Consider two functions fi: [0, 1]R and fa: (1,2) R defined by : fi() = 4x-1 and f(*) =6-2r. Sh(z) , if 0sa S i Sa(x) , if 1 | fS(x)dz]. Let f(x) = The function is 4. None of the above O 1 2 4 O O

Q: Let f: R → R be differentiable. Suppose that the first derivative f' atisfies |f'(y) – f'(x)| < L\y…

A: Given, f'y-f'x≤Ly-x. To show fy-fx-f'xy-x≤Ly-x2.   Consider , fy-fx-f'xy-x=fy-fxy-x-f'x=f'x-f'x=0.…

Q: Let f(x) = 1/n for x = m/n with (m, n) = 1, and f(x) = 0 for irrational x. For irrational u € (0, 1)…

A: The function f defined here assigns a value of 1n to rational numbers mn where m and n are coprime…

Q: Let f(x,y, z) = xtan (Vyz)- sin(e*) , evaluate zy %3D дудхда

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Q: e /// 2 Evaluate 2xz dV where E {(x, y, z) |1 ≤ x ≤ 4, r ≤ y ≤ 2r, 0 <z<x+3y}

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Q: 1 Let f (x, y) =x²y*sin (- x²+y* when (x, y) ± (0,0) and f (0,0) =0. determine if f is continuous in…

A: We have to show given function is continuous at origin

Q: Let h(x, y): = x5 + y sin²x x² + y² 0, if (x, y) = (0,0) if (x, y) = (0,0) (a) Show that h is…

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Q: (C) Using Cauchy-Riemann theorem, show that f' and f" exist everywhere, then find f' and f", where…

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Q: Define h(0,0) in such a way that h is continuous at origin       h(u,v)=ln(nu2-u2v2+nv2/u2+v2

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Q: Q2: Expand f(x,y) = [sin (x y³)]² about (5, 1).

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Q: If U(x, y) = -e-Y sin x andf(z) = U(x, y) + iV(x, y) is analytic, then f(z) = %3D O None of them…

A: Solution

Q: Let z be defined implicitly as a function of r and y by the equation əz 4+ In (ry²) - cos (xy - z) =…

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Q: (0 if xE Q (1 if x E I If f (x) = show that f is not Reimann Integrable

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Q: 6. Show that f(x) = sin x is uniformly continous on R.

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Q: If f(x,y,z) = 2x² − yz + xz², where x = 2 sint, y = t² − t + 1, and z = e−³t, find f't)

A: from chain rule of derivaties…

Q: If f =x-e" and F=x'yi-2y'zj + xy'z'k, find: div(F x grad(yf )) at (4,1,-2)

A: The function                     Then, we have to find

Q: Let h(x, y) = x5 + y sin²x x² + y² 0, { 25 if (x, y) = (0,0) if (x, y) = (0,0) (a) Show that h is…

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Q: Find the continuous lest squares approximation of degree 2 for f(x) = cos(xx) on [0,1]. %3D

A: Let P2(x)=a0+a1x+a2x2 is the continuous least square approximation of degree 2 for the function…

Q: Find numbers a and b, and c, so that f is continuous and differentiable at every point. 4) x- + C,,…

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Q: Is the set {sin nx , cos mx} an orthogonal set with respect to the inner product on the set of all…

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Q: Find o r(t)dt if r(t) = (cos t, sin t, 3t²). %3D O A.O B. (0, 2, ³) OC (0,0, x³) O D. (0, 2, 7) O E.…

A: We have to choose the correct answer from the given option

Q: Show that the function f defined on [0, 1] by the conditions 1 f(x)=2rx when 7=1, 2, 3,...... És…

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Q: Let F(x, y, =) = (6x° In(5y² + 9) + 4=5) i + ( 10yx + 7:)j + (20xz4 + 7y – 37 sin 72) k 5y2 + 9 and…

A: To evaluate

Q: Use the Cauchy-Riemann Equations to show that f(z) = z Im(z) is only differentiable at z = 0 and…

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Q: A bounded, 3B > 0 such that |f(x)| < B Vx € (0, 1), continuous function on (0, 1) with unded…

A: We will find an example