Define the function f(x, v) = 4x³-x³y-4y5 3x5-3y5 at the origin so that it becomes continuous there.

Q: Use the Second Fundamental Theorem of Calculus to find F′(x).

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Q: Q2 (A) At what points f(x, y, z) in which the function f(x, y, z) = -√x²+y² is Z- continuous?

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Q: Determine if = f(x)=Vxis uniformly continous on the interval I [0, +)

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Q: Differentiate fx) with respect to g(x) of yx= xsiny

A: To find the derivative

Q: Fundamental Theorem of Calculus.

A: ∫abfx dx

Q: Find the linearization of the function f(x,y) = V9x² + 16y² at the point (1, 1), and use it to…

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Q: Use the Second Fundamental Theorem of Calculus to find F'(x). F(x) = 1 + t3 dt

A: To find F'x Fx=∫0xt21+t3dt

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

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Q: Differentiate f(x) with respect to g(x) of yx=xsiny

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Q: Where is the function f(r, y) = x/y continuous?

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Q: Find the derivatie: F (+2 %3D (-t*+*z+)た =人

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Q: · Use the definition of the derivative of complex function to show that f(z) is not dif- ferentiable…

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Q: Explain why f (x) = sin (x² + 3r + 1) is a continuous function for all c € R. %3D

A: Let us assume that: g(x)=sin(x)and h(x)=(x2+3x+1)So,…

Q: Determine if the function is continuous at point (0,0): F(x, y)=**y°cos(x*+y³) (x²+y²)2

A: Given that Now take x=0, Now take y=0,

Q: What does it mean for f to be differentiable at a ?

A: we have to define that what we mean by the f is differentiable at a.

Q: Suppose that F(t) is a twice-differentiable function of one variable. Define the function f(x,y) =…

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

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Q: For the function f(x, y) = (cos x)ey +v 1 + x + y. %3D Show the process of using the linearization…

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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: Use the Second Fundamental Theorem of Calculus to find F′(x).

A:

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

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Q: Determine if the function is continuous at point (0,0): F (x, y) _x³y°cos(x³+y³) (x²+y²)²

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Q: Show that the function f defined by (r, y) = (1, –1) (r, y) # (1, –1) f(r, y) = 1² + y is not…

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Q: Is the scalar valued function f(x)=(x^2-1)/(x+1) continuous?

A: This is a problem related to limit and continuity of the function. Please follow the procedure given…

Q: Determine if f is continuous at (0, 0) where: 3.x?y x4 + 2y2' (x, y) # (0,0) f (x, y) = (x, y) =…

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Q: Find a linear approximation of the function f(x) = V1-x at a = 0. %3D

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Q: Find whether the function F-(* - y')i + (v² – 3x)j + (=* - xy)k is irrotational.

A: A vector field F is called irrotational if curl (F) = 0.

Q: What is the domain of the function w = ƒ(x, y, z) = 1/xyz?

A: What is the domain of the function w = ƒ(x, y, z) = 1/xyz?

Q: Find the linearization L(x) of f(x) = cot x at x = 4 %3D -

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Q: I. Show that the function ƒ defined by (1, y) = (1, –1) f(1, v) = { +g (z, y) # (1, –1) + y is not…

A: We will use the sequential characterization of continuity to solve this problem. The detailed…

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: 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: Let f be a continuous function such that f (x) dx 5 and 6° | f(x) dx = 3. What is the value of |…

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

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Q: 1. (y-1) x²+4 Cy-1)² (x,y)# (0,1) f(ox,y)= %3D (x,y) = (0,1)

A: Explained below...

Q: Find the inflection point of the function f(x) = x°In(x). %3D (x, y) = (

A: The given function is fx=x5lnx

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: For the function f(x, y) = xy((x2 − y2) /( x2 + y2) ) define f(0, 0) such that f is continuous at…

A: A function f is continuous at the origin, the function value at the origin must be equal to the…

Q: Is the function f(z) = (x2 - y2) + 2xyi entire? Why or why not?

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Q: Let g(u) be a differentiable function and let f(x, y) = g(x² + y²), find the direction of maximal…

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Q: Show that f(x) := x/3, x e R, is not differentiable at x 0.

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Q: 1. Show that the function x2 – y2 2xy f(x,y) ={ x2 + y? if (x,y) # 0 if (x, y) = 0 is continuous at…

A: If for every ε&gt;0, there exists a δ&gt;0, such that if |x-a|&lt;δ, |y-a|&lt;δ, then the function…

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: Show that the function f(z) = z = is differentiable but not analytic at z = 0

A: Givenf(z)=zz

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

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Q: Suppose that F(x, y, z) = 0 defines z as a function of x and y and F is differentiable. Prove that…

A: This is a problem of the chain rule and Implicit function theorem.

Q: Let f be the function from R to R with f (x) = x2. Is f invertible?

A: