Let F be defined on R x R to R by F(x, y) = x² – y. Show that F is C' on a neighborhood of (0, 0) but there does not exist a continuous e defined on a neighborhood of O such that F (p(y), y] = 0.

Q: If f(t) is a continuous function on the interval a, b), then the integral | f(t) dt is defined to be…

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Q: 2. Let f and g be integrable functions such that f(x) dx = 6 and g(x) dx = –2. Using the Linearity…

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Q: Show that F(x, y) = x² + 3y is not uniformly continuous on the whole plane.

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Q: f is not integrable on [0, 1].

A: It is given that f:[0, 1]→ℝ by f(x)=5x  if x∈ℚ0    if x∉ℚ We know, if lower integral and upper…

Q: If f is bounded and integrable on [a, b] sch that | f(x) | 0). 9.

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Q: Let f be the function defined by S1+x, when 0<x<}, }<x<1 0, when x-1. f(x)= Show that fis integrable…

A: Recall the Result/Theorem A bounded function f which has only a finite number of points of…

Q: Use the definition of the integral n | f(x)dx = lim f(x;)Ax a i=1 (where Ax = (b – a)/n and x; = a +…

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Q: Consider the following theorem. Theorem If f is integrable on [a, b], then = lim f(x;)Ax n- 00 la i…

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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: f(z) = Im z is continuous 0 صواب ihi O

A: True

Q: In Exercises 23-28 explain why fis continuous. 26. f(x, y, z) = sin (xyz – 1) 27. f(x, y, z) = ln(e"…

A: The function is continuous if all partial derivatives exist.

Q: Suppose that f(x) > 0 on [0, a], ƒ(x)< 0 on [a, b], ſo“ f(x)dx = 7, and S" f(x)dx = –3. Evaluate the…

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Q: Verify the Divergonce Theorem for F=(2x-x)+x*y} -aż? taken over the bounded by x -o, X=1, y=0,…

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Q: Let f(x, y, z) = ln(x² + z) + y²e®z – cos(2yz). Find fzyn. Zyx ·

A: Given function is  f(x, y, z)=ln(x2+z)+y2exz-cos(2yz)

Q: Define the map f : R² → R² as f(x, y) = (e" cos(y), æ²y + sin(y)) . (a) Show that f is…

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Q: 3) Let F = (y+ev=,8x+ cos(y*)) Use a theorem to compute ſ F •dT where c is the boundary of the…

A: Given that  F→=y+ex,8x+cosy3 To compute ∫cF→⋅dr→ where c is the boundary of the planar region…

Q: Evaluate the line integral [F-ds where F = 2xyi +(x-y³)j and C is (a) y =x from (0,0) to [10] (1,1)…

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Q: f(z) =|z| is continuous in C 0 صواب ihi O

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Q: Suppose that f(x, y) = x/y³ + 1 on the domain D = {(x, y) | 0 < x < 2, x < y < Then the double…

A: The double integral with respect to x first and then with respect to x .   The picture below shown…

Q: If f is bounded and integrable on [a, b] such that | f(x) | 0). *<k lb-al

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Q: If f is bounded and integrable on [a, b] sch that 1f(x) | 0).

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Q: If f(x, y)is integrable function on the rectangle R = [a, b] x [c, d] where [a, b] on the x- axis…

A: In the question it is asked to show how the two double integrals are equal to each other.

Q: Suppose that f and g are integrable and that -1 f(x) dx = 5 h(x) dx = 4 7, dx = %3D Use the rules to…

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Q: Let f : [0, 1] → R defined by if æ = 0, f (x) - L otherwise is Riemann integrable on [0, 1].

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Q: Compute the Wronskian of a set of functions {vı, y2} = {e* cos(/), e* sin(v¤)} cos (Væ), e* sin(va)}…

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Q: f(2) = u(x, y) + jv(x, y) v(x, y) = sin x cosh y f (0) = 0.

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Q: (a) Show that the limit xżycos y lim (x.y)→(0,0) 2x+ +y² does not exist. (b) If u = x cos y+ ye»,…

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Q: Let f(x, y) = xcosy - ye*y. fry at (1,0)is equal to: -1

A: See the attachment

Q: - Let f(z) = cosh z. (1) Find the image of S := {z = x+iye C: x < 0,0 < y < } under f.

A: Follow the steps.

Q: Since f(x) =1/x is continuous on (0, 1), f(x) is integrable on (0,1).

A: This is a false statement.

Q: A necessary condition for the functional defined by X2 = dx f(x, Y(x),Y'(x)) to have a local minimum…

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Q: Let f be the function defined by SI+x, when 0<x<!, }<x<1 0, when x=1. f(x)=: Show that fis…

A: We know that  A bounded function f which has only a finite number of points of discontinuity in…

Q: If f is bounded and integrable on [a, b] such that | f(x) | 0). *0).

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Q: Define T: (C¹ [0, 1], || · ||…) → (℃ [0, 1], || ||) by Tf = f'. Show that T is not continuous. Is…

A: T:C'[0,1], .∞→C'[0,1], .∞ T(f)=f' Let f(x)=xn then T(x)=nxn-1 T(x)=nxn-1 as n→∞,   T(x)→∞ So T is…

Q: Consider the following theorem. Theorem If f is integrable on [a, b], then F(x) dx = lim s fx,)Ax n-…

A: Given- Consider the following theorem. Theorem  If f is integrable on a, b, then ∫abfx dx = limn→∞…

Q: Let f(x) = (cosh x)² be defined on (0, 0), find %3D d 1/2) . dy

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Q: (a) Show that the limit x*ycosy lim (x.y)-(0,0) does not exist. (b) If u = x cos y+ye", wherex=s+2t…

A: The given function is f(x, y)=x2+y23 at point (0, 1) and direction u→=35, -45. The direction…

Q: 1) If f(x)=x2, find f(x+2)-f(2) .what is bijective h mapping? 2) Does the integral J, dx exist?

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Q: Define f : [0, ∞) → [0, ∞) by f(x) = Vx. (a) (b) Prove that va +b 0. Prove that f is uniformly…

A: We have to prove that   √a+√b≤ √(a + b) for all a, b ≥ 0.

Q: Suppose that f and h are integrable and that: L f(x) = -1, f(x) dx = 5, h (x) dx = 4 , Find: %3D %3D…

A: Answer and explanation is given below..

Q: Let f : C[0,1] → P be given byf (x ) = x (1)Vx e C[0,1]. Show that f is continuous w.r.t the supnorm…

A: f is continuous with respect to sup norm. Let xn be a sequence of continuous functions in 0,1 such…

Q: Consider the following theorem. Theorem If f is integrable on [a, b], then f(x) dx : lim f(x;)Ax n-…

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Q: If f is bounded and integrable on [a, b]such that | f(x) | 0). *0).

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Q: Consider the following theorem. ь — а If f is integrable on [a, b], then ['rx) dx = lim Kx;)Ax,…

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Q: Let f be the function defined by fx)=1+x, when 0<x<}, }<x<1 0, when x . Show that fis integrable…

A: We know that  A bounded function f which has only a finite number of points of discontinuity in…

Q: Evaluate the line integral | F•ds where F = 2xyi +(x- y²)j and C is (a) y= x from (0,0) to (1,1) (b)…

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Q: Suppose that f is continuous on [0, 1], ƒ(0) = 0, and f'(0) exists. Prove that the Lebesgue integral…

A: This is a question of measure theory.

Q: Consider the following theorem. b - a If f is integrable on [a, b], then f(x) dx = lim n- 00…

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