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 origin.

Q: given f(z)=u(x,y)+iv(x,y) is an analytic function where u(x,y)=x^3-3xy^2. find f(z)

A: We have to find .

Q: ay or dx. 2x + y = 7x - 6 to find y a)

A: Since you have asked  multiple questions in single request  so we  will be answering only first…

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: Is f(x)= Vx + 1 continuous at x == -1

A: Draw the graph and check continuty with help of limit of function.

Q: Let f:A-B and X,Y be subsets of B. Then f-¹(XnY) = f'(X) nf-¹(Y). True False

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Q: Is there a function f(x, y) such that fx : one. If not, explain the reason = e* cos y and fy : = e*…

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Q: what is f'of x

A: Find f’(x).

Q: Use Green's Thoerem to evaluate F dr. where F(r, y) = (-3Vx² + 2, 7 tan '(x)) and C is the triangle…

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

Q: Show that A Function with a Single Point of Discontinuity 2xy x² +1,²² 0, is continuous at every…

A:

Q: Let f(x, y) = x sin ry. Find: fyy (x, y) fyz(x, y)

A: f(x,y)=xsinxyfx=sinxy+xycosxyfyx=xcosxy+x(cosxy-xysinxy)=2xcosxy-x2ysinxyfy=x2cosxyfyy=-x3sinxy

Q: Let g(u) be a differentiable function, and let f(x, y) = g(x^2,y^2)   Find the direction of…

A: Given that gu be the differential function and let fx,y=gx2,y2 We have to find the direction of…

Q: Express the following function in the form f(z)=u(x,y)+iv(x,y) 1. cos(z^2)

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Q: Let f(x,y)=tan^-1(y,x). Find and simplify fx and fy. Then find and simplify fxx, fxy, and fyy.

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Q: Let f(x, y) = e²x* %3D Evaluate f(3, 5)

A: Given :   f ( x, y) = exy x4   point ( 3,5)

Q: Assume that f"(x) is continuous on [0, 1] an ƒ(0)=1,ƒ(1) = 3, ƒ'(1) = 7. Find [xf"(x) dx.

A:

Q: Assume that f : (0, 1) → R is a differentiable function with the derivative f' increasing on (0,…

A: Given f:0,1→R is a differentiable function and derivative of f is increasing in 0,1 that is f'x>0…

Q: Assume that f(x, y) is a function given by if (r, y) # (0,0), f(x, y) 0 if (r, y) = (0, 0). Page 1…

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

A: Topic = Limits and  Continuity

Q: 4. Let f(x, y)=2x*y' . Find f,(x,y)

A:

Q: Determine the two-variable function F(x, y) if Fy (x, y) = y sin(xy) + x given that x^2 + πχ2 F(x₁ -…

A: Given That: Partial derivative of a function with respect to y is Fy(x,y)=ysin(πxy)+x with the…

Q: ·b f(b) − ƒ(a) = f'(a)(b − a) – ſå ƒ"(x)(x − b)dx - a

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Q: Check the following functions if Piz) exist not and find f'(z) if exist: ,or D fcz) = 1Zit %3D

A: note : as per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: Let f(x) = x?sin (-) for x + 0 and f(0) = 0. c) Show that f'is not continuous at x = 0.

A: Given function  f(x)=x2 sin1x      for x≠00                  for x=0 we have to show that f' is not…

Q: Let f(x, y, z) = x^2 + y^3 + z^5 and let u =. Find the directional derivative Du f(x, y, z).

A: The directional derivative Du​f(x,y,z) is 64/81​.Explanation:Step 1: To find the directional…

Q: af If f(x,y) = sin-(4x³y?) then dx 12x?y? 1-(4x³y²)² O a. b. 1 4x³y²/(4x³y?)² – 1 c. 12x²y² 4x³y?…

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Q: Let f(x, y) = (5x – y)*. Then %3D дхду Эхдудх dx² dy

A: Partial derivatives: It is a derivative of a function of two or more variables with respect to one…

Q: 2. At what points of R² is the function continuous? (sin(x² + y²) x² + y² f(x, y) = 0, -, if (x, y)…

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Q: Dete‎rmine if f (x, y) = ={(xsi‎n3y)/(x‎2+y6), (x, ‎y) ≠ (0, ‎0) ={1, (x, y) = (0, 0) is‎…

A: Solution: A graph for a function that's smooth without any holes, jumps, or asymptotes is called…

Q: ingetral from 0,2 ingetral from0,5 ingetral from5,7 ingetral from0,9

A: To evaluate the integrals ∫02fxdx,∫05fxdx,∫57fxdx,∫09fxdx from the graph of the function. To…

Q: Consider a function f continuous and differentiable on R with f(2. 5) = -3. 1 and f'(2. 5)=-0. 2.…

A: We have to find

Q: (x³ + bx + a, xs-1 lbx? - Let f(x) = .Then the values of a and b so that the %3D -a, x >-1…

A: The function is: f(x)=x3+bx+a, x≤-1bx2-a,      x>-1 So, the function is continuous at x=-1 if…

Q: Determine if f(x, y) is continuous at (1, 0) where (x - 1)²y² (x − 1)² + y² - 0 f(x, y) = = if (x,…

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Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0). The multivariable limit of this function as it approaches the points (x0, y0) and is denoted as: