Analysis of Functions of a Single Variable a Detailed Development
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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. -
Leibniz and the Infinite
Quaderns d’Història de l’Enginyeria volum xvi 2018 LEIBNIZ AND THE INFINITE Eberhard Knobloch [email protected] 1.-Introduction. On the 5th (15th) of September, 1695 Leibniz wrote to Vincentius Placcius: “But I have so many new insights in mathematics, so many thoughts in phi- losophy, so many other literary observations that I am often irresolutely at a loss which as I wish should not perish1”. Leibniz’s extraordinary creativity especially concerned his handling of the infinite in mathematics. He was not always consistent in this respect. This paper will try to shed new light on some difficulties of this subject mainly analysing his treatise On the arithmetical quadrature of the circle, the ellipse, and the hyperbola elaborated at the end of his Parisian sojourn. 2.- Infinitely small and infinite quantities. In the Parisian treatise Leibniz introduces the notion of infinitely small rather late. First of all he uses descriptions like: ad differentiam assignata quavis minorem sibi appropinquare (to approach each other up to a difference that is smaller than any assigned difference)2, differat quantitate minore quavis data (it differs by a quantity that is smaller than any given quantity)3, differentia data quantitate minor reddi potest (the difference can be made smaller than a 1 “Habeo vero tam multa nova in Mathematicis, tot cogitationes in Philosophicis, tot alias litterarias observationes, quas vellem non perire, ut saepe inter agenda anceps haeream.” (LEIBNIZ, since 1923: II, 3, 80). 2 LEIBNIZ (2016), 18. 3 Ibid., 20. 11 Eberhard Knobloch volum xvi 2018 given quantity)4. Such a difference or such a quantity necessarily is a variable quantity. -
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]. -
The Axiom of Choice and Its Implications
THE AXIOM OF CHOICE AND ITS IMPLICATIONS KEVIN BARNUM Abstract. In this paper we will look at the Axiom of Choice and some of the various implications it has. These implications include a number of equivalent statements, and also some less accepted ideas. The proofs discussed will give us an idea of why the Axiom of Choice is so powerful, but also so controversial. Contents 1. Introduction 1 2. The Axiom of Choice and Its Equivalents 1 2.1. The Axiom of Choice and its Well-known Equivalents 1 2.2. Some Other Less Well-known Equivalents of the Axiom of Choice 3 3. Applications of the Axiom of Choice 5 3.1. Equivalence Between The Axiom of Choice and the Claim that Every Vector Space has a Basis 5 3.2. Some More Applications of the Axiom of Choice 6 4. Controversial Results 10 Acknowledgments 11 References 11 1. Introduction The Axiom of Choice states that for any family of nonempty disjoint sets, there exists a set that consists of exactly one element from each element of the family. It seems strange at first that such an innocuous sounding idea can be so powerful and controversial, but it certainly is both. To understand why, we will start by looking at some statements that are equivalent to the axiom of choice. Many of these equivalences are very useful, and we devote much time to one, namely, that every vector space has a basis. We go on from there to see a few more applications of the Axiom of Choice and its equivalents, and finish by looking at some of the reasons why the Axiom of Choice is so controversial. -
17 Axiom of Choice
Math 361 Axiom of Choice 17 Axiom of Choice De¯nition 17.1. Let be a nonempty set of nonempty sets. Then a choice function for is a function f sucFh that f(S) S for all S . F 2 2 F Example 17.2. Let = (N)r . Then we can de¯ne a choice function f by F P f;g f(S) = the least element of S: Example 17.3. Let = (Z)r . Then we can de¯ne a choice function f by F P f;g f(S) = ²n where n = min z z S and, if n = 0, ² = min z= z z = n; z S . fj j j 2 g 6 f j j j j j 2 g Example 17.4. Let = (Q)r . Then we can de¯ne a choice function f as follows. F P f;g Let g : Q N be an injection. Then ! f(S) = q where g(q) = min g(r) r S . f j 2 g Example 17.5. Let = (R)r . Then it is impossible to explicitly de¯ne a choice function for . F P f;g F Axiom 17.6 (Axiom of Choice (AC)). For every set of nonempty sets, there exists a function f such that f(S) S for all S . F 2 2 F We say that f is a choice function for . F Theorem 17.7 (AC). If A; B are non-empty sets, then the following are equivalent: (a) A B ¹ (b) There exists a surjection g : B A. ! Proof. (a) (b) Suppose that A B. -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
The Exponential Constant E
The exponential constant e mc-bus-expconstant-2009-1 Introduction The letter e is used in many mathematical calculations to stand for a particular number known as the exponential constant. This leaflet provides information about this important constant, and the related exponential function. The exponential constant The exponential constant is an important mathematical constant and is given the symbol e. Its value is approximately 2.718. It has been found that this value occurs so frequently when mathematics is used to model physical and economic phenomena that it is convenient to write simply e. It is often necessary to work out powers of this constant, such as e2, e3 and so on. Your scientific calculator will be programmed to do this already. You should check that you can use your calculator to do this. Look for a button marked ex, and check that e2 =7.389, and e3 = 20.086 In both cases we have quoted the answer to three decimal places although your calculator will give a more accurate answer than this. You should also check that you can evaluate negative and fractional powers of e such as e1/2 =1.649 and e−2 =0.135 The exponential function If we write y = ex we can calculate the value of y as we vary x. Values obtained in this way can be placed in a table. For example: x −3 −2 −1 01 2 3 y = ex 0.050 0.135 0.368 1 2.718 7.389 20.086 This is a table of values of the exponential function ex. -
Calculus Formulas and Theorems
Formulas and Theorems for Reference I. Tbigonometric Formulas l. sin2d+c,cis2d:1 sec2d l*cot20:<:sc:20 +.I sin(-d) : -sitt0 t,rs(-//) = t r1sl/ : -tallH 7. sin(A* B) :sitrAcosB*silBcosA 8. : siri A cos B - siu B <:os,;l 9. cos(A+ B) - cos,4cos B - siuA siriB 10. cos(A- B) : cosA cosB + silrA sirrB 11. 2 sirrd t:osd 12. <'os20- coS2(i - siu20 : 2<'os2o - I - 1 - 2sin20 I 13. tan d : <.rft0 (:ost/ I 14. <:ol0 : sirrd tattH 1 15. (:OS I/ 1 16. cscd - ri" 6i /F tl r(. cos[I ^ -el : sitt d \l 18. -01 : COSA 215 216 Formulas and Theorems II. Differentiation Formulas !(r") - trr:"-1 Q,:I' ]tra-fg'+gf' gJ'-,f g' - * (i) ,l' ,I - (tt(.r))9'(.,') ,i;.[tyt.rt) l'' d, \ (sttt rrJ .* ('oqI' .7, tJ, \ . ./ stll lr dr. l('os J { 1a,,,t,:r) - .,' o.t "11'2 1(<,ot.r') - (,.(,2.r' Q:T rl , (sc'c:.r'J: sPl'.r tall 11 ,7, d, - (<:s<t.r,; - (ls(].]'(rot;.r fr("'),t -.'' ,1 - fr(u") o,'ltrc ,l ,, 1 ' tlll ri - (l.t' .f d,^ --: I -iAl'CSllLl'l t!.r' J1 - rz 1(Arcsi' r) : oT Il12 Formulas and Theorems 2I7 III. Integration Formulas 1. ,f "or:artC 2. [\0,-trrlrl *(' .t "r 3. [,' ,t.,: r^x| (' ,I 4. In' a,,: lL , ,' .l 111Q 5. In., a.r: .rhr.r' .r r (' ,l f 6. sirr.r d.r' - ( os.r'-t C ./ 7. /.,,.r' dr : sitr.i'| (' .t 8. tl:r:hr sec,rl+ C or ln Jccrsrl+ C ,f'r^rr f 9. -
Chapter 4 Notes
4. Exponential and logarithmic functions 4.1 Exponential Functions A function of the form f(x) = ax, a > 0 , a 1 is called an exponential function. Its domain is the set of all real f (x 1) numbers. For an exponential function f we have a . The graph of an exponential function depends f (x) on the value of a. a> 1 0 < a< 1 y y 5 5 4 4 3 3 2 2 (1,a) (-1, 1/a) (-1, 1/a) 1 1 (1,a) x x -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 5 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 Points on the graph: (-1, 1/a), (0,1), (1, a) Properties of exponential functions 1. The domain is the set of all real numbers: Df = R 2. The range is the set of positive numbers: Rf = (0, +). (This means that ax is always positive, that is ax > 0 for all x. The equation ax = negative number has no solution) 3. There are no x-intercepts 4. The y-intercept is (0, 1) 5. The x-axis (line y = 0) is a horizontal asymptote 6. An exponential function is increasing when a > 1 and decreasing when 0 < a < 1 7. An exponential function is one to one, and therefore has the inverse. The inverse of the exponential x function f(x) = a is a logarithmic function g(x) = loga(x) 8. Since an exponential function is one to one we have the following property: If au = av , then u = v. -
Lecture 3 Vector Spaces of Functions
12 Lecture 3 Vector Spaces of Functions 3.1 Space of Continuous Functions C(I) Let I ⊆ R be an interval. Then I is of the form (for some a < b) 8 > fx 2 Rj a < x < bg; an open interval; <> fx 2 Rj a · x · bg; a closed interval; I = > fx 2 Rj a · x < bg :> fx 2 Rj a < x · bg: Recall that the space of all functions f : I ¡! R is a vector space. We will now list some important subspaces: Example (1). Let C(I) be the space of all continuous functions on I. If f and g are continuous, so are the functions f + g and rf (r 2 R). Hence C(I) is a vector space. Recall, that a function is continuous, if the graph has no gaps. This can be formulated in di®erent ways: a) Let x0 2 I and let ² > 0 . Then there exists a ± > 0 such that for all x 2 I \ (x0 ¡ ±; x0 + ±) we have jf(x) ¡ f(x0)j < ² This tells us that the value of f at nearby points is arbitrarily close to the value of f at x0. 13 14 LECTURE 3. VECTOR SPACES OF FUNCTIONS b) A reformulation of (a) is: lim f(x) = f(x0) x!x0 3.2 Space of continuously di®erentiable func- tions C1(I) Example (2). The space C1(I). Here we assume that I is open. Recall that f is di®erentiable at x0 if f(x) ¡ f(x0) f(x0 + h) ¡ f(x0) 0 lim = lim =: f (x0) x!x0 x ¡ x0 h!0 h 0 exists.