Sum Difference Product Quotient Worksheet
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Numerical Differentiation and Integration
Chapter 4 Numerical Di↵erentiation and Integration 4.1 Numerical Di↵erentiation In this section, we introduce how to numerically calculate the derivative of a function. First, the derivative of the function f at x0 is defined as f x0 h f x0 f 1 x0 : lim p ` q´ p q. p q “ h 0 h Ñ This formula gives an obvious way to generate an approximation to f x0 : simply compute 1p q f x0 h f x0 p ` q´ p q h for small values of h. Although this way may be obvious, it is not very successful, due to our old nemesis round-o↵error. But it is certainly a place to start. 2 To approximate f x0 , suppose that x0 a, b ,wheref C a, b , and that x1 x0 h for 1p q Pp q P r s “ ` some h 0 that is sufficiently small to ensure that x1 a, b . We construct the first Lagrange ‰ Pr s polynomial P0,1 x for f determined by x0 and x1, with its error term: p q x x0 x x1 f x P0,1 x p ´ qp ´ qf 2 ⇠ x p q“ p q` 2! p p qq f x0 x x0 h f x0 h x x0 x x0 x x0 h p qp ´ ´ q p ` qp ´ q p ´ qp ´ ´ qf 2 ⇠ x “ h ` h ` 2 p p qq ´ for some ⇠ x between x0 and x1. Di↵erentiating gives p q f x0 h f x0 x x0 x x0 h f 1 x p ` q´ p q Dx p ´ qp ´ ´ qf 2 ⇠ x p q“ h ` 2 p p qq „ ⇢ f x0 h f x0 2 x x0 h p ` q´ p q p ´ q´ f 2 ⇠ x “ h ` 2 p p qq x x0 x x0 h p ´ qp ´ ´ qDx f 2 ⇠ x . -
MPI - Lecture 11
MPI - Lecture 11 Outline • Smooth optimization – Optimization methods overview – Smooth optimization methods • Numerical differentiation – Introduction and motivation – Newton’s difference quotient Smooth optimization Optimization methods overview Examples of op- timization in IT • Clustering • Classification • Model fitting • Recommender systems • ... Optimization methods Optimization methods can be: 1 2 1. discrete, when the support is made of several disconnected pieces (usu- ally finite); 2. smooth, when the support is connected (we have a derivative). They are further distinguished based on how the method calculates a so- lution: 1. direct, a finite numeber of steps; 2. iterative, the solution is the limit of some approximate results; 3. heuristic, methods quickly producing a solution that may not be opti- mal. Methods are also classified based on randomness: 1. deterministic; 2. stochastic, e.g., evolution, genetic algorithms, . 3 Smooth optimization methods Gradient de- scent methods n Goal: find local minima of f : Df → R, with Df ⊂ R . We assume that f, its first and second derivatives exist and are continuous on Df . We shall describe an iterative deterministic method from the family of descent methods. Descent method - general idea (1) Let x ∈ Df . We shall construct a sequence x(k), with k = 1, 2,..., such that x(k+1) = x(k) + t(k)∆x(k), where ∆x(k) is a suitable vector (in the direction of the descent) and t(k) is the length of the so-called step. Our goal is to have fx(k+1) < fx(k), except when x(k) is already a point of local minimum. Descent method - algorithm overview Let x ∈ Df . -
A Quotient Rule Integration by Parts Formula Jennifer Switkes ([email protected]), California State Polytechnic Univer- Sity, Pomona, CA 91768
A Quotient Rule Integration by Parts Formula Jennifer Switkes ([email protected]), California State Polytechnic Univer- sity, Pomona, CA 91768 In a recent calculus course, I introduced the technique of Integration by Parts as an integration rule corresponding to the Product Rule for differentiation. I showed my students the standard derivation of the Integration by Parts formula as presented in [1]: By the Product Rule, if f (x) and g(x) are differentiable functions, then d f (x)g(x) = f (x)g(x) + g(x) f (x). dx Integrating on both sides of this equation, f (x)g(x) + g(x) f (x) dx = f (x)g(x), which may be rearranged to obtain f (x)g(x) dx = f (x)g(x) − g(x) f (x) dx. Letting U = f (x) and V = g(x) and observing that dU = f (x) dx and dV = g(x) dx, we obtain the familiar Integration by Parts formula UdV= UV − VdU. (1) My student Victor asked if we could do a similar thing with the Quotient Rule. While the other students thought this was a crazy idea, I was intrigued. Below, I derive a Quotient Rule Integration by Parts formula, apply the resulting integration formula to an example, and discuss reasons why this formula does not appear in calculus texts. By the Quotient Rule, if f (x) and g(x) are differentiable functions, then ( ) ( ) ( ) − ( ) ( ) d f x = g x f x f x g x . dx g(x) [g(x)]2 Integrating both sides of this equation, we get f (x) g(x) f (x) − f (x)g(x) = dx. -
Laplace Transform
Chapter 7 Laplace Transform The Laplace transform can be used to solve differential equations. Be- sides being a different and efficient alternative to variation of parame- ters and undetermined coefficients, the Laplace method is particularly advantageous for input terms that are piecewise-defined, periodic or im- pulsive. The direct Laplace transform or the Laplace integral of a function f(t) defined for 0 ≤ t< 1 is the ordinary calculus integration problem 1 f(t)e−stdt; Z0 succinctly denoted L(f(t)) in science and engineering literature. The L{notation recognizes that integration always proceeds over t = 0 to t = 1 and that the integral involves an integrator e−stdt instead of the usual dt. These minor differences distinguish Laplace integrals from the ordinary integrals found on the inside covers of calculus texts. 7.1 Introduction to the Laplace Method The foundation of Laplace theory is Lerch's cancellation law 1 −st 1 −st 0 y(t)e dt = 0 f(t)e dt implies y(t)= f(t); (1) R R or L(y(t)= L(f(t)) implies y(t)= f(t): In differential equation applications, y(t) is the sought-after unknown while f(t) is an explicit expression taken from integral tables. Below, we illustrate Laplace's method by solving the initial value prob- lem y0 = −1; y(0) = 0: The method obtains a relation L(y(t)) = L(−t), whence Lerch's cancel- lation law implies the solution is y(t)= −t. The Laplace method is advertised as a table lookup method, in which the solution y(t) to a differential equation is found by looking up the answer in a special integral table. -
Calculus Lab 4—Difference Quotients and Derivatives (Edited from U. Of
Calculus Lab 4—Difference Quotients and Derivatives (edited from U. of Alberta) Objective: To compute difference quotients and derivatives of expressions and functions. Recall Plotting Commands: plot({expr1,expr2},x=a..b); Plots two Maple expressions on one set of axes. plot({f,g},a..b); Plots two Maple functions on one set of axes. plot({f(x),g(x)},x=a..b); This allows us to plot the Maple functions f and g using the form of plot() command appropriate to Maple expressions. If f and g are Maple functions, then f(x) and g(x) are the corresponding Maple expressions. The output of this plot() command is precisely the same as that of the preceding (function version) plot() command. 1. We begin by using Maple to compute difference quotients and, from them, derivatives. Try the following sequence of commands: 1 f:=x->1/(x^2-2*x+2); This defines the function f (x) = . x 2 − 2x + 2 (f(2+h)-f(2))/h; This is the difference quotient of f at the point x = 2. simplify(%); Simplifies the last expression. limit(%,h=0); This gives the derivative of f at the point where x = 2. Exercise 1: Find the difference quotient and derivative of this function at a general point x (hint: make a simple modification of the above steps). This f (x + h) − f (x) means find and f’(x). Record your answers below. h Use this to evaluate the derivative at the points x = -1 and x = 4. (It may help to remember the subs() command here; for example, subs(x=1,e1); means substitute x = 1 into the expression e1). -
3.2 the Derivative As a Function 201
SECT ION 3.2 The Derivative as a Function 201 SOLUTION Figure (A) satisfies the inequality f .a h/ f .a h/ f .a h/ f .a/ C C 2h h since in this graph the symmetric difference quotient has a larger negative slope than the ordinary right difference quotient. [In figure (B), the symmetric difference quotient has a larger positive slope than the ordinary right difference quotient and therefore does not satisfy the stated inequality.] 75. Show that if f .x/ is a quadratic polynomial, then the SDQ at x a (for any h 0) is equal to f 0.a/ . Explain the graphical meaning of this result. D ¤ SOLUTION Let f .x/ px 2 qx r be a quadratic polynomial. We compute the SDQ at x a. D C C D f .a h/ f .a h/ p.a h/ 2 q.a h/ r .p.a h/ 2 q.a h/ r/ C C C C C C C 2h D 2h pa2 2pah ph 2 qa qh r pa 2 2pah ph 2 qa qh r C C C C C C C D 2h 4pah 2qh 2h.2pa q/ C C 2pa q D 2h D 2h D C Since this doesn’t depend on h, the limit, which is equal to f 0.a/ , is also 2pa q. Graphically, this result tells us that the secant line to a parabola passing through points chosen symmetrically about x a is alwaysC parallel to the tangent line at x a. D D 76. Let f .x/ x 2. -
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 -
CHAPTER 3: Derivatives
CHAPTER 3: Derivatives 3.1: Derivatives, Tangent Lines, and Rates of Change 3.2: Derivative Functions and Differentiability 3.3: Techniques of Differentiation 3.4: Derivatives of Trigonometric Functions 3.5: Differentials and Linearization of Functions 3.6: Chain Rule 3.7: Implicit Differentiation 3.8: Related Rates • Derivatives represent slopes of tangent lines and rates of change (such as velocity). • In this chapter, we will define derivatives and derivative functions using limits. • We will develop short cut techniques for finding derivatives. • Tangent lines correspond to local linear approximations of functions. • Implicit differentiation is a technique used in applied related rates problems. (Section 3.1: Derivatives, Tangent Lines, and Rates of Change) 3.1.1 SECTION 3.1: DERIVATIVES, TANGENT LINES, AND RATES OF CHANGE LEARNING OBJECTIVES • Relate difference quotients to slopes of secant lines and average rates of change. • Know, understand, and apply the Limit Definition of the Derivative at a Point. • Relate derivatives to slopes of tangent lines and instantaneous rates of change. • Relate opposite reciprocals of derivatives to slopes of normal lines. PART A: SECANT LINES • For now, assume that f is a polynomial function of x. (We will relax this assumption in Part B.) Assume that a is a constant. • Temporarily fix an arbitrary real value of x. (By “arbitrary,” we mean that any real value will do). Later, instead of thinking of x as a fixed (or single) value, we will think of it as a “moving” or “varying” variable that can take on different values. The secant line to the graph of f on the interval []a, x , where a < x , is the line that passes through the points a, fa and x, fx. -
MATH 162: Calculus II Framework for Thurs., Mar
MATH 162: Calculus II Framework for Thurs., Mar. 29–Fri. Mar. 30 The Gradient Vector Today’s Goal: To learn about the gradient vector ∇~ f and its uses, where f is a function of two or three variables. The Gradient Vector Suppose f is a differentiable function of two variables x and y with domain R, an open region of the xy-plane. Suppose also that r(t) = x(t)i + y(t)j, t ∈ I, (where I is some interval) is a differentiable vector function (parametrized curve) with (x(t), y(t)) being a point in R for each t ∈ I. Then by the chain rule, d ∂f dx ∂f dy f(x(t), y(t)) = + dt ∂x dt ∂y dt 0 0 = [fxi + fyj] · [x (t)i + y (t)j] dr = [fxi + fyj] · . (1) dt Definition: For a differentiable function f(x1, . , xn) of n variables, we define the gradient vector of f to be ∂f ∂f ∂f ∇~ f := , ,..., . ∂x1 ∂x2 ∂xn Remarks: • Using this definition, the total derivative df/dt calculated in (1) above may be written as df = ∇~ f · r0. dt In particular, if r(t) = x(t)i + y(t)j + z(t)k, t ∈ (a, b) is a differentiable vector function, and if f is a function of 3 variables which is differentiable at the point (x0, y0, z0), where x0 = x(t0), y0 = y(t0), and z0 = z(t0) for some t0 ∈ (a, b), then df ~ 0 = ∇f(x0, y0, z0) · r (t0). dt t=t0 • If f is a function of 2 variables, then ∇~ f has 2 components. -
Differentiation
CHAPTER 3 Differentiation 3.1 Definition of the Derivative Preliminary Questions 1. What are the two ways of writing the difference quotient? 2. Explain in words what the difference quotient represents. In Questions 3–5, f (x) is an arbitrary function. 3. What does the following quantity represent in terms of the graph of f (x)? f (8) − f (3) 8 − 3 4. For which value of x is f (x) − f (3) f (7) − f (3) = ? x − 3 4 5. For which value of h is f (2 + h) − f (2) f (4) − f (2) = ? h 4 − 2 6. To which derivative is the quantity ( π + . ) − tan 4 00001 1 .00001 a good approximation? 7. What is the equation of the tangent line to the graph at x = 3 of a function f (x) such that f (3) = 5and f (3) = 2? In Questions 8–10, let f (x) = x 2. 1 2 Chapter 3 Differentiation 8. The expression f (7) − f (5) 7 − 5 is the slope of the secant line through two points P and Q on the graph of f (x).Whatare the coordinates of P and Q? 9. For which value of h is the expression f (5 + h) − f (5) h equal to the slope of the secant line between the points P and Q in Question 8? 10. For which value of h is the expression f (3 + h) − f (3) h equal to the slope of the secant line between the points (3, 9) and (5, 25) on the graph of f (x)? Exercises 1. -
Multivariable Analysis
1 - Multivariable analysis Roland van der Veen Groningen, 20-1-2020 2 Contents 1 Introduction 5 1.1 Basic notions and notation . 5 2 How to solve equations 7 2.1 Linear algebra . 8 2.2 Derivative . 10 2.3 Elementary Riemann integration . 12 2.4 Mean value theorem and Banach contraction . 15 2.5 Inverse and Implicit function theorems . 17 2.6 Picard's theorem on existence of solutions to ODE . 20 3 Multivariable fundamental theorem of calculus 23 3.1 Exterior algebra . 23 3.2 Differential forms . 26 3.3 Integration . 27 3.4 More on cubes and their boundary . 28 3.5 Exterior derivative . 30 3.6 The fundamental theorem of calculus (Stokes Theorem) . 31 3.7 Fundamental theorem of calculus: Poincar´elemma . 32 3 4 CONTENTS Chapter 1 Introduction The goal of these notes is to explore the notions of differentiation and integration in arbitrarily many variables. The material is focused on answering two basic questions: 1. How to solve an equation? How many solutions can one expect? 2. Is there a higher dimensional analogue for the fundamental theorem of calculus? Can one find a primitive? The equations we will address are systems of non-linear equations in finitely many variables and also ordinary differential equations. The approach will be mostly theoretical, schetching a framework in which one can predict how many solutions there will be without necessarily solving the equation. The key assumption is that everything we do can locally be approximated by linear functions. In other words, everything will be differentiable. One of the main results is that the linearization of the equation predicts the number of solutions and approximates them well locally. -
Vector Calculus and Differential Forms with Applications To
Vector Calculus and Differential Forms with Applications to Electromagnetism Sean Roberson May 7, 2015 PREFACE This paper is written as a final project for a course in vector analysis, taught at Texas A&M University - San Antonio in the spring of 2015 as an independent study course. Students in mathematics, physics, engineering, and the sciences usually go through a sequence of three calculus courses before go- ing on to differential equations, real analysis, and linear algebra. In the third course, traditionally reserved for multivariable calculus, stu- dents usually learn how to differentiate functions of several variable and integrate over general domains in space. Very rarely, as was my case, will professors have time to cover the important integral theo- rems using vector functions: Green’s Theorem, Stokes’ Theorem, etc. In some universities, such as UCSD and Cornell, honors students are able to take an accelerated calculus sequence using the text Vector Cal- culus, Linear Algebra, and Differential Forms by John Hamal Hubbard and Barbara Burke Hubbard. Here, students learn multivariable cal- culus using linear algebra and real analysis, and then they generalize familiar integral theorems using the language of differential forms. This paper was written over the course of one semester, where the majority of the book was covered. Some details, such as orientation of manifolds, topology, and the foundation of the integral were skipped to save length. The paper should still be readable by a student with at least three semesters of calculus, one course in linear algebra, and one course in real analysis - all at the undergraduate level.