MATH 162: Calculus II Differentiation
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Section 8.8: Improper Integrals
Section 8.8: Improper Integrals One of the main applications of integrals is to compute the areas under curves, as you know. A geometric question. But there are some geometric questions which we do not yet know how to do by calculus, even though they appear to have the same form. Consider the curve y = 1=x2. We can ask, what is the area of the region under the curve and right of the line x = 1? We have no reason to believe this area is finite, but let's ask. Now no integral will compute this{we have to integrate over a bounded interval. Nonetheless, we don't want to throw up our hands. So note that b 2 b Z (1=x )dx = ( 1=x) 1 = 1 1=b: 1 − j − In other words, as b gets larger and larger, the area under the curve and above [1; b] gets larger and larger; but note that it gets closer and closer to 1. Thus, our intuition tells us that the area of the region we're interested in is exactly 1. More formally: lim 1 1=b = 1: b − !1 We can rewrite that as b 2 lim Z (1=x )dx: b !1 1 Indeed, in general, if we want to compute the area under y = f(x) and right of the line x = a, we are computing b lim Z f(x)dx: b !1 a ASK: Does this limit always exist? Give some situations where it does not exist. They'll give something that blows up. -
Notes on Calculus II Integral Calculus Miguel A. Lerma
Notes on Calculus II Integral Calculus Miguel A. Lerma November 22, 2002 Contents Introduction 5 Chapter 1. Integrals 6 1.1. Areas and Distances. The Definite Integral 6 1.2. The Evaluation Theorem 11 1.3. The Fundamental Theorem of Calculus 14 1.4. The Substitution Rule 16 1.5. Integration by Parts 21 1.6. Trigonometric Integrals and Trigonometric Substitutions 26 1.7. Partial Fractions 32 1.8. Integration using Tables and CAS 39 1.9. Numerical Integration 41 1.10. Improper Integrals 46 Chapter 2. Applications of Integration 50 2.1. More about Areas 50 2.2. Volumes 52 2.3. Arc Length, Parametric Curves 57 2.4. Average Value of a Function (Mean Value Theorem) 61 2.5. Applications to Physics and Engineering 63 2.6. Probability 69 Chapter 3. Differential Equations 74 3.1. Differential Equations and Separable Equations 74 3.2. Directional Fields and Euler’s Method 78 3.3. Exponential Growth and Decay 80 Chapter 4. Infinite Sequences and Series 83 4.1. Sequences 83 4.2. Series 88 4.3. The Integral and Comparison Tests 92 4.4. Other Convergence Tests 96 4.5. Power Series 98 4.6. Representation of Functions as Power Series 100 4.7. Taylor and MacLaurin Series 103 3 CONTENTS 4 4.8. Applications of Taylor Polynomials 109 Appendix A. Hyperbolic Functions 113 A.1. Hyperbolic Functions 113 Appendix B. Various Formulas 118 B.1. Summation Formulas 118 Appendix C. Table of Integrals 119 Introduction These notes are intended to be a summary of the main ideas in course MATH 214-2: Integral Calculus. -
The Mean Value Theorem Math 120 Calculus I Fall 2015
The Mean Value Theorem Math 120 Calculus I Fall 2015 The central theorem to much of differential calculus is the Mean Value Theorem, which we'll abbreviate MVT. It is the theoretical tool used to study the first and second derivatives. There is a nice logical sequence of connections here. It starts with the Extreme Value Theorem (EVT) that we looked at earlier when we studied the concept of continuity. It says that any function that is continuous on a closed interval takes on a maximum and a minimum value. A technical lemma. We begin our study with a technical lemma that allows us to relate 0 the derivative of a function at a point to values of the function nearby. Specifically, if f (x0) is positive, then for x nearby but smaller than x0 the values f(x) will be less than f(x0), but for x nearby but larger than x0, the values of f(x) will be larger than f(x0). This says something like f is an increasing function near x0, but not quite. An analogous statement 0 holds when f (x0) is negative. Proof. The proof of this lemma involves the definition of derivative and the definition of limits, but none of the proofs for the rest of the theorems here require that depth. 0 Suppose that f (x0) = p, some positive number. That means that f(x) − f(x ) lim 0 = p: x!x0 x − x0 f(x) − f(x0) So you can make arbitrarily close to p by taking x sufficiently close to x0. -
Lecture 9: Partial Derivatives
Math S21a: Multivariable calculus Oliver Knill, Summer 2016 Lecture 9: Partial derivatives ∂ If f(x,y) is a function of two variables, then ∂x f(x,y) is defined as the derivative of the function g(x) = f(x,y) with respect to x, where y is considered a constant. It is called the partial derivative of f with respect to x. The partial derivative with respect to y is defined in the same way. ∂ We use the short hand notation fx(x,y) = ∂x f(x,y). For iterated derivatives, the notation is ∂ ∂ similar: for example fxy = ∂x ∂y f. The meaning of fx(x0,y0) is the slope of the graph sliced at (x0,y0) in the x direction. The second derivative fxx is a measure of concavity in that direction. The meaning of fxy is the rate of change of the slope if you change the slicing. The notation for partial derivatives ∂xf,∂yf was introduced by Carl Gustav Jacobi. Before, Josef Lagrange had used the term ”partial differences”. Partial derivatives fx and fy measure the rate of change of the function in the x or y directions. For functions of more variables, the partial derivatives are defined in a similar way. 4 2 2 4 3 2 2 2 1 For f(x,y)= x 6x y + y , we have fx(x,y)=4x 12xy ,fxx = 12x 12y ,fy(x,y)= − − − 12x2y +4y3,f = 12x2 +12y2 and see that f + f = 0. A function which satisfies this − yy − xx yy equation is also called harmonic. The equation fxx + fyy = 0 is an example of a partial differential equation: it is an equation for an unknown function f(x,y) which involves partial derivatives with respect to more than one variables. -
Series: Convergence and Divergence Comparison Tests
Series: Convergence and Divergence Here is a compilation of what we have done so far (up to the end of October) in terms of convergence and divergence. • Series that we know about: P∞ n Geometric Series: A geometric series is a series of the form n=0 ar . The series converges if |r| < 1 and 1 a1 diverges otherwise . If |r| < 1, the sum of the entire series is 1−r where a is the first term of the series and r is the common ratio. P∞ 1 2 p-Series Test: The series n=1 np converges if p1 and diverges otherwise . P∞ • Nth Term Test for Divergence: If limn→∞ an 6= 0, then the series n=1 an diverges. Note: If limn→∞ an = 0 we know nothing. It is possible that the series converges but it is possible that the series diverges. Comparison Tests: P∞ • Direct Comparison Test: If a series n=1 an has all positive terms, and all of its terms are eventually bigger than those in a series that is known to be divergent, then it is also divergent. The reverse is also true–if all the terms are eventually smaller than those of some convergent series, then the series is convergent. P P P That is, if an, bn and cn are all series with positive terms and an ≤ bn ≤ cn for all n sufficiently large, then P P if cn converges, then bn does as well P P if an diverges, then bn does as well. (This is a good test to use with rational functions. -
Chapter 9 Optimization: One Choice Variable
RS - Ch 9 - Optimization: One Variable Chapter 9 Optimization: One Choice Variable 1 Léon Walras (1834-1910) Vilfredo Federico D. Pareto (1848–1923) 9.1 Optimum Values and Extreme Values • Goal vs. non-goal equilibrium • In the optimization process, we need to identify the objective function to optimize. • In the objective function the dependent variable represents the object of maximization or minimization Example: - Define profit function: = PQ − C(Q) - Objective: Maximize - Tool: Q 2 1 RS - Ch 9 - Optimization: One Variable 9.2 Relative Maximum and Minimum: First- Derivative Test Critical Value The critical value of x is the value x0 if f ′(x0)= 0. • A stationary value of y is f(x0). • A stationary point is the point with coordinates x0 and f(x0). • A stationary point is coordinate of the extremum. • Theorem (Weierstrass) Let f : S→R be a real-valued function defined on a compact (bounded and closed) set S ∈ Rn. If f is continuous on S, then f attains its maximum and minimum values on S. That is, there exists a point c1 and c2 such that f (c1) ≤ f (x) ≤ f (c2) ∀x ∈ S. 3 9.2 First-derivative test •The first-order condition (f.o.c.) or necessary condition for extrema is that f '(x*) = 0 and the value of f(x*) is: • A relative minimum if f '(x*) changes its sign y from negative to positive from the B immediate left of x0 to its immediate right. f '(x*)=0 (first derivative test of min.) x x* y • A relative maximum if the derivative f '(x) A f '(x*) = 0 changes its sign from positive to negative from the immediate left of the point x* to its immediate right. -
A Brief Tour of Vector Calculus
A BRIEF TOUR OF VECTOR CALCULUS A. HAVENS Contents 0 Prelude ii 1 Directional Derivatives, the Gradient and the Del Operator 1 1.1 Conceptual Review: Directional Derivatives and the Gradient........... 1 1.2 The Gradient as a Vector Field............................ 5 1.3 The Gradient Flow and Critical Points ....................... 10 1.4 The Del Operator and the Gradient in Other Coordinates*............ 17 1.5 Problems........................................ 21 2 Vector Fields in Low Dimensions 26 2 3 2.1 General Vector Fields in Domains of R and R . 26 2.2 Flows and Integral Curves .............................. 31 2.3 Conservative Vector Fields and Potentials...................... 32 2.4 Vector Fields from Frames*.............................. 37 2.5 Divergence, Curl, Jacobians, and the Laplacian................... 41 2.6 Parametrized Surfaces and Coordinate Vector Fields*............... 48 2.7 Tangent Vectors, Normal Vectors, and Orientations*................ 52 2.8 Problems........................................ 58 3 Line Integrals 66 3.1 Defining Scalar Line Integrals............................. 66 3.2 Line Integrals in Vector Fields ............................ 75 3.3 Work in a Force Field................................. 78 3.4 The Fundamental Theorem of Line Integrals .................... 79 3.5 Motion in Conservative Force Fields Conserves Energy .............. 81 3.6 Path Independence and Corollaries of the Fundamental Theorem......... 82 3.7 Green's Theorem.................................... 84 3.8 Problems........................................ 89 4 Surface Integrals, Flux, and Fundamental Theorems 93 4.1 Surface Integrals of Scalar Fields........................... 93 4.2 Flux........................................... 96 4.3 The Gradient, Divergence, and Curl Operators Via Limits* . 103 4.4 The Stokes-Kelvin Theorem..............................108 4.5 The Divergence Theorem ...............................112 4.6 Problems........................................114 List of Figures 117 i 11/14/19 Multivariate Calculus: Vector Calculus Havens 0. -
Differentiation Rules (Differential Calculus)
Differentiation Rules (Differential Calculus) 1. Notation The derivative of a function f with respect to one independent variable (usually x or t) is a function that will be denoted by D f . Note that f (x) and (D f )(x) are the values of these functions at x. 2. Alternate Notations for (D f )(x) d d f (x) d f 0 (1) For functions f in one variable, x, alternate notations are: Dx f (x), dx f (x), dx , dx (x), f (x), f (x). The “(x)” part might be dropped although technically this changes the meaning: f is the name of a function, dy 0 whereas f (x) is the value of it at x. If y = f (x), then Dxy, dx , y , etc. can be used. If the variable t represents time then Dt f can be written f˙. The differential, “d f ”, and the change in f ,“D f ”, are related to the derivative but have special meanings and are never used to indicate ordinary differentiation. dy 0 Historical note: Newton used y,˙ while Leibniz used dx . About a century later Lagrange introduced y and Arbogast introduced the operator notation D. 3. Domains The domain of D f is always a subset of the domain of f . The conventional domain of f , if f (x) is given by an algebraic expression, is all values of x for which the expression is defined and results in a real number. If f has the conventional domain, then D f usually, but not always, has conventional domain. Exceptions are noted below. -
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. -
Critical Points and Classifying Local Maxima and Minima Don Byrd, Rev
Notes on Textbook Section 4.1 Critical Points and Classifying Local Maxima and Minima Don Byrd, rev. 25 Oct. 2011 To find and classify critical points of a function f (x) First steps: 1. Take the derivative f ’(x) . 2. Find the critical points by setting f ’ equal to 0, and solving for x . To finish the job, use either the first derivative test or the second derivative test. Via First Derivative Test 3. Determine if f ’ is positive (so f is increasing) or negative (so f is decreasing) on both sides of each critical point. • Note that since all that matters is the sign, you can check any value on the side you want; so use values that make it easy! • Optional, but helpful in more complex situations: draw a sign diagram. For example, say we have critical points at 11/8 and 22/7. It’s usually easier to evaluate functions at integer values than non-integers, and it’s especially easy at 0. So, for a value to the left of 11/8, choose 0; for a value to the right of 11/8 and the left of 22/7, choose 2; and for a value to the right of 22/7, choose 4. Let’s say f ’(0) = –1; f ’(2) = 5/9; and f ’(4) = 5. Then we could draw this sign diagram: Value of x 11 22 8 7 Sign of f ’(x) negative positive positive ! ! 4. Apply the first derivative test (textbook, top of p. 172, restated in terms of the derivative): If f ’ changes from negative to positive: f has a local minimum at the critical point If f ’ changes from positive to negative: f has a local maximum at the critical point If f ’ is positive on both sides or negative on both sides: f has neither one For the above example, we could tell from the sign diagram that 11/8 is a local minimum of f (x), while 22/7 is neither a local minimum nor a local maximum. -
Two Fundamental Theorems About the Definite Integral
Two Fundamental Theorems about the Definite Integral These lecture notes develop the theorem Stewart calls The Fundamental Theorem of Calculus in section 5.3. The approach I use is slightly different than that used by Stewart, but is based on the same fundamental ideas. 1 The definite integral Recall that the expression b f(x) dx ∫a is called the definite integral of f(x) over the interval [a,b] and stands for the area underneath the curve y = f(x) over the interval [a,b] (with the understanding that areas above the x-axis are considered positive and the areas beneath the axis are considered negative). In today's lecture I am going to prove an important connection between the definite integral and the derivative and use that connection to compute the definite integral. The result that I am eventually going to prove sits at the end of a chain of earlier definitions and intermediate results. 2 Some important facts about continuous functions The first intermediate result we are going to have to prove along the way depends on some definitions and theorems concerning continuous functions. Here are those definitions and theorems. The definition of continuity A function f(x) is continuous at a point x = a if the following hold 1. f(a) exists 2. lim f(x) exists xœa 3. lim f(x) = f(a) xœa 1 A function f(x) is continuous in an interval [a,b] if it is continuous at every point in that interval. The extreme value theorem Let f(x) be a continuous function in an interval [a,b]. -
Notes for Math 136: Review of Calculus
Notes for Math 136: Review of Calculus Gyu Eun Lee These notes were written for my personal use for teaching purposes for the course Math 136 running in the Spring of 2016 at UCLA. They are also intended for use as a general calculus reference for this course. In these notes we will briefly review the results of the calculus of several variables most frequently used in partial differential equations. The selection of topics is inspired by the list of topics given by Strauss at the end of section 1.1, but we will not cover everything. Certain topics are covered in the Appendix to the textbook, and we will not deal with them here. The results of single-variable calculus are assumed to be familiar. Practicality, not rigor, is the aim. This document will be updated throughout the quarter as we encounter material involving additional topics. We will focus on functions of two real variables, u = u(x;t). Most definitions and results listed will have generalizations to higher numbers of variables, and we hope the generalizations will be reasonably straightforward to the reader if they become necessary. Occasionally (notably for the chain rule in its many forms) it will be convenient to work with functions of an arbitrary number of variables. We will be rather loose about the issues of differentiability and integrability: unless otherwise stated, we will assume all derivatives and integrals exist, and if we require it we will assume derivatives are continuous up to the required order. (This is also generally the assumption in Strauss.) 1 Differential calculus of several variables 1.1 Partial derivatives Given a scalar function u = u(x;t) of two variables, we may hold one of the variables constant and regard it as a function of one variable only: vx(t) = u(x;t) = wt(x): Then the partial derivatives of u with respect of x and with respect to t are defined as the familiar derivatives from single variable calculus: ¶u dv ¶u dw = x ; = t : ¶t dt ¶x dx c 2016, Gyu Eun Lee.