Infinitesimal Calculus H
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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 -
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. -
4.1 AP Calc Antiderivatives and Indefinite Integration.Notebook November 28, 2016
4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 4.1 Antiderivatives and Indefinite Integration Learning Targets 1. Write the general solution of a differential equation 2. Use indefinite integral notation for antiderivatives 3. Use basic integration rules to find antiderivatives 4. Find a particular solution of a differential equation 5. Plot a slope field 6. Working backwards in physics Intro/Warmup 1 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 Note: I am changing up the procedure of the class! When you walk in the class, you will put a tally mark on those problems that you would like me to go over, and you will turn in your homework with your name, the section and the page of the assignment. Order of the day 2 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 Vocabulary First, find a function F whose derivative is . because any other function F work? So represents the family of all antiderivatives of . The constant C is called the constant of integration. The family of functions represented by F is the general antiderivative of f, and is the general solution to the differential equation The operation of finding all solutions to the differential equation is called antidifferentiation (or indefinite integration) and is denoted by an integral sign: is read as the antiderivative of f with respect to x. Vocabulary 3 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 Nov 289:38 AM 4 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 ∫ Use basic integration rules to find antiderivatives The Power Rule: where 1. -
Riemann-Liouville Fractional Calculus of Blancmange Curve and Cantor Functions
Journal of Applied Mathematics and Computation, 2020, 4(4), 123-129 https://www.hillpublisher.com/journals/JAMC/ ISSN Online: 2576-0653 ISSN Print: 2576-0645 Riemann-Liouville Fractional Calculus of Blancmange Curve and Cantor Functions Srijanani Anurag Prasad Department of Mathematics and Statistics, Indian Institute of Technology Tirupati, India. How to cite this paper: Srijanani Anurag Prasad. (2020) Riemann-Liouville Frac- Abstract tional Calculus of Blancmange Curve and Riemann-Liouville fractional calculus of Blancmange Curve and Cantor Func- Cantor Functions. Journal of Applied Ma- thematics and Computation, 4(4), 123-129. tions are studied in this paper. In this paper, Blancmange Curve and Cantor func- DOI: 10.26855/jamc.2020.12.003 tion defined on the interval is shown to be Fractal Interpolation Functions with appropriate interpolation points and parameters. Then, using the properties of Received: September 15, 2020 Fractal Interpolation Function, the Riemann-Liouville fractional integral of Accepted: October 10, 2020 Published: October 22, 2020 Blancmange Curve and Cantor function are described to be Fractal Interpolation Function passing through a different set of points. Finally, using the conditions for *Corresponding author: Srijanani the fractional derivative of order ν of a FIF, it is shown that the fractional deriva- Anurag Prasad, Department of Mathe- tive of Blancmange Curve and Cantor function is not a FIF for any value of ν. matics, Indian Institute of Technology Tirupati, India. Email: [email protected] Keywords Fractal, Interpolation, Iterated Function System, fractional integral, fractional de- rivative, Blancmange Curve, Cantor function 1. Introduction Fractal geometry is a subject in which irregular and complex functions and structures are researched. -
The Legacy of Leonhard Euler: a Tricentennial Tribute (419 Pages)
P698.TP.indd 1 9/8/09 5:23:37 PM This page intentionally left blank Lokenath Debnath The University of Texas-Pan American, USA Imperial College Press ICP P698.TP.indd 2 9/8/09 5:23:39 PM Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. THE LEGACY OF LEONHARD EULER A Tricentennial Tribute Copyright © 2010 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-1-84816-525-0 ISBN-10 1-84816-525-0 Printed in Singapore. LaiFun - The Legacy of Leonhard.pmd 1 9/4/2009, 3:04 PM September 4, 2009 14:33 World Scientific Book - 9in x 6in LegacyLeonhard Leonhard Euler (1707–1783) ii September 4, 2009 14:33 World Scientific Book - 9in x 6in LegacyLeonhard To my wife Sadhana, grandson Kirin,and granddaughter Princess Maya, with love and affection. -
Gottfried Wilhelm Leibnitz (Or Leibniz) Was Born at Leipzig on June 21 (O.S.), 1646, and Died in Hanover on November 14, 1716. H
Gottfried Wilhelm Leibnitz (or Leibniz) was born at Leipzig on June 21 (O.S.), 1646, and died in Hanover on November 14, 1716. His father died before he was six, and the teaching at the school to which he was then sent was inefficient, but his industry triumphed over all difficulties; by the time he was twelve he had taught himself to read Latin easily, and had begun Greek; and before he was twenty he had mastered the ordinary text-books on mathematics, philosophy, theology and law. Refused the degree of doctor of laws at Leipzig by those who were jealous of his youth and learning, he moved to Nuremberg. An essay which there wrote on the study of law was dedicated to the Elector of Mainz, and led to his appointment by the elector on a commission for the revision of some statutes, from which he was subsequently promoted to the diplomatic service. In the latter capacity he supported (unsuccessfully) the claims of the German candidate for the crown of Poland. The violent seizure of various small places in Alsace in 1670 excited universal alarm in Germany as to the designs of Louis XIV.; and Leibnitz drew up a scheme by which it was proposed to offer German co-operation, if France liked to take Egypt, and use the possessions of that country as a basis for attack against Holland in Asia, provided France would agree to leave Germany undisturbed. This bears a curious resemblance to the similar plan by which Napoleon I. proposed to attack England. In 1672 Leibnitz went to Paris on the invitation of the French government to explain the details of the scheme, but nothing came of it. -
Chapter 6 Introduction to Calculus
RS - Ch 6 - Intro to Calculus Chapter 6 Introduction to Calculus 1 Archimedes of Syracuse (c. 287 BC – c. 212 BC ) Bhaskara II (1114 – 1185) 6.0 Calculus • Calculus is the mathematics of change. •Two major branches: Differential calculus and Integral calculus, which are related by the Fundamental Theorem of Calculus. • Differential calculus determines varying rates of change. It is applied to problems involving acceleration of moving objects (from a flywheel to the space shuttle), rates of growth and decay, optimal values, etc. • Integration is the "inverse" (or opposite) of differentiation. It measures accumulations over periods of change. Integration can find volumes and lengths of curves, measure forces and work, etc. Older branch: Archimedes (c. 287−212 BC) worked on it. • Applications in science, economics, finance, engineering, etc. 2 1 RS - Ch 6 - Intro to Calculus 6.0 Calculus: Early History • The foundations of calculus are generally attributed to Newton and Leibniz, though Bhaskara II is believed to have also laid the basis of it. The Western roots go back to Wallis, Fermat, Descartes and Barrow. • Q: How close can two numbers be without being the same number? Or, equivalent question, by considering the difference of two numbers: How small can a number be without being zero? • Fermat’s and Newton’s answer: The infinitessimal, a positive quantity, smaller than any non-zero real number. • With this concept differential calculus developed, by studying ratios in which both numerator and denominator go to zero simultaneously. 3 6.1 Comparative Statics Comparative statics: It is the study of different equilibrium states associated with different sets of values of parameters and exogenous variables. -
History and Mythology of Set Theory
Chapter udf History and Mythology of Set Theory This chapter includes the historical prelude from Tim Button's Open Set Theory text. set.1 Infinitesimals and Differentiation his:set:infinitesimals: Newton and Leibniz discovered the calculus (independently) at the end of the sec 17th century. A particularly important application of the calculus was differ- entiation. Roughly speaking, differentiation aims to give a notion of the \rate of change", or gradient, of a function at a point. Here is a vivid way to illustrate the idea. Consider the function f(x) = 2 x =4 + 1=2, depicted in black below: f(x) 5 4 3 2 1 x 1 2 3 4 1 Suppose we want to find the gradient of the function at c = 1=2. We start by drawing a triangle whose hypotenuse approximates the gradient at that point, perhaps the red triangle above. When β is the base length of our triangle, its height is f(1=2 + β) − f(1=2), so that the gradient of the hypotenuse is: f(1=2 + β) − f(1=2) : β So the gradient of our red triangle, with base length 3, is exactly 1. The hypotenuse of a smaller triangle, the blue triangle with base length 2, gives a better approximation; its gradient is 3=4. A yet smaller triangle, the green triangle with base length 1, gives a yet better approximation; with gradient 1=2. Ever-smaller triangles give us ever-better approximations. So we might say something like this: the hypotenuse of a triangle with an infinitesimal base length gives us the gradient at c = 1=2 itself. -
Generalizations and Properties of the Ternary Cantor Set and Explorations in Similar Sets
Generalizations and Properties of the Ternary Cantor Set and Explorations in Similar Sets by Rebecca Stettin A capstone project submitted in partial fulfillment of graduating from the Academic Honors Program at Ashland University May 2017 Faculty Mentor: Dr. Darren D. Wick, Professor of Mathematics Additional Reader: Dr. Gordon Swain, Professor of Mathematics Abstract Georg Cantor was made famous by introducing the Cantor set in his works of mathemat- ics. This project focuses on different Cantor sets and their properties. The ternary Cantor set is the most well known of the Cantor sets, and can be best described by its construction. This set starts with the closed interval zero to one, and is constructed in iterations. The first iteration requires removing the middle third of this interval. The second iteration will remove the middle third of each of these two remaining intervals. These iterations continue in this fashion infinitely. Finally, the ternary Cantor set is described as the intersection of all of these intervals. This set is particularly interesting due to its unique properties being uncountable, closed, length of zero, and more. A more general Cantor set is created by tak- ing the intersection of iterations that remove any middle portion during each iteration. This project explores the ternary Cantor set, as well as variations in Cantor sets such as looking at different middle portions removed to create the sets. The project focuses on attempting to generalize the properties of these Cantor sets. i Contents Page 1 The Ternary Cantor Set 1 1 2 The n -ary Cantor Set 9 n−1 3 The n -ary Cantor Set 24 4 Conclusion 35 Bibliography 40 Biography 41 ii Chapter 1 The Ternary Cantor Set Georg Cantor, born in 1845, was best known for his discovery of the Cantor set. -
2.8 the Derivative As a Function
2 Derivatives Copyright © Cengage Learning. All rights reserved. 2.8 The Derivative as a Function Copyright © Cengage Learning. All rights reserved. The Derivative as a Function We have considered the derivative of a function f at a fixed number a: Here we change our point of view and let the number a vary. If we replace a in Equation 1 by a variable x, we obtain 3 The Derivative as a Function Given any number x for which this limit exists, we assign to x the number f′(x). So we can regard f′ as a new function, called the derivative of f and defined by Equation 2. We know that the value of f′ at x, f′(x), can be interpreted geometrically as the slope of the tangent line to the graph of f at the point (x, f(x)). The function f′ is called the derivative of f because it has been “derived” from f by the limiting operation in Equation 2. The domain of f′ is the set {x | f′(x) exists} and may be smaller than the domain of f. 4 Example 1 – Derivative of a Function given by a Graph The graph of a function f is given in Figure 1. Use it to sketch the graph of the derivative f′. Figure 1 5 Example 1 – Solution We can estimate the value of the derivative at any value of x by drawing the tangent at the point (x, f(x)) and estimating its slope. For instance, for x = 5 we draw the tangent at P in Figure 2(a) and estimate its slope to be about , so f′(5) ≈ 1.5. -
A Visual Demonstration of the Fundamental Theorem of Calculus Emanuel A
A Visual Demonstration of the Fundamental Theorem of Calculus Emanuel A. Lazar This short paper presents an original, simple demonstration of the Fundamental Theorem of Calculus. This theorem asserts that finding slopes of tangent lines (i.e. differentiation) and finding areas under curves (i.e. integration) are inverse operations save a constant. Can we prove this? Yes. Most textbooks, in fact, present an elegant, simple proof for this theorem. However, the standard proofs usually omit references to curves, slopes, and areas--a weakness that can make the proof unnecessarily abstract and difficult to understand. In the next few pages we demonstrate visually the inverse relationship between differentiation and integration; we will see that finding areas under a curve is really the “opposite” of finding slopes of tangent-lines. Before beginning, I assume that the reader is familiar with Riemann’s Sums and thus the correspondence between the function n D lim h(xk ) x and the area under a curve. Dxç0 k =1 For our demonstration, we will need some arbitrary function. For this, we will use the function f(x) = x2. To the right is a graph of that function: Figure 1. f(x) 1 If we so wanted, we could construct a graph of the slope of every line tangent to f(x) at x. Looking at the above graph we see that a line tangent to f(x) at x = 0.2 would have a slope of 0.4; a line tangent at x = 0.6 would have a slope of 1.2; and a line tangent at x = 1 would have a slope of 2. -
Section 2.7. the Cantor Set and the Cantor-Lebesgue Function
2.7. Cantor Set and Cantor-Lebesgue Function 1 Section 2.7. The Cantor Set and the Cantor-Lebesgue Function Note. In this section, we define the Cantor set which gives us an example of an uncountable set of measure zero. We use the Cantor-Lebesgue Function to show there are measurable sets which are not Borel; so B ( M. The supplement to this section gives these results based on cardinality arguments (but the supplement does not address the Cantor-Lebesgue Function). Definition. Let I = [0, 1]. We iteratively remove the “open middle one-third” of closed subintervals of I as follows. We remove: 1 2 O1 = 3, 3 1 2 7 8 O2 = 9, 9 ∪ 9, 9 1 2 7 8 19 20 25 26 O3 = 27, 27 ∪ 27, 27 ∪ 27, 27 ∪ 27, 27 1 2 7 8 19 20 25 26 55 56 61 62 73 74 79 80 O4 = 81, 81 ∪ 81, 81 ∪ 81, 81 ∪ 81, 81 ∪ 81, 81 ∪ 81, 81 ∪ 81, 81 ∪ 81, 81 . k−1 2k−1 Ok = 2 open intervals of total length 3k . then we get: 2.7. Cantor Set and Cantor-Lebesgue Function 2 1 2 C1 = 0, 3 ∪ 3, 1 1 2 1 2 7 8 C2 = 0, 9 ∪ 9, 3 ∪ 3, 9 ∪ 9, 1 1 2 1 2 7 8 1 2 19 20 7 8 25 26 C3 = 0, 27 ∪ 27, 9 ∪ 9, 27 ∪ 27, 3 ∪ 3, 27 ∪ 27, 9 ∪ 9, 27 ∪ 27, 1 . k 2k Ck = 2 closed intervals of total length 3k . ∞ The Cantor set is C = ∩k=1Ck.