Let f:[0,1][0,1] be defined by f(x)== 2-1 24 Then fis a Riemann integrable function such that (a) [f(x) dx = ² 3 (b) ≤ / < f*°ƒ (x)} &x < ² 2 (c) ff(x) dx=1 for x E 2 (d)} } }

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: Show that the function f defined by (r, y) = (0,0) f(x, y) = { 22 + y (r, y) # (0,0) x + y

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Q: (b) Verify that the following function is harmonic u(x, y) = −e sin x - 2e-siny. Find a harmonic…

A: Here we have to verify that the function is harmonic and then find the harmonic conjugate.

Q: Define f : R2 → R by letting (y² – x)² (y4 + x²) f(x, y) = if (x, y) # (0, 0), and f(0,0) = 1. ||…

A: The given function is f:ℝ2→ℝ defined as: fx,y=y2-x2y4+x2if x,y≠0,01if x,y=0,0 Claim: The given…

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A: Definition continuity of a function f(x). A function f is continuous at x=a when: (i)  f(a) is…

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A: We have to evaluate by Green's theorem.

Q: the function f is not differentiable at x=0.

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Q: Show that the function f defined by (x, y) = (1, –1) f(x, y) = r? + y (x, y) # (1, –1) x + y is not…

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

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Q: Please

A: Given function is

Q: { 2xy? 9x2 + y4 (x, y) # (0,0) (x, y) = (0,0) f (x, y) ·

A: Topic = Limits and  Continuity

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A: The given functions, 1. y=tanx6 2. y=x5+9 3. y=e7x We have to write the functions in composite form…

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A: Solution is in next step

Q: 7. −ydx + (x + √xy)dy = 0

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A: We have given that ,  are both infinitely differentiable.We need to prove that , We will use…

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A: We will find an example

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Q: sin(x² + y?) x2 + y2 (x, y) # (0, 0) Let f(x, y) = %3D 1, (x, y) = (0, 0). Show that f is continuous…

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