(d) Demonstrate mathematical reasoning to prove that that the real-valued function х2 — у? u(х, у) %3D х3 + 2х2 - 3ху? 2у2 + (x² + y2)2 is harmonic. Construct a harmonic conjugate of u(x, y) and write down an entire function f(z) such that Re(f(z)) = u(x,y), where z = x + yi. In your working, you may use the integration formula a? – 3y? dy J (a² + y²)³ y + C (a² + y²)² for any real constant a.

Q: 9) let f(x) = (x +2)² - 12. a) Sketch a "good enough graph" of f. b) Determine the number of…

A: (a) f(x)=(x+2)2−12

Q: Find the domain and b) Sketch the domain of f(x, y)= In(4x² – y) -5 -2 -1 3 4 5 x

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Q: The domain of

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Q: In(2 - a) x² + y² – 16' Determine the domain of the function f(x, y)

A: The domain of a function is the values of the variables of the function that result in a defined…

Q: 2. Sketch the domain of the following functions: a) f(x, y) = b) f(x, y, z)= 2-x-y - eª %2 In(4-x² -…

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Q: Determine whether the function defined by = 2x1y₁ - x1y2 - x2y1 + 2x2y2 is an inner product on R².

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Q: Exercise 5: If f(x) = x5 + 2x³ + 2x, check that f is injective and find (f-1)'(-5).

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Q: Use intermediate value theorem to prove that a cubic  x3+ax2+bx+c=0 for at least one real value of…

A: Intermediate value theorem: If f is function which is continuous at every point of the interval…

Q: Sketch a graph of the function f(0) = 2 sin(-20- - -Зл -5л/2 -2л 0-17) -3/2 -Л +3 -п/2 6 5 4 3 2 1…

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Q: Consider the function f(x) = 2x - 1/ x+3…

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