Let u(x, y) = √√|xy|, (x, y) = C. Show that ou and ou both exist at (0,0), but u is əx ду not real-differentiable at (0,0).

Q: Let  f(x, y) = sin(15x2 + 3y2) Calculate the quadratic approximation of f at the point (0,0)

A: Given that,  fx, y=sin15x2+3y2 The quadratic approximation of the function fx, y at the point x0, y0…

Q: Find the plane tangent to the function z = √y at the point (5, 16). L(x, y) =

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Q: What is frz, fyy, and fyz of f(x, y) = In(y²")?

A: Given:           f(x,y)=lnyx2          fx=∂∂xlnyx2         Treating y as constant variable…

Q: The function f(x, y, z) = x² − y² − z ² increases in the direction 〈六〉 (1, 1, 1). at the point

A: Please check step 2 for solution.

Q: Find the f(z) = u(x, y) + j v(x, y) for which X b) u = e cosy most general analytic function

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Q: Exercise 4. Show that the function f is not differentiable at (0,0) f(x, y) = x³ + y³

A: We have given that ,  fx , y = x3 + y33 Here , ∂f∂x = 3x23x3 + y323 ∂f∂x = x2x3 + y323 At , x , y =…

Q: 2. Determine the directional derivative of o(x, y, z) = re" + y² + xyz at the point (2, 0, 3) in the…

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Q: Use the definition of differentiability to show that f(x, y) = y² – xy+5x is differentiable…

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Q: Let f(x, y, z) = 5x³ - y³ + z². Find the maximum value M for the directional derivative at the point…

A: Our objective is given below:

Q: Assume that f(r, y) is a function given by if (r, y) # (0,0), f(r, y) = 0 if (r, y) = (0, 0).

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Q: z= f(x,y) ; x= r-s1 , y= r2s and zx(0,-1) = -24 ; If zx(0,-1)=-18, what is the value of the…

A: Given that z=f(x,y),     zx(0,-1)=-24 and  zy(0,-1)=-18x=r-s and y=r2s We have to find zr(0,-1).

Q: Let f(2)= iz³ + cos(z²) z² +1 i. Find f'(z). ii. Evaluate ƒ'(–2i) and simplify. iii. State the…

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Q: Let f(x, y) = x/y. Then the directional derivative of f in the direction of √2 point (1, 1) is ○ (1,…

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Q: Show that the function given by f(x,y)= x² + 3xy is differentiable at every poin using the…

A: Introduction: Distinctiveness (differentiability) implies continuity. If a function is…

Q: Find fxxx for f(x, y) = Sin() at (π, π)

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Q: 5. Find the linear approximation of f (x, y, z) = x³y² + 22 at the point (2, 3,4). Use this to…

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Q: Let f(x, y) = 6x – 2ry and the point M(5, 9). Among all unit directions, there is a unit direction…

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Q: Let f(x, y) = 2x² + 4y² + 2xy+ 10x + 3y + 9 Find D = frr · fyy - i

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Q: f[x'+zy2,y²2x2], then prove that [y?-zx] (a?-yz] dy [z-xy] du -o. If

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Q: Show that the function g(x, y) = is not continuous at (0,0). xy 2x² + y²¹ 0, (x, y) = (0,0) (x, y) =…

A: g(x,y)=(xy/(2x2+y2), (x, y) not equal to (0,0)               0                 ,(x, y) = (0,0)

Q: If a field K is algebraically closed, then every polynomiel with co efficients in k has a root in K.…

A: A field K is said to be algebraically closed if every polynomial with coefficients in K has a root…