Assume that f(r, y) is a function given by ( : if (r, y) # (0,0) f(r, y) = 0; if (1, y) = (0, 0) (a) Show whether f(r, y) is continuous at point (0, 0) (b) Prove that f(r, y) < y

Q: At what points (x, y) in the plane are the functions continuous? g(x, y) = 1/(x2 - y)

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Q: Is the function f(x, y) = ||x| - |y|| - |x| - |y| differentiable at (0,0)?

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Q: Choose the gradient of the function f defined by f(x,v,z) = cos(xy +9xz + 13yz). O sin(xy O -cos(…

A: Take gradient and solve

Q: if (x,y) # (0,0) 1. Let f:R2 → R be defined by f(x,y) = Find if f is 0, if (x,y) = (0,0) continuous…

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Q: Let f(x,y) be a differentiable function satisfying the equation = xy. %3D ду af and After the change…

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Q: 11. Verify that the conclusion of Clairaut's Theorem holds for the function: f(x, y) = e*Y sin(y)

A: The given function is: fx,y=exy siny To verify that the conclusion of Clairaut's theorem hold for…

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

A:

Q: 2. Find a function f(x, y) which is NOT linear in either a or y and for which the tangent plane at…

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Q: : Find the derivative of the function at Po in the direction of the vector AC. f(x, y, z) = x² + 2y²…

A: To find the directional derivative of the given function.

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A:

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

A: Given function is f(z)=z*2z, z&gt;00,        z=0 we have to prove, the above function is not…

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A: The solution is given as

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: Let A = {(x,y) = RXR; 0<max{|x-3], ly- 2|}<1}, then: *

A:

Q: Find the derivative of the function at Po in the direction of u. f(x, y, z) = In(x 2 + 3y 2 + 9z 2),…

A: Given that      f(x,y,z)=ln(x2+3y2+9z2), P0(3,3,3), u=3i+4j Here we have to find the derivative of…

Q: Find a function f (x, y, z) such that ∇f =〈z, 2y,x〉.

A: Given that, ∇f=z,2y,x We have to find a function f(x, y, z) such that ∇f=z,2y,x. Let us consider,…

Q: Use a linear approximation to estimate f(0,1), given that y=f(X) is a differentiable  with f(0)=1…

A: Consider the given information

Q: Let f(x, y, z) = x In y - exz.Calculate the directional derivative of fat point (-5, 1, -2) and…

A: In this question, we find the directional derivative of the function f(x,y,z)=xln(y)-exz3The line…

Q: Suppose that f(æ, y) f(x, y) in the directional (1, – 1) and at the point = xy. The directional…

A: Here, the given function is: f(x,y)=xy So, ∇f=∂f∂x,∂f∂y⇒∇f=y,x⇒∇f(4,-3)=-3,4

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 the set of points at which the function xyz f(x, y, z) 4x2 + 5y2 - is continuous. 1. { }…

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Q: Q2 (A) At what points f(x, y, z) in which the function f(x, y, z) = continuous? is

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Q: Let f be a function with continuous first derivatives, defined by f (cos(xy), x2 + y³) xy If we take…

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Q: x + y + 1 f(x, y) = , x ln (y² х — 1 -

A: A function f=f1, f2, ⋯, fn is said to be continuous at a if and only if the all the components f1,…

Q: At what points (x, y, z) in space are the functions in continuous? ƒ(x, y, z) = x2 + y2 - 2z2

A: Solution:-  At what points (x, y, z) in space are the functions in continuous? Let (m,n,p) is the…

Q: 2 - What is the directional derivative of the function f(x,y,z) = x3 - 2xy +z at point Ao= (1,1,0)…

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Q: prove that xy^2/x^2+y^2 is not differentiable at 0,0 when  xy^2/x^2+y^2 if (x,y) not equal to (0,0)…

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Q: Verify that the conclusion of Clairaut's Theorem holds for the function: f (x, y) = e*y sin(y)

A:

Q: Find the directional derivative of the function at the point in the direction given. f (x, y, z) =…

A:

Q: Let f(1,y) = (r+ay°,r²y°,2r + by), g(u, v,w) = (3u² + bv² + w°, uvw). Find the derivative of…

A: Let f(x, y)=x3+ay2, x2y3, 2x+by,  gu, v, w=3u2+bv2+w3, uvw. We need to find the derivative of fgu,…

Q: I. Show that the function f defined by (r, y) = (1, –1) f(r.y) 2 + y (x, y) # (1, -1) is not…

A:

Q: Suppose f and g are two differentiable functions such that f(9(x)) = x? and supposed the equation of…

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

A:

Q: If f(x, y, 2) = cos(x - y)+e² , compute directional derivative of f at (5, 5, 0) in the direction of…

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&gt;0}…

Q: 3. Let f(x, y) = In 2x + 4y, find the directional derivative of f(x, y) in the direction of at the…

A:

Q: Suppose that f: R2R is a function given by S x²+y² f(x, y) 0 for xy = 0. Show that f(x, y) is…

A: We have to show the given function  f(x, y) is continuous  at ( 0,0)

Q: Determine f„ (0, T, 1), fy(0, , 1), f¿(0, r, 1) given the function z sin(3xz+ y²)

A: The given function,        zsin3xz+y2

Q: Find the derivative of the function at Po in the direction of A. f(x,y,z) = 3 ex cos (yz),…

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Q: Show that the function f(x, y) = 3x - 4y² is differentiable at point (−1, 2)

A: answer is in next step. please give a like !!!

Q: Use the definition of directional derivative to solve Duf(xo, yo, zo) at point P(xo, yozo) where…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

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: Find the direotional derivative of the scalar function f-x'+ y' +2z', at the given point (1,1,2), in…

A: Explanation of the answer is as follows

Q: Consider function f(x, y) = x cos(TxYy) + x³y. .3,

A:

Q: If (x, y, z) = cos(x – y)+ e² compute directional derivative of Í at (3, 3, 0) in the direction of ŷ…

A: Option C is correct option.

Q: that F(x, y, z)= Divergence Theoren

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Q: What is the maximum value of the directional derivative of f(x, t) = f (x, t) = x² + t²/8 – 10 at…

A: Note: The directional derivative of a function is the rate at which the function changes at a…

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.