1.(45 pts)(Chapter 3.2. Ex.# 49) Show that if f1(x) is O(g1 (x)), f2(æ) is O(92(x)), and f2(x) # 0 and g2(x) # 0 for all real numbers x > 0 then (fi/f2)(x) is O((g1/92)(x)). Verify this result by giving concrete examples of fi(x), f2(x), g1(x) and g2(x).

Q: Find the wronskian function of e^x, е^2х, and e^3x * O 2e^6x e^6x O 3e^6x

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Q: In GF(2), find the inverse of x^3+x+1 modulo x^5+ +x^2+1. Show your complete sölution Oax^3+x…

A: For some finite field  F and fx ,gx∈Fx s.t gx≠0 On applying successive division algorithm:…

Q: 1. Show that, if x(t) and x[n] are even, then x(t) = 2 x(t) dt and > x[n] = x[0] + 2 x[n]. n=-k n=1

A: If a function f(x) is an even function then f(-x)=f(x). Since x(t) , x[n] are even functions…

Q: 6.8. Suppose the cubic spline s E S3(N3) is such that s(x) = 0 for x E [xo, x1] and for xe[x2, x3].…

A: Given: The cubic spline s∈S3Ω3 sx=0 for x∈x0,x1 and for x∈x2,x3   To show: sx=0 for x∈x1,x2 (a.) by…

Q: ,2k 101. Let F(x) = 2k . k! k=0 (b) Show that y = F(x) is a solution of y" = xy' + y. y(0) = 1,…

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Q: 2. Find all the critical numbers a. f(x) = 2secx (0,2n) b. g(x) = (x² – 4)3

A: Critical Numbers : Critical numbers are the numbers where the function is defined and its derivative…

Q: Suppose f" (x) = N,(h) + a,h? + azh* + azhº + ... %3D The value of N2(h) using Richardson's…

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Q: 4. Define f: [0, 1] → R by if x E Qn [0, 1] if x ¢ Qn [0, 1] f(x) = (a) Prove that S° f = 0; s*f =…

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Q: 1) A course repeater claims that his brother who did some Calculus at college years ago, but…

A: (1) ⇒f(x)=(2x2-1)(x2+3)x2+1⇒f'(x)=(x2+1)ddx(2x2-1)(x2+3)-(2x2-1)(x2+3)ddx(x2+1)(x2+1)2    (∵using…

Q: What is the largest positive constant K such that if 0 O A. e O B. 1 O C. e1 O D. e?

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Q: 2. Use the Root Test to determine absolutely or diverge. 4n+3\ n (a)E-1( Zn=1 Зп-5) (b) E (-In (e²…

A: As we have to use the root test , So first I have written the Root test then using them for both the…

Q: x2n+1 El-1)"; 2n +1 n=0

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Q: The critical number(s) of f(r) = (x+ 1)/3 is (are)

A: Given,        The critical numbers of fx=x+113

Q: 1 1 Prove that: L' a exp(a't) erfe( avt), where erfc is complementary a+v error function.

A: Formula Used:- L1t=πsLeat=1s-aLerf(at)=as(s+a)Leatf(t)=F(s-a)                   where…

Q: 1+ By using the definition In(1+x) = / 1 -dt, show that for a certain range of values of r, t…

A: Solution:

Q: -10 -9 -8 -7 5 -4 -3 3 4 5 9 10 f (x) = x(x² 4) O f (2) = x(x² + 4) %3D Submit Answer a) = (r)f…

A: we have to find out the equation from the graph

Q: A manufacturing company employs two devices to inspect output for quality control purposes. The…

A: Given: For X n = 4 p = 0.993 X ~ Bin (4, 0.993) For Y n = 4 p = 0.997 Y ~ Bin (4, 0.997) X and Y are…

Q: 1. Differentiate the following with respect to x: a. y= 5x - + 5Vx + 3p ex + In(x) - 7 (9)

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Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

A: We have given  y'=2x-3y+1   ,y(1)=6 we know that Euler's formula…

Q: 1. Let flxx) = x³ + y3 - 3x2-3y2 - 9y. (a) Find all of the critical points of f.

A: Part (a): (i) Find derivative of x and y. (ii) Equate to zero, to get critical points.

Q: 3.Eliminate the arbitrary constant on the given equation: xsiny – x²y- C

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Q: Suppose f"(x) = N;(h) + a,h + azh² + azh³ + -…- ... The value of N2(h) using Richardson's…

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Q: Note that the exact ( analytical) solution was given in the tables. Yeuler |&] (%) t Ytrue 0.0 0.05…

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Q: 2. Let G = Z,xZ4, and find [G:H] for: a) H = {0}xZ4. %3D

A: We have to find [G:H]

Q: Suppose f"(x) = N (h) + a,h + azh2 + azh + -. The value of N½(h) using Richardson's extrapolation…

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Q: If the Wronskian of the DE : x³y" + xp(x)y' + 3y = 0 is μ(x) er. Find p(x) = -2x O 2x e O -2 x4 O…

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Q: Exercise 9.3.12. In the formula n = x – y = (x – y)( +y), what is the smallest possible value for a…

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Q: 4. Prove that if f(x) = agx" + a,x"-1 +..+ an-1x = 0 has a positive root x = xo, then the equation…

A: Note that f0=0 and that fx0=0. So using mean value theorem there must be some 0<λ<x0:…

Q: Find the x values of the two critical points of: 3(x- 1)23 6x – x² - 5 if x > 2. if x < 2 f(x)%3D X…

A: we have the given function is ;  f(x)=3(x-1)2/3if x≤26x-x2-5if x&gt;2

Q: 5. (a) Solve the cauchy problem given by x²ux + xyuy + xu = Xx-y where (x, y) + (0,0) if u(x, xln x)…

A: Given x2ux+xyuy+xu=x-y where x,y≠0,0 if ux,xlnx=-lnx, x&gt;0

Q: 4.8. Show that Dirichlet's function 1, if x is a rational number, { otherwise f(x) = 0, is not of a…

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Q: Suppose the function f(x)=3x^4+ax^3+bx^2 has a critical point at x=2, for which f(2)= −80. Determine…

A: Suppose the function f(x)=3x4+ax3+bx2 has a critical point at x=2, for whichf(2)= −80. Determine a,…

Q: Suppose f(x) = N₁ (h) + a₂h + a₂h³ + ... The value of N₂(h) using Richardson's extrapolation is: O O…

A: Replacing the h by h/2 and deduce the formula

Q: 2 In x 1 e* 2 1 In 2x -1

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Q: 12. Find the general solution using Reduction of Order: a) y" = In x

A: you have asked multiple questions in a single request we would be answering only the first question…

Q: Given that X has pmf f(x)=1/(2)^(x+1) , x=0,1,2,... and et<2 , use mgf technique to…

A: Given: X has the pmf as follows, f(x)=1/(2)^(x+1) , x=0,1,2,...et&lt;2

Q: Formally prove the T(n) = (5n – 4)° is O(n³). Show all work and all steps to receive full credit.

A: Order: T(n)=(5n-4)3 T(n)=(5n)3-3.(5n)2.4+3.(5n).(4)2-(4)3T(n)=125n3-300n2+240n-64

Q: 2. Consider the function f(x)=sin(x) on the interval [0,1]. Use theorem(3.3) to determine the step…

A: Given function is f(x)=sinx Interval is [0,1] Solution a: For linear interpolation N=1 So…

Q: Consider the following NLP problem: f(æ) = -x+ 2x1 – 122 – T2 +1 max s.t. I2 > 1. Determine if x =…

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Q: d If g(x)=- S* [e²-In (21+1) ]dt, find g(x). -2 2 (2x + 1)2 4 B (2x + 1)² Ⓒe-In(2x+1) 4 (2x + 1)² di…

A: A function is given by, gx=ddx∫-2xet-ln2t+1dt. The objective is to find the second derivative g''x…

Q: Theorem 8.3.3 If X ~ t(k), then E(X2r + 1) = 0 and µz, = k"- (2r – 1)(2r – 3) ... .1_ (k– 2)(k – 4).…

A: Answer: For the given theorem, The proof is:

Q: Suppose f(x) = N(h) + a,h2 + azh + agh +.. %3D The value of N,(h) using Richardson's extrapolation…

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Q: 7] The Wronskian of the functions 1, x, and e -2x is (а) е' -2x (2x+1) (b) 4e-2x (c) -e(2x +1) (d)…

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Q: 5. Let a > 0, find the "generalized solution" to a" + 3x' + 2x = 8(t – a) with x(0) = 1 and æ'(0) =…

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Q: For f(x) = 1 –x2on [−2, 1], do the hypotheses and conclusion of Rolle’s Theorem hold?

A: Rolle's Theorem: Rolle's theorem states that if a function f is continuous and differentiable on the…

Q: The Newton-Raphson method fails if Of" (x₂) = 0 Of (1₂) = 0 O f' (1₂) = 0 0 = (º¹),f O

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Q: Given G(x) = jin(2t+1)dt , find G '(x). O In(2x + 1) O In(2x + 1) - In3 O In(2t + 1) + C 2 2x+1

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Q: 3. If f(x) = -x, g(x) = 2x + 1 and h(x) = x2, find fogoh 1 point %3D %3D О А. х2 - 1 О в. 2х2 - 1 О…

A: We use fogoh=f(g(h(x))).

Q: Solve the following linear independent and Wronskian. 2. Are the functions y, = e* and y2 = x…

A: Here we have to determine whether the given functions are linearly independent.  We have,  y1=ex…

Q: The function f(x) = (6x+5)e^4x has one critical number. X= ?

A: Proceed as shown below...