2. Consider an operator of the form (Lf) (x) = Ef(x;)g;(x) where a O for all a E [a,b] and for all i.

Q: 3. Find the maximum and minimum values of a 1 āz - on |z| ≤ 1 (where a with la| < 1 is fixed)

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Q: A. Expand, that is, perform the indicated operations on the differential operator D. 1. xD(D + x) 2.…

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Q: 0 0 0 1 0 0 0 1 0 ) Let A =

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Q: (2) Let F(x, y) = (cos(y)+2x, -x sin y+1). Show that F is path independent by finding %3D a…

A: For the given vector function to be independent, it must be derived from a potential scalar function…

Q: 59. Prove the general formula iwollo oi ons dx 1 x - In x – b + C %3D (x – a)(x – b) a – b - - where…

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Q: 22 dt at x = 4. ind the linearization of f(x) = 7 - 4 +t

A: Consider the given function fx=7-∫7x+3224+tdt

Q: If L(x)=mx+b is the linearization of f(x)=2x³ - x +3 at x-1, then m= Using L to approximate 2(1.01)…

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Q: 3. (a) Show all functions satisfying u"(x) = 0 are linear by integrating the expression of u"(x) %3D…

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Q: Consider the function f(x) = In(x). %3D 2.1 Show that x(lIn(x) – 1) is an antiderivative of the…

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Q: If the Wronskian of the three functions r.ev) is equal to zero, then the function y satisfies the DE…

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Q: The expanded form of the differential operator D(xD-1) is A XD²+1 B XD² C XD²-1 (D) XD²-D

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Q: The expanded form of the differential operator D(xD – 1) is A xD²-1 B xD? - D xD2+1 (D XD2

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Q: (a) Find if y = y (ln r). (b) The function f(z) (In r) is an invertible function (you do not need to…

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Q: 3) Calculate Wronskian and evaluate for which values a, b ER ne following sets of functions are…

A: Given functions are: 1) sinax,cosbx 2) sinhax,coshbx Calculate the Wronskian and evaluate for which…

Q: 6. Let 1 f (x) - cos x + sin(2x), x E [0, 1] 2 Find the least squares polynomial approximation of…

A: Let fx=12cos x-sin2x, x∈0,1. To find quadratic fitting  fx≈P2x=a0+a1x+a2x2 Consider, s0=∫011dx=1

Q: A continuous r.v. F(x1= === 2 × -IN 15 X has CDF X Co OBX C1 18X 16 17 **** 18

A: Given: Fx=0            x&lt;012x        0≤x&lt;1k            x≥1

Q: Prove that for the gamma function [(x) = ftx-¹e-tdt that for n a positive integer then I(n) = (n-1)!

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Q: If two functions f(x) = x2 - x and g(x) = x3- x2 +x 1 combine to form h(x), where h(x) f(x) find the…

A: Find your answer below

Q: d x"J,(x)=-x"Jns(x) d, n+1 X.

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Q: If the Wronskian of the three functions ex v is equal to zero, then the function y satisfies the DE…

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Q: Define f : X → K by f(x) = x'(1), x € X. Again, the linear functional f is not continuous. By 6.4,…

A: Given: f:X→K by fx=x'1, x∈X To prove f is not continuous using the given corollary.

Q: 7) Suppose w=rt+rs' and r= tan-' (2x² +3y), s=x²y' +3y and t = x²y² .Use the chain rule (from…

A: Given w=r2t+rs3r=tan−1(2x2+3y)s=x2y3+3yt=x2y2

Q: The expanded form of the differential operator D(xD -1) is XD² – 1 A В xD² (c) xD² +1 C D xD² – D

A: To find the expanded form of differential operator, D(xD-1)

Q: 1 1 Select all the antiderivatives of f(x)=-+ %3D x² +1

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Q: 6. Find the Wronskian for the set of functions {e¬x, xe¯*, (x + 1)e¬*}.

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Q: 8. Given for C[0,1] the inner-product function (f,g)=[f(x)g(x)dx , find ||f|| if f(x) = x+1.

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Q: 8. Prove that the differential operator M: C² [a,b] → C[a,b] defined by d²u du Mu= 2+3u dx is a…

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Q: let f(x) = { ax2+x+1 if x is less than or equal to 1                 { bx if x >1   find a and b…

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Q: Find for x = x(t) defined implicitly by 25+25 x In(t) = e25xt Also evaluate dt at (t, x) = (1,1).

A: Given,25 + 25xlnt = e25xtTaking 'log' with base 'e' on both side, we getln25 + 25xlnt = lne25xtln25…

Q: 1. Determine whether the functions f1(x) = cos 3x, f2(x) = x, and f3(x) = cos² x are linearly…

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Q: 2/3 The integral S(x+2y)eu-)drdy can be written and evaluated after using the transformation u =…

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Q: 2. Give an example of a regulated function f on [0, 1] such that f(x) > 0 for all æ e (0, 1), So…

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Q: If G(x) is an antiderivative of g(x) and F(x) = G(x)- 5, then F(x) is also an antiderivative of…

A: See below the answer

Q: 3) Calculate Wronskian and evaluate for which values a, b, c e R the following sets of functions are…

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Q: 10

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Q: 2n Jnsi (x) = " J,(x) – J(4)

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Q: 1 Justify the Cauchy-Riemann equations for f(2) = e² where f(z) is holomorphic.

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Q: д. L. = a(x, y)+b(x, y) +c(x, y). – d(x, y) -

A: Proof: Consider V be any vector space of all functions of two variable x  and y whose partial…

Q: (11) Let G(x) = 1 In(1+t²)dt. (a) Find G(1).

A: Solution Given G(x)=∫1xπ ln(1+t2)dt.We need to find the G(1)

Q: Let z(x, y) be the function implicitly defined as the solution to 2x + y + z + sin(xyz) = 4 + that…

A: ddxsinnx =cosnxddxnx =ncosnx

Q: Let f1(x, y) - e cos(5y), f2(x, y) = x² – 25 y². Find Vf, and Vf2, and show that %3D (fv) = 0 for…

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Q: 3 Evaluate lim et dt. e-t dt. 20 O a) -2 O b) 2 e O) 2 O d) 2 e e) None of these

A: Giveen query is to find indicated limit.

Q: Find T; (x): Taylor polynomial of degree 5 of the function f(x) = cos(x) at a = 0. T3(x) = %3D

A: f(x)=cos x at a=0, we have to find Taylor polynomial of degree 5.

Q: Find a linear differential operator that annihilates the given function. f(x) = 3x2 - x sin 3x +…

A: Given function is fx=3x2−xsin3x+x2e−xcos4x We have to find linear differential operator that…

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: If f(x) - g(x) = 1, then f and g have the same antiderivative. %3D O True False

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Q: 14. Use the Cauchy-Riemann Equations to show that f'(z) does not exist at any point. f(2) = Z b)…

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Q: Verify Young's Theorem z = In(2x + y) y? x2 +e(x+2y) y

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Q: Find the linearization L(x) of y = e6« ln(x) at a = 1. L(x) =

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Q: y = In[(2x – 1)(x + 3)] with Respect to x by using the chain rule.

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