xY (a) Prove that the function f(x,y) is differentiable everywhere in its do- x² + y? main.

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

A:

Q: What does it mean for a function of two variables, f (x, y),to be differentiable at a point (a, b)?

A: A function differentiable at a point has a derivative that is defined at that point. This is defined…

Q: xy Prove that the function f(x, y) is differentiable everywhere in its do- x² + y? main.

A:

Q: 2. Use the First Fundamental Theorem of Calculus to differentiate the function f(r) = VI+ tª dt.

A: We will use the first fundamental theorem of calculus to evaluate the derivative.   The detailed…

Q: b) Let f be a differentiable function of one variable, and let z = f(3x² – 5y²). Show that az əz…

A: Given  z = 3x2 - 5y2

Q: . Is there an analytic function f(z) whose real part is u(x, y) = x + e*cos(y) ? If yes, find the…

A: We will find out the required solution.

Q: (b) Test the continuity of the function f(x) = sgn(lx|) at x = 0. Is f(x) %3D differentiable at x 0?…

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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: Solution is given below:

Q: Integrate to find F as a function of x, and demonstrate the Second Fundamental Theorem of Calculus…

A: We need to Integrate fo find F as a function of x.

Q: : Apply Rolle's theorem for the function f(z) = 7x² – 28z+ 21 over the interval [1,3]. er.

A: The given function is, f(x)=7x2-28x+21 over the interval [1,3]. Observe that, the given function…

Q: Consider the function f defined by 1²+y² V12+y?+1-1 ,2 f(r, y) = %3D lim(7.y)¬(0,0) f(x, y) =

A:

Q: → [0, ∞) defined by f (x) = Vx + 2. Prove that f is 17. Consider the function f: (-2, 0) surjective.

A:

Q: hav is that not the function defined by ху f(x, y) √x² + y² if differentiable at (x, y) + (0;0) if…

A: We have to show that f(x,y) is not differentiable at (0,0)

Q: (a) integrate to find F as a function of x, and (b) demonstrate the Second Fundamental Theorem of…

A: Given- Fx=∫0xtt2+1dt To Integrate-To find F as a function of x, and  demonstrate the Second…

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: 5. Use the Second Fundamental Theorem of Calculus to find F'(x). х 1 F(x) = S* dt.

A:

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: Find the linearization L(x) of f (x) = tan x at a =

A:

Q: - Find the Linearization of f (x) = sinx at a 5. Provide your answer as L(x) =? %3D %3D

A:

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: Let f(2) = x* – i(y + 1)°, (z = x + iy), (a) Determine the points where f is C-differentiable, and…

A: Given :-  let f z = x4 - i  y + 13 where z = x + iy (a) The points where f is ℂ - differentiable ,…

Q: What does it mean for a function ƒ(x, y) to be differentiable? What does the Increment Theorem say…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

Q: Find the linearization of the function a) z = cos(sin y – x); at (-2,0) and use it to approximate…

A: Suppose f(x,y) be a differentiable function at (a,b). Then linearization of f(x,y) at (a,b) is given…

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

A: To find F'x using the second fundamental theorem of calculus: Fx=∫-1xt4+1dt Solution: The second…

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

A:

Q: Let f : R → R with f(x) = 3x² + 2. Prove that f is differentiable at the point –1 %3D

A: f(x) =3x2+2 f'(x) =limh→0f(-1+h) -f(-1)h        =limh→03(-1+h)2+2-3-2h        =limh→03+3h2-6h -3h…

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: Find a point zo where the function f (z) = (z – 8 – 10i) (z – 8 – 10i) is differentiable. What is…

A:

Q: Let f (z) = (11z+ 10) (i – z). What is the real part of the derivative f' (1 + i)?

A:

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

A:

Q: Prove that f(x, y) = x sin y is differentiable everywhere.

A: Answer

Q: Find the derivative of the function at Po in the direction of A. f(x, y, z) = xy + yz + zx, Po(1,…

A:

Q: 2. Use the First Fundamental Theorem of Calculus to differentiate the function -5x f(1) = | V1+ tª…

A: To differentiate the function by using first fundamental theorem of calculus f(x)=integration (x to…

Q: Show that f(z) = x³+i(1−y)³ is differentiable only at x = 0, y = 1 and f'(z) = 0.

A:

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

A: The given function is fx=x5lnx

Q: Q1. (a) Find the points (x ,y, z) in space, where the function f(x, y,z) = /x² +y² -1 Is continuous.

A: Given :- The points x,y,z in space , where the function f x,y,z = x2 +y2 -1 is continuous.

Q: xY 2. (a) Prove that the function f(x, y) is differentiable everywhere in its do- %3D x2 + y? main.

A: Solution is given below:

Q: Show the function f(z) = 1+2i+2Re(z) is differentiable or not differentiable at any points.

A: Solution

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

A:

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

A:

Q: Let L(r) be the linearization of f(x) = sin x + 1 at the point x = . Find L(π/2).

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Q: Find the linearization L(x) of the function at a. f(x) = sin(x), a = %3D L(x) = %3D

A:

Q: Find and sketch the domain of the function. f(x, y, z) = ln(64 − 4x2 − 16y2 − z2)

A: Given function is      f(x,y,z) = ln(64-4x2-16y2-z2) We have to find the domain of this function.

Q: Consider the function z = f(x,y). If fx(a,b) and fy (a, b) both exist, and they are equal, then the…

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

A: Givenf(z)=zz

Q: Find the linearization of f (x) = xe¯* at a = 0. (Hint: Remember thne product rule.)

A: Given : fx = xe-x2

Q: Consider the following. - fx sec² t dt (a) Integrate to find F as a function of x. F(x) = F(x) = (b)…

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Q: 34. Show that f(x) = Jã is not differentiable at x = 0.

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Q: Find the direction in which the function f (x, y, z) = xy - In z is increasing most rapidly at the…

A: We will use directional derivative here