Let ƒ(x, y) =(x2-y2)/(x2+y2)  for(x, y) ≠ (0, 0). Is it possible to define ƒ(0, 0) in a way that makes ƒ continuous at the origin? Why

Q: (a) Find the points (x ,y, z) in space, where the function f(x, y, z) = Vx² +y -1 Is continuous.

A: It is well known that the equation z=x2+y2 represents a circle with centre at the origin. The…

Q: Let x and y be differentiable functions of t, and lets = √(x2 + y2) be the distance between the…

A:

Q: Value of divergence of function (x2y2)î - (z³y 4x) ĵ + 4y² xk at (1, 2, 1) is

A: Given vector is (x2y2)i^ - (z3y-4x)j^ + 4y2xk^

Q: 3. Let f(x, y) +y with x()=r and y(t) sin t. Find the derivative of w = %3D %3D %3D f(x, y) with…

A: Asked derivatives is solved.

Q: Find the linearization at x = a. IT f(x) = sin³ (x), a = 4

A: Given,            The linearization at x=a.             fx=sin3x, a=π4We know that,             The…

Q: How is the linearization of f (x, y) at (a, b) defined?

A: given,          any function fx,y at a,b

Q: Find f-(x, y) and fy(x, y). f(z, y) = (y° tan(5z)) f-(x, y) fy(x, y) =

A:

Q: Let f(x, y) = exy and P(1, 0). Find the maximum and minimum rates of change of f at the point P. In…

A: Given function is: f(x,y)=exy and point P(1,0)

Q: Let x and y be differentiable functions of t and lets = sqrt(x2 + y2) be the distance between the…

A: Given that, s=x2+y2

Q: Let F(x, y) =x2+xyz+z , find at the point (1,1,1)

A:

Q: (2) Let f(x, y) = x² – y². Find the tangent line to the level curve f(x, y) = -3 at (1,2).

A:

Q: Find the orthogonal trajectories of the family of curves: y = k (csc x + cotx) Ⓒy² = 2 (C- cos x) O…

A:

Q: Let F(x, y, z) = (sin(xy), cos(xy), sin(xyz)). What is the divergence of F at the point (T, 2, 0)?…

A: Given : F→x,y,z = sinxy, cosxy, sinxyz

Q: What is the linearization L(x, y) of a function f (x, y) at a point (a, b)? How is L(x, y) connected…

A: Linearization L(x,y) of a function f(x,y) is given by the formula

Q: Use the limit definition of partial derivatives to find f,(x, y) and f, fy(x, y). f(x, y) = x2 – 6xy…

A: Using partial derivative,

Q: Prove that the function (z*)2 1리 > 0 0, z = 0 f (z) = is not analytic at z = 0 by calculating all…

A: Calculate the derivative of fz at z=0.  f'0=limZ→0fz-f0z-0       =limZ→0z*2/z-0z-0       =limZ→0z*z2

Q: Let f(x, y) = (x° + ay´, xʻy°, 2x+by), g(u,v,w) = (3u² + bv² + w°, uvw). Find the derivative of…

A: f(x,y)=(x3+ay2,x2y3,2x+by)g(u,v,w)=(3u2+bv2+w3,uvw)To Find: Derivative of f(g(u,v,w)) using the…

Q: (3) Consider the derivative dz = 2xydx + (x² + 2y)dy Evaluate whether this is a total derivative, by…

A: We will check whether the given function is total derivative or not. We also integrate dz pathwise.

Q: Show that the function x² – y² 2ху f(x,y) = { x² + y² if (x,y)# 0 if (x,y) = 0 is continuous at…

A:

Q: Compute the partial derivative of the function f(x, y, z) = e'-z cos(y) +ze-/(1+y²) with respect to…

A:

Q: The partial derivative of f(x, y) with respect to y at the point (x,y) is f(x+h,y)-f(x,y) h B f(x…

A: We have to Write the Partial derivative of f(x, y) with respect to y at the point (x0,y0).

Q: f(x,y)=ln(5x+4y). Find the directional derivative of f at (1,0) in the direction of the line…

A:

Q: Determine whether the statement is true or false. If f has continuous partial derivatives of all…

A: since curl of divergence i.e

Q: Find the partial derivative of f/x and f/y f(x,y)=x/x+y

A:

Q: Let f(x, y,z) = z-sin(xy), x = t, y = ln t , z = e^(t-1) . F

A:

Q: Suppose that the partial derivatives of a function f(x.y.z) at point on the helix x=cost, y=sint,…

A: In this question, we have given the helix and first derivatives of f. x=cost, y=sint, and z=t and…

Q: Find the orthogonal trajectories of the family of curves: y = k (csc x + cotx) Ⓒy²= 2 (C- cos x) O y…

A: This question is related to differential equations, we will solve it using given information.

Q: Plot the function f(x, y) = x² – y² over the square region bounded by the lines y =x - 2, y =x+ 2, y…

A: Given the function fx,y=x2-y2

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: Find the derivative of the function at Po in the direction of A. f(x, y, z) = x2 + 2y2 – 3z?, Po(1,…

A: We can solve this as follows:

Q: Assume the second derivatives of ƒ are continuous throughout the xy-plane and ƒx(0, 0) = ƒy(0, 0) =…

A: Consider the provided information, fxx0,0=5,fyy0,0=3  and fxy0,0=-4 The Second Derivative Test to…

Q: The second order partial derivative of f with respect to z (mean fz) where f(x, y, z) = x'0 sin(3y +…

A: Given: fx,y,z=x10 sin3y+10z To determine: fzz

Q: |(y) + (0,0) (x. y) (0,0) Show that the function f (x, y). is continuous at the origin however =…

A: consider the equation  f(x,y)=∑xyx2-y2x2+y2 ,0  x,y  #(0,0)xyx2+y2 where f(0,0)

Q: let f(x,y)= xexy  Compute all the first and and second partial derivatives of f.  Find the tangent…

A:

Q: Determine the set of points at which the function is continuous. cos(ry) er – y² ' b) f(x,y) =

A: The given function is: f(x,y)=cos(xy)ex-y2

Q: : Find fz for f(x, y, z) = xcos(y + 3z) + xyz at the point ( 1,3,-1)

A:

Q: The gradient of f(x, y) = xy at (x, y) = (3, 1) is

A:

Q: Find the maximum rate of change of f(x, y, z) = tan(2x + 7y + 3z) at the point (– 2, – 3, – 6). - -

A:

Q: The maximum directional derivative of f (x, y, z) = x² sin(zy) – yx + 10 | at (-1, 2, 0) is

A:

Q: At what points (x, y) in the plane are the functions continuous? ƒ(x, y) = (x + y)/(x - y)

A: Continuity: Given function is : f(x,y)=x+yx-y The function f(x,y) is defined on S={(x,y)∈R:x,y>0}…

Q: Assume the second derivatives of ƒ are continuous throughout the xy-plane and ƒx(0, 0) = ƒy(0, 0) =…

A: Maxima or minima: Let f(x, y) is any function. 1) Evaluate fxx, y=0 and fy0,0 for critical point. 2)…

Q: Let f(r,y, z) = x² + 3x°y + z°, g(u, v) = (u² + v², uv, 3u – 2v). Find the derivative of f(g(u, v))…

A:

Q: Compute the partial derivative: f(x, y) = sin (a- 6y) f,(0, T) =

A:

Q: Find fz for f(x, y, z) = xcos(y + 3z) + xyz at the point ( 1,3,-1)

A: As per the company guidelines we can solve first question. I hope that the given solution will help…

Q: Let f(x,y) = (x² + (G))°. Determine the directional derivative of f at the point (1,0) in the…

A: Explanation of the answer is as follows

Q: /alue of divergence of function (x2y2)î - (z³y – 4x) ĵ + 4y2 xk at (1, 2, 1) is

A: Let f = (x2y2)i^ - (z3y-4x)j^ + 4y2xk^

Q: Let x and y be differentiable functions of t, and lets = √(x2 + y2) be the distance between the…

A: Given equation is s=x2+y2. Compute the derivative of s with respect to t.…

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: