POWER SERIES
... The infant lost weight for approximately the first month and then gained for the 2nd and 3rd. The infant lost weight for approximately .56 month and then gained thereafter. The infant gained weight for approximately the first month and then lost weight during the 2nd and 3rd months. The infant lost ...
... The infant lost weight for approximately the first month and then gained for the 2nd and 3rd. The infant lost weight for approximately .56 month and then gained thereafter. The infant gained weight for approximately the first month and then lost weight during the 2nd and 3rd months. The infant lost ...
Strict topologies on spaces of vector
... functions which vanish at infinity (i.e. S = S0 ), we get the strict topology on Γb (π), as defined here. Or, if S is the space of functions with compact support, we get the κ-topology. Along these lines, as in [9], we point out that several definitions for the strict topology β on Cb (X) have been ...
... functions which vanish at infinity (i.e. S = S0 ), we get the strict topology on Γb (π), as defined here. Or, if S is the space of functions with compact support, we get the κ-topology. Along these lines, as in [9], we point out that several definitions for the strict topology β on Cb (X) have been ...
USING FUNCTIONAL ANALYSIS AND SOBOLEV SPACES TO
... for any x ∈ E, we have f (x) = g(x), which means f = g. Therefore, φ is also an injection. We claim φ is a homeomorphism. It suffices to show that φ and φ−1 are continuous. Referring to Lemma 3.7 and noticing that ∀x ∈ E, the function E ∗ → R sending f to f (x) is continuous, we know φ is continuous ...
... for any x ∈ E, we have f (x) = g(x), which means f = g. Therefore, φ is also an injection. We claim φ is a homeomorphism. It suffices to show that φ and φ−1 are continuous. Referring to Lemma 3.7 and noticing that ∀x ∈ E, the function E ∗ → R sending f to f (x) is continuous, we know φ is continuous ...
Trig Unified Syllabus - North Allegheny School District
... Students planning to continue their study of mathematics in calculus, statistics or other disciplines, as well as those taking trigonometry as their final mathematics course, will benefit from the content in this class. Textbook Lial, Hornsby, Schneider, Trigonometry, Boston: Addison Wesley Company, ...
... Students planning to continue their study of mathematics in calculus, statistics or other disciplines, as well as those taking trigonometry as their final mathematics course, will benefit from the content in this class. Textbook Lial, Hornsby, Schneider, Trigonometry, Boston: Addison Wesley Company, ...
Math 163 Notes Section 5.3
... A function is concave upward on an interval a, b if the graph of the function lies above its tangent line at each point of a, b . A function is concave downward on a, b if the graph of the function lies below its tangent line at each point of a, b . A point where a graph changes concavit ...
... A function is concave upward on an interval a, b if the graph of the function lies above its tangent line at each point of a, b . A function is concave downward on a, b if the graph of the function lies below its tangent line at each point of a, b . A point where a graph changes concavit ...
Week 2 - NUI Galway
... The most important example of a piecewise defined function is the absolute value function. ...
... The most important example of a piecewise defined function is the absolute value function. ...
MATH 409, Fall 2013 [3mm] Advanced Calculus I
... Theorem Any function continuous on a closed bounded interval [a, b] is also uniformly continuous on [a, b]. Proof: Assume that a function f : [a, b] → R is not uniformly continuous on [a, b]. We have to show that f is not continuous on [a, b]. By assumption, there exists ε > 0 such that for any δ > ...
... Theorem Any function continuous on a closed bounded interval [a, b] is also uniformly continuous on [a, b]. Proof: Assume that a function f : [a, b] → R is not uniformly continuous on [a, b]. We have to show that f is not continuous on [a, b]. By assumption, there exists ε > 0 such that for any δ > ...
Honors Precalculus Topics
... multiple-choice questions and free response questions. Partial credit may be awarded on some items. ...
... multiple-choice questions and free response questions. Partial credit may be awarded on some items. ...
College Algebra Chapter 2 Functions and Graphs
... a month. Once the sales person reaches $50,000 in total sales, he earns his base salary plus a 4.3% commission on all sales of $50,000 or more. Write a piecewisedefined function (in dollars) to model the expected total monthly salary as a function of the amount of sales, x. ...
... a month. Once the sales person reaches $50,000 in total sales, he earns his base salary plus a 4.3% commission on all sales of $50,000 or more. Write a piecewisedefined function (in dollars) to model the expected total monthly salary as a function of the amount of sales, x. ...
Muthuvel, R.
... Quizzes: There will be many short quizzes (about 5 minutes long) at the beginning of many class periods. These will be worth 5 points each. Quizzes must be taken at the time they are scheduled in class. ...
... Quizzes: There will be many short quizzes (about 5 minutes long) at the beginning of many class periods. These will be worth 5 points each. Quizzes must be taken at the time they are scheduled in class. ...
Discrete distributions: empirical, Bernoulli, binomial, Poisson
... possible outcomes (usually viewed as success or failure) occurring with probability p and 1-p, respectively. A Bernoulli trial is an instantiation of a Bernoulli event. So long as the probability of success or failure remains the same from trial to trial (i.e., each trial is independent of the other ...
... possible outcomes (usually viewed as success or failure) occurring with probability p and 1-p, respectively. A Bernoulli trial is an instantiation of a Bernoulli event. So long as the probability of success or failure remains the same from trial to trial (i.e., each trial is independent of the other ...
3.3 Derivatives of Logarithmic and Exponential Functions (10/21
... 3.3 Derivatives of Logarithmic and Exponential Functions In this section we will be using the product rule, quotient rule, and chain rule to differentiate functions, but our functions will involve exponentials and logarithms, so we need to discuss their derivatives. The proofs of these can be fo ...
... 3.3 Derivatives of Logarithmic and Exponential Functions In this section we will be using the product rule, quotient rule, and chain rule to differentiate functions, but our functions will involve exponentials and logarithms, so we need to discuss their derivatives. The proofs of these can be fo ...
Trigonometric Functions The Unit Circle
... We learned that to every real number, we can assign a point (the terminal point) on the unit circle. We use the x and y coordinates of this point to define several functions. Let P (x, y) be the point on the unit circle defined by t. The trigonometric functions are defined as follows: 1. The functio ...
... We learned that to every real number, we can assign a point (the terminal point) on the unit circle. We use the x and y coordinates of this point to define several functions. Let P (x, y) be the point on the unit circle defined by t. The trigonometric functions are defined as follows: 1. The functio ...
Weak topologies Weak-type topologies on vector spaces. Let X be a
... topology on an arbitrary Cartesian product P = Πi∈I Ti of topological spaces is the weakest topology on P for which the canonical projections πk : P → Tk , πk ((xi )i∈I ) = xk , are all continuous. An open basis of the product topology consists of the sets A = Πi∈I Ai ...
... topology on an arbitrary Cartesian product P = Πi∈I Ti of topological spaces is the weakest topology on P for which the canonical projections πk : P → Tk , πk ((xi )i∈I ) = xk , are all continuous. An open basis of the product topology consists of the sets A = Πi∈I Ai ...
quasi - mackey topology - Revistas académicas, Universidad
... a continuous linear map such that T (B) is relatively weakly compact in E2 for every bounded set B in E1 . Then (i) the adjoint map T 0 : (E20 , τ (E20 , E2 )) → (E10 , β(E10 , E1 )) is continuous; (ii) the adjoiint map of T 0 in (i), T 00 : (E100 , τ (E100 , E1 )) → (E2 , τ (E2 , E20 )) is also con ...
... a continuous linear map such that T (B) is relatively weakly compact in E2 for every bounded set B in E1 . Then (i) the adjoint map T 0 : (E20 , τ (E20 , E2 )) → (E10 , β(E10 , E1 )) is continuous; (ii) the adjoiint map of T 0 in (i), T 00 : (E100 , τ (E100 , E1 )) → (E2 , τ (E2 , E20 )) is also con ...
Topological Vector Spaces I: Basic Theory
... (ii) Whenever (αλ ) and (xλ ) are nets in K and X , respectively, such that αλ → α (in K)) and xλ → x (in X )), it follows that (αλ xλ ) → (αx). Exercise 2. Show that a linear topology on X is Hausdorff, if and only if the singleton set {0} is closed. Example 2. Let I be an arbitrary non-empty set. ...
... (ii) Whenever (αλ ) and (xλ ) are nets in K and X , respectively, such that αλ → α (in K)) and xλ → x (in X )), it follows that (αλ xλ ) → (αx). Exercise 2. Show that a linear topology on X is Hausdorff, if and only if the singleton set {0} is closed. Example 2. Let I be an arbitrary non-empty set. ...
Lesson 3-8: Derivatives of Inverse Functions, Part 1
... If f is differentiable at every point of an interval I and is never zero on I, then f has an dx inverse and f 1 is differentiable at every point of the interval f I ...
... If f is differentiable at every point of an interval I and is never zero on I, then f has an dx inverse and f 1 is differentiable at every point of the interval f I ...
f(x)
... 13) Is f defined at x = 2 ? Is f continuous at x = 2? 14) At what values of x is f continuous? 15) What value should be assigned to f(2) to make the extended function continuous at x = 2 . 16) What new value should be assigned to f(1) to make the extended function continuous at x = 1 . 17) Is it pos ...
... 13) Is f defined at x = 2 ? Is f continuous at x = 2? 14) At what values of x is f continuous? 15) What value should be assigned to f(2) to make the extended function continuous at x = 2 . 16) What new value should be assigned to f(1) to make the extended function continuous at x = 1 . 17) Is it pos ...
Distribution (mathematics)
Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function (which is historically called a ""function"" even though it is not considered a genuine function mathematically).The practical use of distributions can be traced back to the use of Green functions in the 1830's to solveordinary differential equations, but was not formalized until much later. According to Kolmogorov & Fomin (1957), generalized functions originated in the work of Sergei Sobolev (1936) on second-order hyperbolic partial differential equations, and the ideas were developed in somewhat extended form by Laurent Schwartz in the late 1940s. According to his autobiography, Schwartz introduced the term ""distribution"" by analogy with a distribution of electrical charge, possibly including not only point charges but also dipoles and so on. Gårding (1997) comments that although the ideas in the transformative book by Schwartz (1951) were not entirely new, it was Schwartz's broad attack and conviction that distributions would be useful almost everywhere in analysis that made the difference.The basic idea in distribution theory is to reinterpret functions as linear functionals acting on a space of test functions. Standard functions act by integration against a test function, but many other linear functionals do not arise in this way, and these are the ""generalized functions"". There are different possible choices for the space of test functions, leading to different spaces of distributions. The basic space of test function consists of smooth functions with compact support, leading to standard distributions. Use of the space of smooth, rapidly decreasing test functions gives instead the tempered distributions, which are important because they have a well-defined distributional Fourier transform. Every tempered distribution is a distribution in the normal sense, but the converse is not true: in general the larger the space of test functions, the more restrictive the notion of distribution. On the other hand, the use of spaces of analytic test functions leads to Sato's theory of hyperfunctions; this theory has a different character from the previous ones because there are no analytic functions with non-empty compact support.