14 Further Calculus 14 FURTHER CALCULUS
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Curve Sketching and Relative Maxima and Minima the Goal
Curve Sketching and Relative Maxima and Minima The Goal The purpose of the first derivative sign diagram is to indicate where a function is increasing and decreasing. The purpose of this is to identify where the relative maxima and minima are, since we know that if a function switches from increasing to decreasing at some point and remains continuous there it has to have a relative maximum there (likewise for switching in the other direction for a relative minimum). This is known as the First Derivative Test. It is possible to come up with functions that are NOT continuous at a certain point and may not even switch direction but still have a relative maximum or minimum there, but we can't really deal with those here. Plot x - Floor x , x, 0, 4 , AspectRatio ® Automatic 1.0 0.8 @ @ D 8 < D 0.6 0.4 0.2 1 2 3 4 The purpose of the second derivative sign diagram is to fill in the shape of the function a little more for sketching purposes, since changes in curvature happen at different locations than changes in direction and are distinctive features of the curve. Addition- ally, in certain circumstances (when the second derivative exists and is not zero) the sign of the second derivative at a critical value will also indicate whether there is a maximum there (downward curvature, f''(x) < 0) or a minimum (upward curvature, f''(x) > 0). This is known as the Second Derivative Test. Also, the change in direction of the curvature also corresponds to the locations of the steepest slope, which turns out to be useful in certain applications. -
Tutorial 8: Solutions
Tutorial 8: Solutions Applications of the Derivative 1. We are asked to find the absolute maximum and minimum values of f on the given interval, and state where those values occur: (a) f(x) = 2x3 + 3x2 12x on [1; 4]. − The absolute extrema occur either at a critical point (stationary point or point of non-differentiability) or at the endpoints. The stationary points are given by f 0(x) = 0, which in this case gives 6x2 + 6x 12 = 0 = x = 1; x = 2: − ) − Checking the values at the critical points (only x = 1 is in the interval [1; 4]) and endpoints: f(1) = 7; f(4) = 128: − Therefore the absolute maximum occurs at x = 4 and is given by f(4) = 124 and the absolute minimum occurs at x = 1 and is given by f(1) = 7. − (b) f(x) = (x2 + x)2=3 on [ 2; 3] − As before, we find the critical points. The derivative is given by 2 2x + 1 f 0(x) = 3 (x2 + x)1=2 and hence we have a stationary point when 2x + 1 = 0 and points of non- differentiability whenever x2 + x = 0. Solving these, we get the three critical points, x = 1=2; and x = 0; 1: − − Checking the value of the function at these critical points and the endpoints: f( 2) = 22=3; f( 1) = 0; f( 1=2) = 2−4=3; f(0) = 0; f(3) = (12)2=3: − − − Hence the absolute minimum occurs at either x = 1 or x = 0 since in both − these cases the minimum value is 0, while the absolute maximum occurs at x = 3 and is f(3) = (12)2=3. -
Lecture Notes – 1
Optimization Methods: Optimization using Calculus-Stationary Points 1 Module - 2 Lecture Notes – 1 Stationary points: Functions of Single and Two Variables Introduction In this session, stationary points of a function are defined. The necessary and sufficient conditions for the relative maximum of a function of single or two variables are also discussed. The global optimum is also defined in comparison to the relative or local optimum. Stationary points For a continuous and differentiable function f(x) a stationary point x* is a point at which the slope of the function vanishes, i.e. f ’(x) = 0 at x = x*, where x* belongs to its domain of definition. minimum maximum inflection point Fig. 1 A stationary point may be a minimum, maximum or an inflection point (Fig. 1). Relative and Global Optimum A function is said to have a relative or local minimum at x = x* if f ()xfxh**≤+ ( )for all sufficiently small positive and negative values of h, i.e. in the near vicinity of the point x*. Similarly a point x* is called a relative or local maximum if f ()xfxh**≥+ ( )for all values of h sufficiently close to zero. A function is said to have a global or absolute minimum at x = x* if f ()xfx* ≤ ()for all x in the domain over which f(x) is defined. Similarly, a function is D Nagesh Kumar, IISc, Bangalore M2L1 Optimization Methods: Optimization using Calculus-Stationary Points 2 said to have a global or absolute maximum at x = x* if f ()xfx* ≥ ()for all x in the domain over which f(x) is defined. -
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 Theory of Stationary Point Processes By
THE THEORY OF STATIONARY POINT PROCESSES BY FREDERICK J. BEUTLER and OSCAR A. Z. LENEMAN (1) The University of Michigan, Ann Arbor, Michigan, U.S.A. (3) Table of contents Abstract .................................... 159 1.0. Introduction and Summary ......................... 160 2.0. Stationarity properties for point processes ................... 161 2.1. Forward recurrence times and points in an interval ............. 163 2.2. Backward recurrence times and stationarity ................ 167 2.3. Equivalent stationarity conditions .................... 170 3.0. Distribution functions, moments, and sample averages of the s.p.p ......... 172 3.1. Convexity and absolute continuity .................... 172 3.2. Existence and global properties of moments ................ 174 3.3. First and second moments ........................ 176 3.4. Interval statistics and the computation of moments ............ 178 3.5. Distribution of the t n .......................... 182 3.6. An ergodic theorem ........................... 184 4.0. Classes and examples of stationary point processes ............... 185 4.1. Poisson processes ............................ 186 4.2. Periodic processes ........................... 187 4.3. Compound processes .......................... 189 4.4. The generalized skip process ....................... 189 4.5. Jitte processes ............................ 192 4.6. ~deprendent identically distributed intervals ................ 193 Acknowledgments ............................... 196 References ................................... 196 Abstract An axiomatic formulation is presented for point processes which may be interpreted as ordered sequences of points randomly located on the real line. Such concepts as forward recurrence times and number of points in intervals are defined and related in set-theoretic Q) Presently at Massachusetts Institute of Technology, Lexington, Mass., U.S.A. (2) This work was supported by the National Aeronautics and Space Administration under research grant NsG-2-59. 11 - 662901. Aeta mathematica. 116. Imprlm$ le 19 septembre 1966. -
The First Derivative and Stationary Points
Mathematics Learning Centre The first derivative and stationary points Jackie Nicholas c 2004 University of Sydney Mathematics Learning Centre, University of Sydney 1 The first derivative and stationary points dy The derivative of a function y = f(x) tell us a lot about the shape of a curve. In this dx section we will discuss the concepts of stationary points and increasing and decreasing functions. However, we will limit our discussion to functions y = f(x) which are well behaved. Certain functions cause technical difficulties so we will concentrate on those that don’t! The first derivative dy The derivative, dx ,isthe slope of the tangent to the curve y = f(x)atthe point x.Ifwe know about the derivative we can deduce a lot about the curve itself. Increasing functions dy If > 0 for all values of x in an interval I, then we know that the slope of the tangent dx to the curve is positive for all values of x in I and so the function y = f(x)isincreasing on the interval I. 3 For example, let y = x + x, then y dy 1 =3x2 +1> 0 for all values of x. dx That is, the slope of the tangent to the x curve is positive for all values of x. So, –1 0 1 y = x3 + x is an increasing function for all values of x. –1 The graph of y = x3 + x. We know that the function y = x2 is increasing for x>0. We can work this out from the derivative. 2 If y = x then y dy 2 =2x>0 for all x>0. -
EE2 Maths: Stationary Points
EE2 Maths: Stationary Points df Univariate case: When we maximize f(x) (solving for the x0 such that dx = 0) the d df gradient is zero at these points. What about the rate of change of gradient: dx ( dx ) at the minimum x0? For a minimum the gradient increases as x0 ! x0 + ∆x (∆x > 0). It follows d2f d2f that dx2 > 0. The opposite is true for a maximum: dx2 < 0, the gradient decreases upon d2f positive steps away from x0. For a point of inflection dx2 = 0. r @f @f Multivariate case: Stationary points occur when f = 0. In 2-d this is ( @x ; @y ) = 0, @f @f namely, a generalization of the univariate case. Recall that df = @x dx + @y dy can be written as df = ds · rf where ds = (dx; dy). If rf = 0 at (x0; y0) then any infinitesimal step ds away from (x0; y0) will still leave f unchanged, i.e. df = 0. There are three types of stationary points of f(x; y): Maxima, Minima and Saddle Points. We'll draw some of their properties on the board. We will now attempt to find ways of identifying the character of each of the stationary points of f(x; y). Consider a Taylor expansion about a stationary point (x0; y0). We know that rf = 0 at (x0; y0) so writing (∆x; ∆y) = (x − x0; y − y0) we find: − ∆f = f(x; y)[ f(x0; y0) ] 1 @2f @2f @2f ' 0 + (∆x)2 + 2 ∆x∆y + (∆y)2 : (1) 2! @x2 @x@y @y2 Maxima: At a maximum all small steps away from (x0; y0) lead to ∆f < 0. -
Applications of Differentiation Applications of Differentiation– a Guide for Teachers (Years 11–12)
1 Supporting Australian Mathematics Project 2 3 4 5 6 12 11 7 10 8 9 A guide for teachers – Years 11 and 12 Calculus: Module 12 Applications of differentiation Applications of differentiation – A guide for teachers (Years 11–12) Principal author: Dr Michael Evans, AMSI Peter Brown, University of NSW Associate Professor David Hunt, University of NSW Dr Daniel Mathews, Monash University Editor: Dr Jane Pitkethly, La Trobe University Illustrations and web design: Catherine Tan, Michael Shaw Full bibliographic details are available from Education Services Australia. Published by Education Services Australia PO Box 177 Carlton South Vic 3053 Australia Tel: (03) 9207 9600 Fax: (03) 9910 9800 Email: [email protected] Website: www.esa.edu.au © 2013 Education Services Australia Ltd, except where indicated otherwise. You may copy, distribute and adapt this material free of charge for non-commercial educational purposes, provided you retain all copyright notices and acknowledgements. This publication is funded by the Australian Government Department of Education, Employment and Workplace Relations. Supporting Australian Mathematics Project Australian Mathematical Sciences Institute Building 161 The University of Melbourne VIC 3010 Email: [email protected] Website: www.amsi.org.au Assumed knowledge ..................................... 4 Motivation ........................................... 4 Content ............................................. 5 Graph sketching ....................................... 5 More graph sketching .................................. -
Lecture 3: 20 September 2018 3.1 Taylor Series Approximation
COS 597G: Toward Theoretical Understanding of Deep Learning Fall 2018 Lecture 3: 20 September 2018 Lecturer: Sanjeev Arora Scribe: Abhishek Bhrushundi, Pritish Sahu, Wenjia Zhang Recall the basic form of an optimization problem: min (or max) f(x) subject to x 2 B; where B ⊆ Rd and f : Rd ! R is the objective function1. Gradient descent (ascent) and its variants are a popular choice for finding approximate solutions to these sort of problems. In fact, it turns out that these methods also work well empirically even when the objective function f is non-convex, most notably when training neural networks with various loss functions. In this lecture we will explore some of the basics ideas behind gradient descent, try to understand its limitations, and also discuss some of its popular variants that try to get around these limitations. We begin with the Taylor series approximation of functions which serves as a starting point for these methods. 3.1 Taylor series approximation We begin by recalling the Taylor series for univariate real-valued functions from Calculus 101: if f : R ! R is infinitely differentiable at x 2 R then the Taylor series for f at x is the following power series (∆x)2 (∆x)k f(x) + f 0(x)∆x + f 00(x) + ::: + f (k)(x) + ::: 2! k! Truncating this power series at some power (∆x)k results in a polynomial that approximates f around the point x. In particular, for small ∆x, (∆x)2 (∆x)k f(x + ∆x) ≈ f(x) + f 0(x)∆x + f 00(x) + ::: + f (k)(x) 2! k! Here the error of the approximation goes to zero at least as fast as (∆x)k as ∆x ! 0. -
Motion in a Straight Line Motion in a Straight Line – a Guide for Teachers (Years 11–12)
1 Supporting Australian Mathematics Project 2 3 4 5 6 12 11 7 10 8 9 A guide for teachers – Years 11 and 12 Calculus: Module 17 Motion in a straight line Motion in a straight line – A guide for teachers (Years 11–12) Principal author: Dr Michael Evans AMSI Peter Brown, University of NSW Associate Professor David Hunt, University of NSW Dr Daniel Mathews, Monash University Editor: Dr Jane Pitkethly, La Trobe University Illustrations and web design: Catherine Tan, Michael Shaw Full bibliographic details are available from Education Services Australia. Published by Education Services Australia PO Box 177 Carlton South Vic 3053 Australia Tel: (03) 9207 9600 Fax: (03) 9910 9800 Email: [email protected] Website: www.esa.edu.au © 2013 Education Services Australia Ltd, except where indicated otherwise. You may copy, distribute and adapt this material free of charge for non-commercial educational purposes, provided you retain all copyright notices and acknowledgements. This publication is funded by the Australian Government Department of Education, Employment and Workplace Relations. Supporting Australian Mathematics Project Australian Mathematical Sciences Institute Building 161 The University of Melbourne VIC 3010 Email: [email protected] Website: www.amsi.org.au Assumed knowledge ..................................... 4 Motivation ........................................... 4 Content ............................................. 6 Position, displacement and distance ........................... 6 Constant velocity ..................................... -
1 Overview 2 a Characterization of Convex Functions
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 simplex method for finding optimal solutions. In this lecture we return to convex optimization and give a characterization of optimality. The main theorem we will prove today shows that for a convex function f : S ! R defined over a convex set S, a stationary point (the point x¯ for which rf(x¯)) is a global minimum. We we will review basic definitions and properties from multivariate calculus, prove some important properties of convex functions, and obtain various characterizations of optimality. 2 A Characterization of Convex Functions Recall the definitions we introduced in the second lecture of convex sets and convex functions: Definition. A set S is called a convex set if any two points in S contain their line, i.e. for any x1; x2 2 S we have that λx1 + (1 − λ)x2 2 S for any λ 2 [0; 1]. n Definition. For a convex set S ⊆ R , we say that a function f : S ! R is convex on S if for any two points x1; x2 2 S and any λ 2 [0; 1] we have that: f (λx1 + (1 − λ)x2) ≤ λf(x1) + 1 − λ f(x2): We will first give an important characterization of convex function. To so, we need to characterize multivariate functions via their Taylor expansion. 2.1 First-order Taylor approximation n n Definition. The gradient of a function f : R ! R at point x 2 R denoted rf(x¯) is: @f(x) @f(x)| rf(x) := ;:::; @x1 @xn First-order Taylor expansion. -
A Collection of Problems in Differential Calculus
A Collection of Problems in Differential Calculus Problems Given At the Math 151 - Calculus I and Math 150 - Calculus I With Review Final Examinations Department of Mathematics, Simon Fraser University 2000 - 2010 Veselin Jungic · Petra Menz · Randall Pyke Department Of Mathematics Simon Fraser University c Draft date December 6, 2011 To my sons, my best teachers. - Veselin Jungic Contents Contents i Preface 1 Recommendations for Success in Mathematics 3 1 Limits and Continuity 11 1.1 Introduction . 11 1.2 Limits . 13 1.3 Continuity . 17 1.4 Miscellaneous . 18 2 Differentiation Rules 19 2.1 Introduction . 19 2.2 Derivatives . 20 2.3 Related Rates . 25 2.4 Tangent Lines and Implicit Differentiation . 28 3 Applications of Differentiation 31 3.1 Introduction . 31 3.2 Curve Sketching . 34 3.3 Optimization . 45 3.4 Mean Value Theorem . 50 3.5 Differential, Linear Approximation, Newton's Method . 51 i 3.6 Antiderivatives and Differential Equations . 55 3.7 Exponential Growth and Decay . 58 3.8 Miscellaneous . 61 4 Parametric Equations and Polar Coordinates 65 4.1 Introduction . 65 4.2 Parametric Curves . 67 4.3 Polar Coordinates . 73 4.4 Conic Sections . 77 5 True Or False and Multiple Choice Problems 81 6 Answers, Hints, Solutions 93 6.1 Limits . 93 6.2 Continuity . 96 6.3 Miscellaneous . 98 6.4 Derivatives . 98 6.5 Related Rates . 102 6.6 Tangent Lines and Implicit Differentiation . 105 6.7 Curve Sketching . 107 6.8 Optimization . 117 6.9 Mean Value Theorem . 125 6.10 Differential, Linear Approximation, Newton's Method . 126 6.11 Antiderivatives and Differential Equations .