Worksheet 3 Solutions
... dx = arcsin(x) + C. We make the 1 − x2 substitution u = 2x , du = ln 2 · 2x dx and our integral becomes ...
... dx = arcsin(x) + C. We make the 1 − x2 substitution u = 2x , du = ln 2 · 2x dx and our integral becomes ...
Concentration inequalities
... distribution on the sphere has some interesting consequences, one of which has to do with data compression. Given a set of m data points in Rn , an obvious way to try to compress them is to project onto a lower dimensional subspace. How much information is lost? If the only features in the original ...
... distribution on the sphere has some interesting consequences, one of which has to do with data compression. Given a set of m data points in Rn , an obvious way to try to compress them is to project onto a lower dimensional subspace. How much information is lost? If the only features in the original ...
Homework 22 Solutions
... As a simple example, let f (x) = g(x) = 1. This F (x) = G(x) = x and the above equation would read: Z 1 · 1 dx = x · x but we know that Z 1 dx = x, so the statement is not true! ...
... As a simple example, let f (x) = g(x) = 1. This F (x) = G(x) = x and the above equation would read: Z 1 · 1 dx = x · x but we know that Z 1 dx = x, so the statement is not true! ...
Logarithm and inverse function
... f(x) = bx = y, for any given positive y. The expression logb(x) depends on both b and x, but the term logarithm function (or logarithmic function) refers to a function of the form logb(x) in which the base b is fixed and x is variable, thus yielding a function that assigns to any x its logarithm log ...
... f(x) = bx = y, for any given positive y. The expression logb(x) depends on both b and x, but the term logarithm function (or logarithmic function) refers to a function of the form logb(x) in which the base b is fixed and x is variable, thus yielding a function that assigns to any x its logarithm log ...
Summer Calculus BC Homework
... everything you can do in Calculus in ( x, y ) coordinates we can also do in polar ( r , ) coordinates as well. ...
... everything you can do in Calculus in ( x, y ) coordinates we can also do in polar ( r , ) coordinates as well. ...
Chapter 4 Continuous Random Variables
... Generating arbitrary random numbers can be done with the help of the following theorem. Theorem 2. Let X be a random variable with an invertible CDF FX (x), i.e., FX−1 exists. If Y = FX (X), then Y ∼ Uniform(0, 1). Proof. First, we know that if U ∼ Uniform(0, 1), then fU (u) = 1 for 0 ≤ u ≤ 1 and so ...
... Generating arbitrary random numbers can be done with the help of the following theorem. Theorem 2. Let X be a random variable with an invertible CDF FX (x), i.e., FX−1 exists. If Y = FX (X), then Y ∼ Uniform(0, 1). Proof. First, we know that if U ∼ Uniform(0, 1), then fU (u) = 1 for 0 ≤ u ≤ 1 and so ...
educative commentary on jee 2014 advanced mathematics papers
... are true. This does not quite prove that these are the only correct alternatives. That has to be done only by counter-examples for (B) and (C). An easy counterexample which will work for both is to take both f and g to be the constant functions k for some k 6= 0. ...
... are true. This does not quite prove that these are the only correct alternatives. That has to be done only by counter-examples for (B) and (C). An easy counterexample which will work for both is to take both f and g to be the constant functions k for some k 6= 0. ...
(1,1)fyy - KSU Web Home
... The critical points occur at values (x, y) such that ex cos (y) = 0 −ex sin (y) = 0. Since ex is never zero and cos (y) and sin (y) are never zero for the same value of y, there are no critical points for this function. 13. For the function f (x, y) = x sin (y) , we have fx = sin (y) fy = x cos (y) ...
... The critical points occur at values (x, y) such that ex cos (y) = 0 −ex sin (y) = 0. Since ex is never zero and cos (y) and sin (y) are never zero for the same value of y, there are no critical points for this function. 13. For the function f (x, y) = x sin (y) , we have fx = sin (y) fy = x cos (y) ...
For this assignment, we must write three definitions of a term we
... Calculus is a branch of mathematics that deals with calculation of continuous change by working with infinitely small numbers. The word calculus origins from Latin word, calculus, which means small pebbles used for counting on a counting frame. Although in general calculus means methods of calculati ...
... Calculus is a branch of mathematics that deals with calculation of continuous change by working with infinitely small numbers. The word calculus origins from Latin word, calculus, which means small pebbles used for counting on a counting frame. Although in general calculus means methods of calculati ...
A Note on Formalizing Undefined Terms in Real Analysis
... first teaching goal — and teach effective high-level formulas manipulation — our second teaching goal. The two teaching goals may require different sets of exercises (or different approaches to them). For instance, our running example could be proposed as an exercise twice. The first time, just afte ...
... first teaching goal — and teach effective high-level formulas manipulation — our second teaching goal. The two teaching goals may require different sets of exercises (or different approaches to them). For instance, our running example could be proposed as an exercise twice. The first time, just afte ...
Page 107
... that f is strictly increasing on the interval (− π2 , 0]. Consider the numerator g(x) = x−sin(x) and the denominator h(x) = 1 − cos(x). Note that on (− π2 , 0), g 0 (x) = 1 − cos(x) > 0, so by Page 104 Exercise 5, g is strictly increasing. Further, h0 (x) = sin(x) < 0, so h is decreasing. As a resul ...
... that f is strictly increasing on the interval (− π2 , 0]. Consider the numerator g(x) = x−sin(x) and the denominator h(x) = 1 − cos(x). Note that on (− π2 , 0), g 0 (x) = 1 − cos(x) > 0, so by Page 104 Exercise 5, g is strictly increasing. Further, h0 (x) = sin(x) < 0, so h is decreasing. As a resul ...
this document - KSU Web Home
... bound on the area. They are not very good, since they are far apart. The next example will show that we can do better. Example 244 Same as the previous example, using 8 subintervals. In this case, a = 0, b = 4, n = 8. If we select xi to be the right end point of each subinterval, then x1 = 0:5, ...
... bound on the area. They are not very good, since they are far apart. The next example will show that we can do better. Example 244 Same as the previous example, using 8 subintervals. In this case, a = 0, b = 4, n = 8. If we select xi to be the right end point of each subinterval, then x1 = 0:5, ...
On Sequent Calculi for Intuitionistic Propositional Logic
... calculus in reverse, i.e. by starting from a given sequent and using the rules of the calculus in backward direction, with a hope to arrive at axioms (i.e. initial sequents). Then it may be a problem to ensure termination of the procedure. For example the contraction rule, if used in reverse, allows ...
... calculus in reverse, i.e. by starting from a given sequent and using the rules of the calculus in backward direction, with a hope to arrive at axioms (i.e. initial sequents). Then it may be a problem to ensure termination of the procedure. For example the contraction rule, if used in reverse, allows ...
Math 107H Topics for the first exam Integration Antiderivatives
... So, typically, using twice as many intervals (i.e., doing twice the work) gives us an estimate about 16 times closer to the real value of the integral. The importance of these estimates of the error is that they give us a means to decide beforehand how many subintervals to work with, in order to gua ...
... So, typically, using twice as many intervals (i.e., doing twice the work) gives us an estimate about 16 times closer to the real value of the integral. The importance of these estimates of the error is that they give us a means to decide beforehand how many subintervals to work with, in order to gua ...
Inverses (Farrand-Shultz) - Tools for the Common Core Standards
... operations ( ∗ ) are likely to include addition ( a ∗ b = a + b ) or multiplication ( a ∗ b = a ⋅ b ). For a collection of functions with a specified domain, taking values in a number system , the operation might be pointwise multiplication ( f ∗ g = f ⋅ g ). For a collection of functions with a spe ...
... operations ( ∗ ) are likely to include addition ( a ∗ b = a + b ) or multiplication ( a ∗ b = a ⋅ b ). For a collection of functions with a specified domain, taking values in a number system , the operation might be pointwise multiplication ( f ∗ g = f ⋅ g ). For a collection of functions with a spe ...
From Axioms to Analytic Rules in Nonclassical Logics
... and ∨ (disjunction). ¬α and α ↔ β will be used as abbreviations for α → 0 and (α → β) ∧ (β → α). (Our notation should not be confused with that of linear logic, where symbol 0 is used for ⊥ and vice versa.) Henceforth metavariables α, β, . . . will denote formulas, Π, Θ will stand for stoups, i.e., ...
... and ∨ (disjunction). ¬α and α ↔ β will be used as abbreviations for α → 0 and (α → β) ∧ (β → α). (Our notation should not be confused with that of linear logic, where symbol 0 is used for ⊥ and vice versa.) Henceforth metavariables α, β, . . . will denote formulas, Π, Θ will stand for stoups, i.e., ...
Finding Limits Numerically and Graphically
... Consider the function f x sin 1x . Note that the domain of f is all real numbers except 0. What can we say about lim sin 1x x→0 When we try to graph this function for values of x near zero, our graphing calculator has problems. Graph f over the intervals −2 ≤ y ≤ 2 and (1) −3 ≤ x ≤ 3, (2) −1 ≤ x ...
... Consider the function f x sin 1x . Note that the domain of f is all real numbers except 0. What can we say about lim sin 1x x→0 When we try to graph this function for values of x near zero, our graphing calculator has problems. Graph f over the intervals −2 ≤ y ≤ 2 and (1) −3 ≤ x ≤ 3, (2) −1 ≤ x ...
Function - Auburn University`s new IIS development environment on
... If a function f has domain A and codomain B we write f : A → B and say that f is a function from A to B. Strictly speaking, in order to define a function, one must specify the domain and the codomain as we did above when we said that f was a function from the set R of real numbers to itself. However ...
... If a function f has domain A and codomain B we write f : A → B and say that f is a function from A to B. Strictly speaking, in order to define a function, one must specify the domain and the codomain as we did above when we said that f was a function from the set R of real numbers to itself. However ...
De nition and some Properties of Generalized Elementary Functions
... even if the functions are non-elementary, very often the studying of these non-elementary functions lead to generalized elementary functions. According to the denition of generalized elementary functions given in this note, there are functions that are usually considered to be non-elementary, but a ...
... even if the functions are non-elementary, very often the studying of these non-elementary functions lead to generalized elementary functions. According to the denition of generalized elementary functions given in this note, there are functions that are usually considered to be non-elementary, but a ...
REVIEW FOR FINAL EXAM April 08, 2014 • Final Exam Review Session:
... I. Local maximum/minimal values: a. A function f has a local maximum value at point (a, b) if f (x, y) ≤ f (a, b) for all (x, y) in some small open disk centered at (a, b). b. A function f has a local minimum value at point (a, b) if f (x, y) ≥ f (a, b) for all (x, y) in some small open disk centere ...
... I. Local maximum/minimal values: a. A function f has a local maximum value at point (a, b) if f (x, y) ≤ f (a, b) for all (x, y) in some small open disk centered at (a, b). b. A function f has a local minimum value at point (a, b) if f (x, y) ≥ f (a, b) for all (x, y) in some small open disk centere ...
Section 3.2 Rolle`s Theorem and the Mean Value Theorem Rolle`s
... Case 1: If f x d for all x in a, b, f is constant on the interval and, by Theorem 2.2, fx 0 for all x in a, b. Case 2: Suppose f x > d for some x in a, b. By the Extreme Value Theorem, you know that f has a maximum at some c in the interval. Moreover, because f c > d, this maximum ...
... Case 1: If f x d for all x in a, b, f is constant on the interval and, by Theorem 2.2, fx 0 for all x in a, b. Case 2: Suppose f x > d for some x in a, b. By the Extreme Value Theorem, you know that f has a maximum at some c in the interval. Moreover, because f c > d, this maximum ...