ON A FUNCTIONAL EQUATION BASED UPON A RESULT OF GASPARD MONGE
|
|
- Frank Palmer
- 6 years ago
- Views:
Transcription
1 ON A FUNCTIONAL EQUATION BASED UPON A RESULT OF GASPARD MONGE C. Alsina, M. Sablik, J. Sikorska Abstract. We present and solve completely a functional equation motivated by a classical result of Gaspard Monge. 1. INTRODUCTION Gaspard Monge ( ) was a great geometer. His ideas and results on Descriptive Geometry are, still today, a rigourous way to deal with graphical constructions. Our aim in this paper is to focus our attention on a property discovered by Monge concerning some triangles associated to any straight line of positive slope (in the rst quadrant). Precisely, let 0 denote the origin of a cartesian reference in the plane and let A = (A 1 ;A ) and B = (B 1 ;B ) with midpoint M (see gure 1). Then Monge noted that the areas S;S 1 and S of the triangles 0AB, A 1 MB 1 and A MB, respectively, satisfy the relationjs 1 S j = S. Figures 1 and 1
2 Based upon this equality we want to examine for which strictly increasing functions we can assure a similar result. Figure 1 suggests a possible generalization shown in Figure. Of course whenf is an a±ne function then gure reduces to gure 1, i.e., to Monge's case. Following the notations of gure, Monge's result may be stated as follows, for any x y, x +y 1 (y x)f 1 + y Z y (f(y) f(x))x = f(t)dt+ 1 x xf(x) 1 yf(y) (1) Our chief concern in this paper is to solve (1), showing that only a±ne functions may present this behaviour. Note that the formulation of (1) forces that the expression on the right hand side is non-negative, i.e., that we are in the situation where f(y) y x f(x), i.e., f(t)=t is non-increasing. But as we will see in our arguments this fact will be derived from (1).. MAIN RESULT We begin by considering a lemma which will be of interest in the next theorem. LEMMA 1. Let f be a function from R + into R + such that f(t)=t is nonincreasing on (0;1). Then, for all x y we have x + y x + y (f(y) f(x)) (y x)f : (3) Proof. Obviously (3) holds for x = y. So let us consider the case 0 < x < y which yields f(x) f(y), i.e., yf(x) xf(y). x y Therefore, since x < x+y < y, x +y x + y (f(y) f(x)) = 1 x + y 1 (yf(y) yf(x)) (yf(y) xf(y)) y y = x + y 1 x +y (y x)f(y) = (y x)f(y) y y x + y x+y x +y (y x)f = (y x)f : x+y
3 Now we can show the following result concerning (1): THEOREM 1. A strictly increasing function f from R + into R + satis es (1) if and only if f(x) = ax + b, for some arbitrary constants a and b such that a > 0 and b 0. Proof. Let us assume that f satis es (1), i.e., for x y x + y (y x)f x + y Z y (f(y) f(x)) = f(t)dt +xf(x) yf(y): x (1) Since f is strictly increasing when y tends to x from the right we have x + y lim f = lim f(y) = f(x+) y!x+ y!x+ so taking limits in (1) when y! x+ we obtain 0 xjf(x+) f(x)j = x(f(x) f(x+)); i.e., f(x+) f(x) and since f(x) f(x+) we deduce f(x) = f(x+). Analogously, taking limits when x!y from the left we deduce f(y) = f(y ). Therefore f is continuous, and being strictly increasingf(t) is di erentiable almost everywhere and the same applies to f(t) t. If F (t) denotes a primitive function of f then F is almost everywhere di erentiable (F 0 = f) non-decreasing and (absolutely) continuous on R + and we can write Z y Presenting (1) in the form: x +y f x + y f(y) f(x) y x x f(t)dt = F(y) F(x): when x tends to y from the left we deduce = F (y) F(x) y x yf(y) xf(x) ; y x 0 jf(y) yf 0 (y)j = f(y) (yf 0 (y) + f(y)) = f(y) yf 0 (y) 3
4 so d dy f(y) y = yf0 (y) f(y) y 0; and f(y)=y is non-increasing on (0;1). By virtue of Lemma 1, (3) holds, so we can omit the absolute value in (1) and present our equation in the form: x +y (y x)f x + y (f(y) f(x)) = (F(y) F(x)) +xf(x) yf(y); or equivalently F(y) F(x) = (y x) f µ x + y + f(x) + f(y) : (4) Since F andf are di erentiable, taking partial derivatives in (4) we obtain x +y f(x) +f(y) 1 x + y f(x) = f + +(y x) f0 + 1 f0 (x) and f(y) = f µ x + y + f(x) + f(y) 1 + (y x) f0 x + y + 1 f0 (y) so subtraction of the rst equality from the second one yields x +y f(y) + f(x) = f + f(x) + f(y) + (y x) 1 (f0 (y) f 0 (x)); i.e., x +y f = f(x) +f(y) + 1 (y x)(f0 (x) f 0 (y)): (5) Fixing x = x 0 > 0 into the above equation (5) we immediately see that for any y in [x 0 ;1) f 0 (y) = f(x 0) + f(y) + 1 (y x 0)f 0 (x 0 ) f((x 0 + y)=) ; (y x 0 )= whence f 0 is also di erentiable in [x 0 ;1). Since x 0 is arbitrary, f 00 exists in R + so we can take again partial derivatives in (5) to obtain: x + y f 0 = f 0 (x) 1 (f0 (x) f 0 (y)) + 1 (y x)f00 (x); 4
5 and x + y f 0 = f 0 (y) + 1 (f0 (x) f 0 (y)) 1 (y x)f00 (y); i.e., f 00 (x) = f 00 (y) so xingy = x 0 we get thatf 00 is constant, i.e., f 00 (x) = k and k = k yields k = 0. That f(x) = ax + b with a > 0;b 0. The theorem is proved. Remark 1. Let us note that one can obtain the same solutions of (4) with no regularity assumptions on f or F by means of results to be found in (Sablik, 000), (Sablik, 004) and (Pawlikowska, 00). However these general results are stated in a very general framework of group theory and therefore here we decided to give a short and direct proof. Similarly one can consider the functional equation x +y 1 (y x)f 1 + y (f(y) f(x))x = 1 Z y yf(y) 1 xf(x) f(t)dt; x (6) which is based on the case where the graph of f is below the line g(t) = (f(y)=y) t. Then one obtains the following THEOREM. Fixed [a;b] with a > 0, a strictly increasing function f from [a;b] into R + such that f(a) > 0, satis es (6) if and only if f(x) = mx + n for some arbitrary constants m;n in R, (m > 0). In amore general situation if f is a continuous strictly increasing function from [a;b] into R + with a > 0 and f(a) > 0 and f can oscillate above and below the line y(t) = (f(b)=b), then one can consider the sets ½ A(b) := x [a;b] : f(x) f(b) ¾ b x > 0 ; ½ B(b) := x [a;b] : f(x) f(b) ¾ b x < 0 ; ½ C(b) := x [a;b] : f(x) f(b) ¾ b x : A(b) and B(b) are open sets so they can be represented as union of open intervals. For each of such intervals easier results can be applied and for C(b) we have the result of Monge. So each consideration leads to a linear 5
6 function f(x) = mx + n on an interval which together with continuity of f gives f(x) = f(b) f(a) b a (x a) + f(a). ACKNOWLEDGMENT The author thanks Prof. J. Garcia-Roig his remarks and to Prof. A. Monreal for making the gures. REFERENCES [1] ACZ EL, J., 1966, Lectures on Functional Equations and Their Applications. Academic Press, New York. [] MONGE, G., 1897, Application de l'analyse a la G eom etrie. Imprimerie de H. Perronneau, Paris [3] MONGE, G., 1996, Geometr ³a descriptiva. Edici on facsimil. Colegio Ingenieros de Caminos, Canales y Puertos, Madrid. [4] I. Pawlikowska, 00, Charakteryzacja odwzorowa n poprzez twierdzenia o warto sci sredniej (in Polish), Doctoral thesis, Uniwersytet Sl»aski. [5] M. Sablik, 000, Taylor's theorem and functional equations. Aequationes Math. 60, no. 3, 58{67. [6] M. Sablik, 004, Remark to a result of C. Alsina. 4nd ISFE. Aequationes Math. C. Alsina M. Sablik and T. Sikorska Sec. Matemµatiques. ETSAB-UPC, mssablik@us.edu.pl Diagonal 649, 0808 Barcelona, Spain. claudio.alsina@upc.es 6
Engg. Math. I. Unit-I. Differential Calculus
Dr. Satish Shukla 1 of 50 Engg. Math. I Unit-I Differential Calculus Syllabus: Limits of functions, continuous functions, uniform continuity, monotone and inverse functions. Differentiable functions, Rolle
More informationLecture 6: Contraction mapping, inverse and implicit function theorems
Lecture 6: Contraction mapping, inverse and implicit function theorems 1 The contraction mapping theorem De nition 11 Let X be a metric space, with metric d If f : X! X and if there is a number 2 (0; 1)
More informationConstructing Approximations to Functions
Constructing Approximations to Functions Given a function, f, if is often useful to it is often useful to approximate it by nicer functions. For example give a continuous function, f, it can be useful
More informationFUNCTIONAL EQUATIONS. Ozgur Kircak
FUNCTIONAL EQUATIONS Ozgur Kircak April 8, 2011 2 Contents 1 FUNCTIONS BASICS 5 1.1 EXERCISES............................. 6 2 CAUCHY FUNCTIONAL EQUATION 9 2.1 EXERCISES............................. 12
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 information6x 2 8x + 5 ) = 12x 8
Example. If f(x) = x 3 4x + 5x + 1, then f (x) = 6x 8x + 5 Observation: f (x) is also a differentiable function... d dx ( f (x) ) = d dx ( 6x 8x + 5 ) = 1x 8 The derivative of f (x) is called the second
More information6x 2 8x + 5 ) = 12x 8. f (x) ) = d (12x 8) = 12
AMS/ECON 11A Class Notes 11/6/17 UCSC *) Higher order derivatives Example. If f = x 3 x + 5x + 1, then f = 6x 8x + 5 Observation: f is also a differentiable function... d f ) = d 6x 8x + 5 ) = 1x 8 dx
More informationFunctional Equations
Functional Equations Henry Liu, 22 December 2004 henryliu@memphis.edu Introduction This is a brief set of notes on functional equations. It is one of the harder and less popular areas among Olympiad problems,
More information2019 Spring MATH2060A Mathematical Analysis II 1
2019 Spring MATH2060A Mathematical Analysis II 1 Notes 1. CONVEX FUNCTIONS First we define what a convex function is. Let f be a function on an interval I. For x < y in I, the straight line connecting
More informationSolutions of Math 53 Midterm Exam I
Solutions of Math 53 Midterm Exam I Problem 1: (1) [8 points] Draw a direction field for the given differential equation y 0 = t + y. (2) [8 points] Based on the direction field, determine the behavior
More informationMean Value Theorem. Increasing Functions Extreme Values of Functions Rolle s Theorem Mean Value Theorem FAQ. Index
Mean Value Increasing Functions Extreme Values of Functions Rolle s Mean Value Increasing Functions (1) Assume that the function f is everywhere increasing and differentiable. ( x + h) f( x) f Then h 0
More informationMath M111: Lecture Notes For Chapter 3
Section 3.1: Math M111: Lecture Notes For Chapter 3 Note: Make sure you already printed the graphing papers Plotting Points, Quadrant s signs, x-intercepts and y-intercepts Example 1: Plot the following
More informationWHEN LAGRANGEAN AND QUASI-ARITHMETIC MEANS COINCIDE
Volume 8 (007, Issue 3, Article 71, 5 pp. WHEN LAGRANGEAN AND QUASI-ARITHMETIC MEANS COINCIDE JUSTYNA JARCZYK FACULTY OF MATHEMATICS, COMPUTER SCIENCE AND ECONOMETRICS, UNIVERSITY OF ZIELONA GÓRA SZAFRANA
More informationMathematics 426 Robert Gross Homework 9 Answers
Mathematics 4 Robert Gross Homework 9 Answers. Suppose that X is a normal random variable with mean µ and standard deviation σ. Suppose that PX > 9 PX
More informationTangent Lines and Derivatives
The Derivative and the Slope of a Graph Tangent Lines and Derivatives Recall that the slope of a line is sometimes referred to as a rate of change. In particular, we are referencing the rate at which the
More informationIowa State University. Instructor: Alex Roitershtein Summer Homework #5. Solutions
Math 50 Iowa State University Introduction to Real Analysis Department of Mathematics Instructor: Alex Roitershtein Summer 205 Homework #5 Solutions. Let α and c be real numbers, c > 0, and f is defined
More informationMATH 202B - Problem Set 5
MATH 202B - Problem Set 5 Walid Krichene (23265217) March 6, 2013 (5.1) Show that there exists a continuous function F : [0, 1] R which is monotonic on no interval of positive length. proof We know there
More information1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989),
Real Analysis 2, Math 651, Spring 2005 April 26, 2005 1 Real Analysis 2, Math 651, Spring 2005 Krzysztof Chris Ciesielski 1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer
More informationMATH 409 Advanced Calculus I Lecture 16: Mean value theorem. Taylor s formula.
MATH 409 Advanced Calculus I Lecture 16: Mean value theorem. Taylor s formula. Points of local extremum Let f : E R be a function defined on a set E R. Definition. We say that f attains a local maximum
More informationSOME FUNCTION CLASSES RELATED TO THE CLASS OF CONVEX FUNCTIONS
SOME FUNCTION CLASSES RELATED TO THE CLASS OF CONVEX FUNCTIONS A. M. BRUCKNER AND E. OSTROW l Introduction* A real-valued function / defined on the positive real line [0, oo) is said to be convex if for
More information2tdt 1 y = t2 + C y = which implies C = 1 and the solution is y = 1
Lectures - Week 11 General First Order ODEs & Numerical Methods for IVPs In general, nonlinear problems are much more difficult to solve than linear ones. Unfortunately many phenomena exhibit nonlinear
More informationChapter 1: Precalculus Review
: Precalculus Review Math 115 17 January 2018 Overview 1 Important Notation 2 Exponents 3 Polynomials 4 Rational Functions 5 Cartesian Coordinates 6 Lines Notation Intervals: Interval Notation (a, b) (a,
More informationSCORE BOOSTER JAMB PREPARATION SERIES II
BOOST YOUR JAMB SCORE WITH PAST Polynomials QUESTIONS Part II ALGEBRA by H. O. Aliu J. K. Adewole, PhD (Editor) 1) If 9x 2 + 6xy + 4y 2 is a factor of 27x 3 8y 3, find the other factor. (UTME 2014) 3x
More informationMetric Space Topology (Spring 2016) Selected Homework Solutions. HW1 Q1.2. Suppose that d is a metric on a set X. Prove that the inequality d(x, y)
Metric Space Topology (Spring 2016) Selected Homework Solutions HW1 Q1.2. Suppose that d is a metric on a set X. Prove that the inequality d(x, y) d(z, w) d(x, z) + d(y, w) holds for all w, x, y, z X.
More informationMATH 116, LECTURE 13, 14 & 15: Derivatives
MATH 116, LECTURE 13, 14 & 15: Derivatives 1 Formal Definition of the Derivative We have seen plenty of limits so far, but very few applications. In particular, we have seen very few functions for which
More informationAverage rates of change to instantaneous rates of change Math 102 Section 106
Average rates of change to instantaneous rates of change Math 102 Section 106 Cole Zmurchok September 14, 2016 Math 102: Announcements Office Hours today: 3-4 pm Math Annex 1118 and Thursday: 3-4 pm in
More informationx x 1 x 2 + x 2 1 > 0. HW5. Text defines:
Lecture 15: Last time: MVT. Special case: Rolle s Theorem (when f(a) = f(b)). Recall: Defn: Let f be defined on an interval I. f is increasing (or strictly increasing) if whenever x 1, x 2 I and x 2 >
More information(x + 3)(x 1) lim(x + 3) = 4. lim. (x 2)( x ) = (x 2)(x + 2) x + 2 x = 4. dt (t2 + 1) = 1 2 (t2 + 1) 1 t. f(x) = lim 3x = 6,
Math 140 MT1 Sample C Solutions Tyrone Crisp 1 (B): First try direct substitution: you get 0. So try to cancel common factors. We have 0 x 2 + 2x 3 = x 1 and so the it as x 1 is equal to (x + 3)(x 1),
More informationMATHEMATICS XII. Topic. Revision of Derivatives Presented By. Avtar Singh Lecturer Paramjit Singh Sidhu June 19,2009
MATHEMATICS XII 1 Topic Revision of Derivatives Presented By Avtar Singh Lecturer Paramjit Singh Sidhu June 19,2009 19 June 2009 Punjab EDUSAT Society (PES) 1 Continuity 2 Def. In simple words, a function
More information1 Functions of Several Variables 2019 v2
1 Functions of Several Variables 2019 v2 11 Notation The subject of this course is the study of functions f : R n R m The elements of R n, for n 2, will be called vectors so, if m > 1, f will be said to
More informationReal Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi
Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.
More informationMath 163: Lecture notes
Math 63: Lecture notes Professor Monika Nitsche March 2, 2 Special functions that are inverses of known functions. Inverse functions (Day ) Go over: early exam, hw, quizzes, grading scheme, attendance
More informationMATH 215/255 Solutions to Additional Practice Problems April dy dt
. For the nonlinear system MATH 5/55 Solutions to Additional Practice Problems April 08 dx dt = x( x y, dy dt = y(.5 y x, x 0, y 0, (a Show that if x(0 > 0 and y(0 = 0, then the solution (x(t, y(t of the
More informationThe stability of functional equation min{f(x + y), f (x - y)} = f(x) -f(y)
RESEARCH Open Access The stability of functional equation min{f(x + y), f (x - y)} = f(x) -f(y) Barbara Przebieracz Correspondence: barbara. przebieracz@us.edu.pl Instytut Matematyki, Uniwersytet Śląski
More informationh(x) lim H(x) = lim Since h is nondecreasing then h(x) 0 for all x, and if h is discontinuous at a point x then H(x) > 0. Denote
Real Variables, Fall 4 Problem set 4 Solution suggestions Exercise. Let f be of bounded variation on [a, b]. Show that for each c (a, b), lim x c f(x) and lim x c f(x) exist. Prove that a monotone function
More informationMATH 2053 Calculus I Review for the Final Exam
MATH 05 Calculus I Review for the Final Exam (x+ x) 9 x 9 1. Find the limit: lim x 0. x. Find the limit: lim x + x x (x ).. Find lim x (x 5) = L, find such that f(x) L < 0.01 whenever 0 < x
More informationHW3 - Due 02/06. Each answer must be mathematically justified. Don t forget your name. 1 2, A = 2 2
HW3 - Due 02/06 Each answer must be mathematically justified Don t forget your name Problem 1 Find a 2 2 matrix B such that B 3 = A, where A = 2 2 If A was diagonal, it would be easy: we would just take
More informationSection 3.7. Rolle s Theorem and the Mean Value Theorem
Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate of change and the average rate of change of
More informationLimits, Continuity, and the Derivative
Unit #2 : Limits, Continuity, and the Derivative Goals: Study and define continuity Review limits Introduce the derivative as the limit of a difference quotient Discuss the derivative as a rate of change
More informationSOLUTIONS FOR 2011 APMO PROBLEMS
SOLUTIONS FOR 2011 APMO PROBLEMS Problem 1. Solution: Suppose all of the 3 numbers a 2 + b + c, b 2 + c + a and c 2 + a + b are perfect squares. Then from the fact that a 2 + b + c is a perfect square
More informationMATH Max-min Theory Fall 2016
MATH 20550 Max-min Theory Fall 2016 1. Definitions and main theorems Max-min theory starts with a function f of a vector variable x and a subset D of the domain of f. So far when we have worked with functions
More informationSection 4.2 The Mean Value Theorem
Section 4.2 The Mean Value Theorem Ruipeng Shen October 2nd Ruipeng Shen MATH 1ZA3 October 2nd 1 / 11 Rolle s Theorem Theorem (Rolle s Theorem) Let f (x) be a function that satisfies: 1. f is continuous
More informationMath Review ECON 300: Spring 2014 Benjamin A. Jones MATH/CALCULUS REVIEW
MATH/CALCULUS REVIEW SLOPE, INTERCEPT, and GRAPHS REVIEW (adapted from Paul s Online Math Notes) Let s start with some basic review material to make sure everybody is on the same page. The slope of a line
More informationOn a Problem of Alsina Again
Journal of Mathematical Analysis and Applications 54, 67 635 00 doi:0.006 jmaa.000.768, available online at http: www.idealibrary.com on On a Problem of Alsina Again Janusz Matkowski Institute of Mathematics,
More informationMath 141: Section 4.1 Extreme Values of Functions - Notes
Math 141: Section 4.1 Extreme Values of Functions - Notes Definition: Let f be a function with domain D. Thenf has an absolute (global) maximum value on D at a point c if f(x) apple f(c) for all x in D
More informationSlide 1. Slide 2. Slide 3 Remark is a new function derived from called derivative. 2.2 The derivative as a Function
Slide 1 2.2 The derivative as a Function Slide 2 Recall: The derivative of a function number : at a fixed Definition (Derivative of ) For any number, the derivative of is Slide 3 Remark is a new function
More informationMATH The Derivative as a Function - Section 3.2. The derivative of f is the function. f x h f x. f x lim
MATH 90 - The Derivative as a Function - Section 3.2 The derivative of f is the function f x lim h 0 f x h f x h for all x for which the limit exists. The notation f x is read "f prime of x". Note that
More informationf( X 3-3x )dx = x 2 --3x +C (A) 3x 2-3+C (B) 4x 4-6x 2 +C (C) x 4 3x 2 x 4 5x 3 +15x 2 +20x+25 5x 2 +15x+25 (E) 5 (D) 225 (B) (A)
. f( X -x )dx = x -+C x -6x +C x x x --x+c ---+C x --x +C. If f(x)=x +x +x+5 and g(x)=5, then g(f(x))= 5 5x +5x+5 5 5x +5x +0x+5. The slope of the line tangent to the graph of y = In (x ) at x = e is e
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Spring 2018, WEEK 3 JoungDong Kim Week 3 Section 2.5, 2.6, 2.7, Continuity, Limits at Infinity; Horizontal Asymptotes, Derivatives and Rates of Change. Section 2.5 Continuity
More informationcos t 2 sin 2t (vi) y = cosh t sinh t (vii) y sin x 2 = x sin y 2 (viii) xy = cot(xy) (ix) 1 + x = sin(xy 2 ) (v) g(t) =
MATH1003 REVISION 1. Differentiate the following functions, simplifying your answers when appropriate: (i) f(x) = (x 3 2) tan x (ii) y = (3x 5 1) 6 (iii) y 2 = x 2 3 (iv) y = ln(ln(7 + x)) e 5x3 (v) g(t)
More informationLecture two. January 17, 2019
Lecture two January 17, 2019 We will learn how to solve rst-order linear equations in this lecture. Example 1. 1) Find all solutions satisfy the equation u x (x, y) = 0. 2) Find the solution if we know
More informationMA 137: Calculus I for the Life Sciences
MA 137: Calculus I for the Life Sciences David Murrugarra Department of Mathematics, University of Kentucky http://www.ms.uky.edu/~ma137/ Spring 2018 David Murrugarra (University of Kentucky) MA 137: Lecture
More informationMaths Higher Prelim Content
Maths Higher Prelim Content Straight Line Gradient of a line A(x 1, y 1 ), B(x 2, y 2 ), Gradient of AB m AB = y 2 y1 x 2 x 1 m = tanθ where θ is the angle the line makes with the positive direction of
More informationSolutions to Problem Sheet for Week 11
THE UNIVERSITY OF SYDNEY SCHOOL OF MATHEMATICS AND STATISTICS Solutions to Problem Sheet for Week MATH9: Differential Calculus (Advanced) Semester, 7 Web Page: sydney.edu.au/science/maths/u/ug/jm/math9/
More information1 The Existence and Uniqueness Theorem for First-Order Differential Equations
1 The Existence and Uniqueness Theorem for First-Order Differential Equations Let I R be an open interval and G R n, n 1, be a domain. Definition 1.1 Let us consider a function f : I G R n. The general
More information116 Problems in Algebra
116 Problems in Algebra Problems Proposer: Mohammad Jafari November 5, 011 116 Problems in Algebra is a nice work of Mohammad Jafari. Tese problems have been published in a book, but it is in Persian (Farsi).
More informationSolutions to Homework # 1 Math 381, Rice University, Fall (x y) y 2 = 0. Part (b). We make a convenient change of variables:
Hildebrand, Ch. 8, # : Part (a). We compute Subtracting, we eliminate f... Solutions to Homework # Math 38, Rice University, Fall 2003 x = f(x + y) + (x y)f (x + y) y = f(x + y) + (x y)f (x + y). x = 2f(x
More informationLecture 13 - Wednesday April 29th
Lecture 13 - Wednesday April 29th jacques@ucsdedu Key words: Systems of equations, Implicit differentiation Know how to do implicit differentiation, how to use implicit and inverse function theorems 131
More information3.Applications of Differentiation
3.Applications of Differentiation 3.1. Maximum and Minimum values Absolute Maximum and Absolute Minimum Values Absolute Maximum Values( Global maximum values ): Largest y-value for the given function Absolute
More informationSlope Fields: Graphing Solutions Without the Solutions
8 Slope Fields: Graphing Solutions Without the Solutions Up to now, our efforts have been directed mainly towards finding formulas or equations describing solutions to given differential equations. Then,
More informationVCE. VCE Maths Methods 1 and 2 Pocket Study Guide
VCE VCE Maths Methods 1 and 2 Pocket Study Guide Contents Introduction iv 1 Linear functions 1 2 Quadratic functions 10 3 Cubic functions 16 4 Advanced functions and relations 24 5 Probability and simulation
More informationFind the indicated derivative. 1) Find y(4) if y = 3 sin x. A) y(4) = 3 cos x B) y(4) = 3 sin x C) y(4) = - 3 cos x D) y(4) = - 3 sin x
Assignment 5 Name Find the indicated derivative. ) Find y(4) if y = sin x. ) A) y(4) = cos x B) y(4) = sin x y(4) = - cos x y(4) = - sin x ) y = (csc x + cot x)(csc x - cot x) ) A) y = 0 B) y = y = - csc
More information1.5 The Derivative (2.7, 2.8 & 2.9)
1.5. THE DERIVATIVE (2.7, 2.8 & 2.9) 47 1.5 The Derivative (2.7, 2.8 & 2.9) The concept we are about to de ne is not new. We simply give it a new name. Often in mathematics, when the same idea seems to
More informationMAC 1105-College Algebra LSCC, S. Nunamaker
MAC 1105-College Algebra LSCC, S. Nunamaker Chapter 1-Graphs, Functions, and Models 1.1 Introduction to Graphing I. Reasons for using graphs A. Visual presentations enhance understanding. B. Visual presentations
More informationClass Meeting # 1: Introduction to PDEs
MATH 18.152 COURSE NOTES - CLASS MEETING # 1 18.152 Introduction to PDEs, Spring 2017 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u =
More informationMS 3011 Exercises. December 11, 2013
MS 3011 Exercises December 11, 2013 The exercises are divided into (A) easy (B) medium and (C) hard. If you are particularly interested I also have some projects at the end which will deepen your understanding
More informationCourse 212: Academic Year Section 1: Metric Spaces
Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........
More informationSOLUTIONS FOR PROBLEMS 1-30
. Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).
More informationDefinitions & Theorems
Definitions & Theorems Math 147, Fall 2009 December 19, 2010 Contents 1 Logic 2 1.1 Sets.................................................. 2 1.2 The Peano axioms..........................................
More informationMath 111: Calculus. David Perkinson
Math : Calculus David Perkinson Fall 207 Contents Week, Monday: Introduction: derivatives, integrals, and the fundamental theorem. 5 Week, Wednesday: Average speed, instantaneous speed. Definition of the
More informationCHAPTER ONE FUNCTIONS AND GRAPHS. In everyday life, many quantities depend on one or more changing variables eg:
CHAPTER ONE FUNCTIONS AND GRAPHS 1.0 Introduction to Functions In everyday life, many quantities depend on one or more changing variables eg: (a) plant growth depends on sunlight and rainfall (b) speed
More informationMA 114 Worksheet # 17: Integration by trig substitution
MA Worksheet # 7: Integration by trig substitution. Conceptual Understanding: Given identity sin θ + cos θ =, prove that: sec θ = tan θ +. Given x = a sin(θ) with a > and π θ π, show that a x = a cos θ.
More informationTransition Density Function and Partial Di erential Equations
Transition Density Function and Partial Di erential Equations In this lecture Generalised Functions - Dirac delta and heaviside Transition Density Function - Forward and Backward Kolmogorov Equation Similarity
More information1 Overview. 2 A Characterization of Convex Functions. 2.1 First-order Taylor approximation. AM 221: Advanced Optimization Spring 2016
AM 221: Advanced Optimization Spring 2016 Prof. Yaron Singer Lecture 8 February 22nd 1 Overview In the previous lecture we saw characterizations of optimality in linear optimization, and we reviewed the
More informationMA 460 Supplement: Analytic geometry
M 460 Supplement: nalytic geometry Donu rapura In the 1600 s Descartes introduced cartesian coordinates which changed the way we now do geometry. This also paved for subsequent developments such as calculus.
More informationAnnouncements. Topics: Homework:
Topics: Announcements - section 2.6 (limits at infinity [skip Precise Definitions (middle of pg. 134 end of section)]) - sections 2.1 and 2.7 (rates of change, the derivative) - section 2.8 (the derivative
More informationf ( x) = L ( the limit of f(x), as x approaches a,
Math 1205 Calculus Sec. 2.4: The Definition of imit I. Review A. Informal Definition of imit 1. Def n : et f(x) be defined on an open interval about a except possibly at a itself. If f(x) gets arbitrarily
More information(x 3)(x + 5) = (x 3)(x 1) = x + 5. sin 2 x e ax bx 1 = 1 2. lim
SMT Calculus Test Solutions February, x + x 5 Compute x x x + Answer: Solution: Note that x + x 5 x x + x )x + 5) = x )x ) = x + 5 x x + 5 Then x x = + 5 = Compute all real values of b such that, for fx)
More informationMATH 409 Advanced Calculus I Lecture 12: Uniform continuity. Exponential functions.
MATH 409 Advanced Calculus I Lecture 12: Uniform continuity. Exponential functions. Uniform continuity Definition. A function f : E R defined on a set E R is called uniformly continuous on E if for every
More informationMath Precalculus I University of Hawai i at Mānoa Spring
Math 135 - Precalculus I University of Hawai i at Mānoa Spring - 2013 Created for Math 135, Spring 2008 by Lukasz Grabarek and Michael Joyce Send comments and corrections to lukasz@math.hawaii.edu Contents
More informationPreliminary Exam 2018 Solutions to Morning Exam
Preliminary Exam 28 Solutions to Morning Exam Part I. Solve four of the following five problems. Problem. Consider the series n 2 (n log n) and n 2 (n(log n)2 ). Show that one converges and one diverges
More informationMath 308 Exam I Practice Problems
Math 308 Exam I Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture and all suggested homework problems..
More informationChapter 8: Taylor s theorem and L Hospital s rule
Chapter 8: Taylor s theorem and L Hospital s rule Theorem: [Inverse Mapping Theorem] Suppose that a < b and f : [a, b] R. Given that f (x) > 0 for all x (a, b) then f 1 is differentiable on (f(a), f(b))
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 information1.1 GRAPHS AND LINEAR FUNCTIONS
MATHEMATICS EXTENSION 4 UNIT MATHEMATICS TOPIC 1: GRAPHS 1.1 GRAPHS AND LINEAR FUNCTIONS FUNCTIONS The concept of a function is already familiar to you. Since this concept is fundamental to mathematics,
More informationExample: Plot the points (0, 0), (2, 1), ( 1, 3), , ( 1, 0), (3, 0) on the Cartesian plane: 5
Graphing Equations: An Ordered Pair of numbers is two numbers (x, y) that is used to represent coordinate of points in the Cartesian plane. The first number is the x coordinate and the second number is
More informationDynamical Systems. August 13, 2013
Dynamical Systems Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 13, 2013 Dynamical Systems are systems, described by one or more equations, that evolve over time.
More informationTopology, Math 581, Fall 2017 last updated: November 24, Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski
Topology, Math 581, Fall 2017 last updated: November 24, 2017 1 Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski Class of August 17: Course and syllabus overview. Topology
More informationSPACES ENDOWED WITH A GRAPH AND APPLICATIONS. Mina Dinarvand. 1. Introduction
MATEMATIČKI VESNIK MATEMATIQKI VESNIK 69, 1 (2017), 23 38 March 2017 research paper originalni nauqni rad FIXED POINT RESULTS FOR (ϕ, ψ)-contractions IN METRIC SPACES ENDOWED WITH A GRAPH AND APPLICATIONS
More informationMath 1270 Honors ODE I Fall, 2008 Class notes # 14. x 0 = F (x; y) y 0 = G (x; y) u 0 = au + bv = cu + dv
Math 1270 Honors ODE I Fall, 2008 Class notes # 1 We have learned how to study nonlinear systems x 0 = F (x; y) y 0 = G (x; y) (1) by linearizing around equilibrium points. If (x 0 ; y 0 ) is an equilibrium
More informationVolumes of Solids of Revolution Lecture #6 a
Volumes of Solids of Revolution Lecture #6 a Sphereoid Parabaloid Hyperboloid Whateveroid Volumes Calculating 3-D Space an Object Occupies Take a cross-sectional slice. Compute the area of the slice. Multiply
More informationCHRISTENSEN ZERO SETS AND MEASURABLE CONVEX FUNCTIONS1 PAL FISCHER AND ZBIGNIEW SLODKOWSKI
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 79, Number 3, July 1980 CHRISTENSEN ZERO SETS AND MEASURABLE CONVEX FUNCTIONS1 PAL FISCHER AND ZBIGNIEW SLODKOWSKI Abstract. A notion of measurability
More information. For each initial condition y(0) = y 0, there exists a. unique solution. In fact, given any point (x, y), there is a unique curve through this point,
1.2. Direction Fields: Graphical Representation of the ODE and its Solution Section Objective(s): Constructing Direction Fields. Interpreting Direction Fields. Definition 1.2.1. A first order ODE of the
More informationAnalysis II - few selective results
Analysis II - few selective results Michael Ruzhansky December 15, 2008 1 Analysis on the real line 1.1 Chapter: Functions continuous on a closed interval 1.1.1 Intermediate Value Theorem (IVT) Theorem
More informationM311 Functions of Several Variables. CHAPTER 1. Continuity CHAPTER 2. The Bolzano Weierstrass Theorem and Compact Sets CHAPTER 3.
M311 Functions of Several Variables 2006 CHAPTER 1. Continuity CHAPTER 2. The Bolzano Weierstrass Theorem and Compact Sets CHAPTER 3. Differentiability 1 2 CHAPTER 1. Continuity If (a, b) R 2 then we write
More informationSet, functions and Euclidean space. Seungjin Han
Set, functions and Euclidean space Seungjin Han September, 2018 1 Some Basics LOGIC A is necessary for B : If B holds, then A holds. B A A B is the contraposition of B A. A is sufficient for B: If A holds,
More information. CALCULUS AB. Name: Class: Date:
Class: _ Date: _. CALCULUS AB SECTION I, Part A Time- 55 Minutes Number of questions -8 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION Directions: Solve each of the following problems, using
More information5. Polynomial Functions and Equations
5. Polynomial Functions and Equations 1. Polynomial equations and roots. Solving polynomial equations in the chemical context 3. Solving equations of multiple unknowns 5.1. Polynomial equations and roots
More informationModule 2: Reflecting on One s Problems
MATH55 Module : Reflecting on One s Problems Main Math concepts: Translations, Reflections, Graphs of Equations, Symmetry Auxiliary ideas: Working with quadratics, Mobius maps, Calculus, Inverses I. Transformations
More informationLecture 7 3.5: Derivatives - Graphically and Numerically MTH 124
Procedural Skills Learning Objectives 1. Given a function and a point, sketch the corresponding tangent line. 2. Use the tangent line to estimate the value of the derivative at a point. 3. Recognize keywords
More information