Theorem: (i) If Fa is a closed set for each a er then Fa is closed. aer (ii) If Fk is a closed set for k = 1,2,..., n then U Fx is closed. k=1 Proof:Exercise

Q: Solve this Question by using Laplace y" + 4y' + 5y = 125t2 y(0) = 0,y'(0) = 0 %3D

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Q: Show that the equation e -x+2=0 has a root between 2 and 3. Use Fixed-Point method to find the value…

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Q: Solve the following problem in co, 00) x (o.0") U(x,0) = J(x) for XE(0,0) u(o,t)=0 for tE [O, oo).…

A: Given that      ut-∆u=0     in 0,∞ ×0,∞⇒  ut=∆u,        in 0,∞ ×0,∞     (1) Let…

Q: Test the series below for convergence using the Ratio Test. (- 1)"92n+1 Σ (2n + 1)! n=0 The limit of…

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Q: 18. A manufacturer wants to make cylindrical cans. Each can must be able to hold 1200 cm³ of liquid.…

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Q: 7. A commercial fishery is estimated to have carrying capacity 40 tons of salmon. Suppose the annual…

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Q: Question 3. Let S(t) - 4+ 3t,0 <t<3 and let the half-range sine series representation of its odd…

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Q: Examl-Section 1: (a) Use appropriate Lagrange interpolating polynomials of degrees one (n-1), to…

A: Numerical analysis, Lagrange interpolation.

Q: Let A = [1 5 −4 −3 0 0 1 −2 1 0 0 0 0 0 1] and w = [120−2−1] (a) Is w in Nul A? Justify your…

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Q: al equation sin(t) with y( solution for this hen t belongs to ||

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Q: x' + y = 0 y' – x = 0 x(0) = 1 y(0) = 0.

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Q: Which of the following rings are principle ideal domains? Select all that apply: O The ring Z/34. O…

A: This question related to abstract algebra's topic principle ideal domain.

Q: Find the solution. dy y =xy2 dx 3x (A y = c – x3 B y = Cx – x 3 (c) y = c+x3 - 3x (D y = c+x3 3.

A: We have dydx-yx=xy2 Consider: 1y2dydx-yxy2=x⇒1y2dydx-1xy=x                  ....(1) Let 1y=t This…

Q: Suppose f is differentiable at a and f(a) + 0. (a) Prove |f| is differentiable at a by using that…

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Q: Let f : A B be a function. Prove that f is surjective if and only if, for every pair of functions g…

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Q: rind the inverse Laplace transform of e 7s F(s) = s2 + Os – 9

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Q: sequence 519 57 . What is the formula of the general term of this sequence, or a,? What is the value…

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Q: The Hermite polynomials Hn(x) are special functions that are extremely im- portant in quantum…

A: For first proof differentiate exponential function and for second part Differentiating series

Q: Determine whether the infinite geometric series is convergent or divergent. If it is convergent,…

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Q: For the following problems, use the Maclaurin series for sin r, which is given by: (-1)*r2k+1 (2k +…

A: The given problem is to find the Maclaurin series of sin(x2) and evaluate the definite integral of…

Q: Q2/ Defined x as 100 linearly spaced values between -2n to 2n. Defined y1 and y2 as sine and cosine…

A: To define x as 100 linearly spaced values between -2π to 2π . And define y1 and y2 .

Q: For any two sets A, B define A ⊗ B as the elements that are unique to A and unique to B. Is, A ⊗ (B…

A: Given that A and B are two set,  Define A ⊗ B as the elements that are unique to A and unique to B.…

Q: Carbon dioxide emissions are those stemming from the burning of fossil fuels and the manufacture of…

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Q: z + 16 f (2) = (z – 3) (z – 16)? Let C be the circle |2| = 9, in the anti-clockwise direction.…

A: It is given that the complex function is fz=z+16z-3z-162. We have to find the residue of fz at z=3…

Q: (a) Suppose f'(x) > g'(x) for all x, and that f(a) = g(a). Show that f(x) > g(x) for x > a and f (x)…

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Q: (Euclidean Algorithm) Let n be any natural number and m be a positive number. Then show that there…

A: The given problem is to prove the given statement of Euclidean Algorithm. Given statement, Let n be…

Q: I. Find the first three terms of the sequence whose nth term is given by the formula. Show the…

A: First three terms of the sequence

Q: Find the derivative of the function. y =    8x2 − 5 2 − x   3    y' =

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Q: b) The partitioned matrix defined is as R= I4 314 fartitioned it exists Find in the fom of matrix.…

A: Given partitioned matrix is  R=FI403I4

Q: Q1. Use Cramer's rule to solve for y without solving for the un- knowns x, z, and w. 4x + y+ z+ w= 6…

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Q: Let I= 0. ST 10 x²eWdxdy. y a) Sketch the region of integration. b) Setup the integral I in the…

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Q: Prove that (R x Q) N (Q × R) = Q x Q

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Q: - J. (x) – ||

A: We know that  Jp-1(x)+Jp+1(x)=2pxJp(x) Put p=1

Q: [Question 6] (17 points) Consider the following data set 3. f(x) -14.25 5.88 1.75 0.88 10 Use Least…

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Q: r. = x (n) |1,

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Q: ou are given a transition matrix P. Find the steady-state distribution vector. [ 3/4 1/4 %3D 4/5 1/5

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Q: B.2) Logic Equivalence 4 points Let p and q be two propositions. Use the laws of logic equivalence…

A: To show the given propositions are logically equivalent we will use the fact that two propositions…

Q: Exercise 14.3.25. Write the symmetries of a square in cycle notation.

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Q: Suppose that f (x) = x·g(x) for some function g that is continuous at 0. Prove that f is…

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Q: What is the time domain relationship for a functions with the following s-domain functions: s+6 F(s)…

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Q: Let T : M2×2 → M2×2 be a mapping defined by T (A) = AT − A for A ∈ M2×2. (a) Show that T is a linear…

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Q: Sketch and identify the graph of the equation x² + y² – 2y – 3 = 0 in 3D.

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Q: Consider the initial value problem: y'. y+t=-7, 0s t s 0.6, y(0) =-7 The approximation of y (0.6) by…

A: Intial value problem by Euler's method

Q: Find the inverse of the ff. using Gauss-Jordan Method 1 2 2 а.) А%3 1 1 3 2 1 1 b.) A= 2 3 1 1 3. 1.

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Q: #1. A function, F: R × R →R × R has been defined as follows: F(x, y) = (3y – 1, 1 - x) for all (x,…

A: Ordered pair (p,q)=(r,s) imply p=r and q=s.

Q: 3 Let A be the following linear functional on C([0, 1]): xf (Va) da, f e C([0, 1]). Prove that A is…

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Q: a) Consider the sequence 19 57* 12 27 }. What is the formula of the general term of this sequence,…

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Q: a) If A and B are each countable sets, show that AUB is countable. (Hint: For each set, consider a…

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Q: = eax %3D

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Q: Consider that, we have the following initial value problem: y' = e-y, y(0) = 1, 0<t<1 ; Use…

A: The given initial value problem is y'=et-y with condition y0=1and 0≤t≤1. Also step size h=0.5 To use…