Problem 3. For real numbers x, we defined [z] as the unique integer such that[x] ≤ x < [x] +1.Prove the following properties:(a) [x + n] = [x] +n for every integer n.(b) [x+y] is equal to [x] + [y] or [x] + [y] + 1.(c) [2x] = [x] + [x]

Let, x be a real number. Then the greatest integer function [x] as the unique integer, is defined as…

Answered: Problem 3. For real numbers x, we…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Given the function ƒ (x) = x³ − 3x² + 3x − 1, determine the real zeros and state the multiplicity of any repeated zeros. O { 1 mult. 3} O {1 mult. 2, 1/2} O {0, 2, -1} O {-1 mult. 3}
x+4 / 1/x + x/4
If the remainder is zero (0) when f(x) is divided by x-4, this means that x-4 is a ____
2) a. Show that x+5 x² +5[x² | is 0(x²) b. Explain what it means for a function to be (1) and give an example of such a function. (
[1 1 [A]{X} = {b} Where [A] 1 2 If 1 and l2 1 -1 (X1) {b} = }1}. Find {X}: X2 (x3) Select one: a. {X} = {X2 = (X1) O b. {X} = }X2 = (x3) (X1 X2 (x3) O c. {X} = {5 O d. {X} = }X2 O e. None of these 花为

2. In each item below, you are given sets A and B, and a formula for a (non?)function f: A → B. In each case, determine whether or not the formula gives a well-defined function. Fully justify your answers. (a) A = Q = B, f(r) = a root in B of the polynomial X² - r (b) A = C = B, f(2)= a root in B of the polynomial X² – z (c) A = P(R) - {0}¹, B = R₁ ƒ(E) = the smallest element of E (d) A = P(R), B = R, ƒ(E) = the smallest element of E if E admits a smallest ele- ment, and f(E)= 0 otherwise A function f : A → B is injective (or one-to-one) if it satisfies the following condition: for all a₁, a2 € A, if f(a₁) = f(a₂), then a₁ = a₂; f is surjective (or onto) if f(A) = B, or, more explicitly, if for each b E B there exists a € A such that f(a) = b; f is bijective if f is both injective and surjective. Here are some examples. • The squaring function f : R → R given by f(x) = x² is not injective because f(-1) = 1 = f(1); f also fails to be surjective because -1 is not in the range of f. The…
Can the Fundamental Theorem of Calculus be used to find ○ No, f(x) = √₂² 0 Só ○ Yes, f(x) = √₂² Yes, f (x) 2 2x²-x-3 √x = • No, f(x) = 2 2x²-x-3 √√x 2 Só 2x²-x-3 √x √₂² So 0 2 2x²-x-3 √x 2 2x²-x-3 dx? find so √x dx is not continuous on the given interval and its antiderivative does not exist. dx is continuous on the given interval and its antiderivative exists. dx is continuous on the given interval but its antiderivative does not exist. dx is not continuous on the given interval but its antiderivative exists.

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Problem 3. For real numbers x, we defined [z] as the unique integer such that [2] ≤ x < [x] +1. Prove the following properties: (a) [r+ n] = [x] +n for every integer n. (b) [r+y] is equal to [x] + [y] or [x] + [y] + 1. (c) [2x] = [x] + [+]