in war—which may leave poor Hardy spinning in his grave. It’s also central to online commerce. Every time you buy a book from Amazon, check your grades on WebSIS, or use a PayPal account, you are relying on number theoretic algorithms. Number theory also provides an excellent environment for us to practice and apply the proof techniques that we developed in Chapters 2 and 3. Since we’ll be focusing on properties of the integers, we’ll adopt the default convention in this chapter that variables range over the set of integers, Z. 4.1 Divisibility The nature of number theory emerges as soon as we consider the divides relation a divides b iff ak b for some k: D The notation, a b, is an abbreviation for “a divides b.” If a b, then we also j j say that b is a multiple of a. -
IVC Factsheet Functions Comp Inverse
Imperial Valley College Math Lab Functions: Composition and Inverse Functions FUNCTION COMPOSITION In order to perform a composition of functions, it is essential to be familiar with function notation. If you see something of the form “푓(푥) = [expression in terms of x]”, this means that whatever you see in the parentheses following f should be substituted for x in the expression. This can include numbers, variables, other expressions, and even other functions. EXAMPLE: 푓(푥) = 4푥2 − 13푥 푓(2) = 4 ∙ 22 − 13(2) 푓(−9) = 4(−9)2 − 13(−9) 푓(푎) = 4푎2 − 13푎 푓(푐3) = 4(푐3)2 − 13푐3 푓(ℎ + 5) = 4(ℎ + 5)2 − 13(ℎ + 5) Etc. A composition of functions occurs when one function is “plugged into” another function. The notation (푓 ○푔)(푥) is pronounced “푓 of 푔 of 푥”, and it literally means 푓(푔(푥)). In other words, you “plug” the 푔(푥) function into the 푓(푥) function. Similarly, (푔 ○푓)(푥) is pronounced “푔 of 푓 of 푥”, and it literally means 푔(푓(푥)). In other words, you “plug” the 푓(푥) function into the 푔(푥) function. WARNING: Be careful not to confuse (푓 ○푔)(푥) with (푓 ∙ 푔)(푥), which means 푓(푥) ∙ 푔(푥) . EXAMPLES: 푓(푥) = 4푥2 − 13푥 푔(푥) = 2푥 + 1 a. (푓 ○푔)(푥) = 푓(푔(푥)) = 4[푔(푥)]2 − 13 ∙ 푔(푥) = 4(2푥 + 1)2 − 13(2푥 + 1) = [푠푚푝푙푓푦] … = 16푥2 − 10푥 − 9 b. (푔 ○푓)(푥) = 푔(푓(푥)) = 2 ∙ 푓(푥) + 1 = 2(4푥2 − 13푥) + 1 = 8푥2 − 26푥 + 1 A function can even be “composed” with itself: c. -
Symmetric Rigidity for Circle Endomorphisms with Bounded Geometry
SYMMETRIC RIGIDITY FOR CIRCLE ENDOMORPHISMS WITH BOUNDED GEOMETRY JOHN ADAMSKI, YUNCHUN HU, YUNPING JIANG, AND ZHE WANG Abstract. Let f and g be two circle endomorphisms of degree d ≥ 2 such that each has bounded geometry, preserves the Lebesgue measure, and fixes 1. Let h fixing 1 be the topological conjugacy from f to g. That is, h ◦ f = g ◦ h. We prove that h is a symmetric circle homeomorphism if and only if h = Id. Many other rigidity results in circle dynamics follow from this very general symmetric rigidity result. 1. Introduction A remarkable result in geometry is the so-called Mostow rigidity theorem. This result assures that two closed hyperbolic 3-manifolds are isometrically equivalent if they are homeomorphically equivalent [18]. A closed hyperbolic 3-manifold can be viewed as the quotient space of a Kleinian group acting on the open unit ball in the 3-Euclidean space. So a homeomorphic equivalence between two closed hyperbolic 3-manifolds can be lifted to a homeomorphism of the open unit ball preserving group actions. The homeomorphism can be extended to the boundary of the open unit ball as a boundary map. The boundary is the Riemann sphere and the boundary map is a quasi-conformal homeomorphism. A quasi-conformal homeomorphism of the Riemann 2010 Mathematics Subject Classification. Primary: 37E10, 37A05; Secondary: 30C62, 60G42. Key words and phrases. quasisymmetric circle homeomorphism, symmetric cir- arXiv:2101.06870v1 [math.DS] 18 Jan 2021 cle homeomorphism, circle endomorphism with bounded geometry preserving the Lebesgue measure, uniformly quasisymmetric circle endomorphism preserving the Lebesgue measure, martingales. -
SHEET 14: LINEAR ALGEBRA 14.1 Vector Spaces
SHEET 14: LINEAR ALGEBRA Throughout this sheet, let F be a field. In examples, you need only consider the field F = R. 14.1 Vector spaces Definition 14.1. A vector space over F is a set V with two operations, V × V ! V :(x; y) 7! x + y (vector addition) and F × V ! V :(λ, x) 7! λx (scalar multiplication); that satisfy the following axioms. 1. Addition is commutative: x + y = y + x for all x; y 2 V . 2. Addition is associative: x + (y + z) = (x + y) + z for all x; y; z 2 V . 3. There is an additive identity 0 2 V satisfying x + 0 = x for all x 2 V . 4. For each x 2 V , there is an additive inverse −x 2 V satisfying x + (−x) = 0. 5. Scalar multiplication by 1 fixes vectors: 1x = x for all x 2 V . 6. Scalar multiplication is compatible with F :(λµ)x = λ(µx) for all λ, µ 2 F and x 2 V . 7. Scalar multiplication distributes over vector addition and over scalar addition: λ(x + y) = λx + λy and (λ + µ)x = λx + µx for all λ, µ 2 F and x; y 2 V . In this context, elements of F are called scalars and elements of V are called vectors. Definition 14.2. Let n be a nonnegative integer. The coordinate space F n = F × · · · × F is the set of all n-tuples of elements of F , conventionally regarded as column vectors. Addition and scalar multiplication are defined componentwise; that is, 2 3 2 3 2 3 2 3 x1 y1 x1 + y1 λx1 6x 7 6y 7 6x + y 7 6λx 7 6 27 6 27 6 2 2 7 6 27 if x = 6 . -
Two Fundamental Theorems About the Definite Integral
Two Fundamental Theorems about the Definite Integral These lecture notes develop the theorem Stewart calls The Fundamental Theorem of Calculus in section 5.3. The approach I use is slightly different than that used by Stewart, but is based on the same fundamental ideas. 1 The definite integral Recall that the expression b f(x) dx ∫a is called the definite integral of f(x) over the interval [a,b] and stands for the area underneath the curve y = f(x) over the interval [a,b] (with the understanding that areas above the x-axis are considered positive and the areas beneath the axis are considered negative). In today's lecture I am going to prove an important connection between the definite integral and the derivative and use that connection to compute the definite integral. The result that I am eventually going to prove sits at the end of a chain of earlier definitions and intermediate results. 2 Some important facts about continuous functions The first intermediate result we are going to have to prove along the way depends on some definitions and theorems concerning continuous functions. Here are those definitions and theorems. The definition of continuity A function f(x) is continuous at a point x = a if the following hold 1. f(a) exists 2. lim f(x) exists xœa 3. lim f(x) = f(a) xœa 1 A function f(x) is continuous in an interval [a,b] if it is continuous at every point in that interval. The extreme value theorem Let f(x) be a continuous function in an interval [a,b]. -
Video Lessons for Illustrative Mathematics: Grades 6-8 and Algebra I
Video Lessons for Illustrative Mathematics: Grades 6-8 and Algebra I The Department, in partnership with Louisiana Public Broadcasting (LPB), Illustrative Math (IM), and SchoolKit selected 20 of the critical lessons in each grade level to broadcast on LPB in July of 2020. These lessons along with a few others are still available on demand on the SchoolKit website. To ensure accessibility for students with disabilities, all recorded on-demand videos are available with closed captioning and audio description. Updated on August 6, 2020 Video Lessons for Illustrative Math Grades 6-8 and Algebra I Background Information: Lessons were identified using the following criteria: 1. the most critical content of the grade level 2. content that many students missed due to school closures in the 2019-20 school year These lessons constitute only a portion of the critical lessons in each grade level. These lessons are available at the links below: • Grade 6: http://schoolkitgroup.com/video-grade-6/ • Grade 7: http://schoolkitgroup.com/video-grade-7/ • Grade 8: http://schoolkitgroup.com/video-grade-8/ • Algebra I: http://schoolkitgroup.com/video-algebra/ The tables below contain information on each of the lessons available for on-demand viewing with links to individual lessons embedded for each grade level. • Grade 6 Video Lesson List • Grade 7 Video Lesson List • Grade 8 Video Lesson List • Algebra I Video Lesson List Video Lessons for Illustrative Math Grades 6-8 and Algebra I Grade 6 Video Lesson List 6th Grade Illustrative Mathematics Units 2, 3, -
Generalizations of the Riemann Integral: an Investigation of the Henstock Integral
Generalizations of the Riemann Integral: An Investigation of the Henstock Integral Jonathan Wells May 15, 2011 Abstract The Henstock integral, a generalization of the Riemann integral that makes use of the δ-fine tagged partition, is studied. We first consider Lebesgue’s Criterion for Riemann Integrability, which states that a func- tion is Riemann integrable if and only if it is bounded and continuous almost everywhere, before investigating several theoretical shortcomings of the Riemann integral. Despite the inverse relationship between integra- tion and differentiation given by the Fundamental Theorem of Calculus, we find that not every derivative is Riemann integrable. We also find that the strong condition of uniform convergence must be applied to guarantee that the limit of a sequence of Riemann integrable functions remains in- tegrable. However, by slightly altering the way that tagged partitions are formed, we are able to construct a definition for the integral that allows for the integration of a much wider class of functions. We investigate sev- eral properties of this generalized Riemann integral. We also demonstrate that every derivative is Henstock integrable, and that the much looser requirements of the Monotone Convergence Theorem guarantee that the limit of a sequence of Henstock integrable functions is integrable. This paper is written without the use of Lebesgue measure theory. Acknowledgements I would like to thank Professor Patrick Keef and Professor Russell Gordon for their advice and guidance through this project. I would also like to acknowledge Kathryn Barich and Kailey Bolles for their assistance in the editing process. Introduction As the workhorse of modern analysis, the integral is without question one of the most familiar pieces of the calculus sequence.