If f : R" →R is C' (R") is convex then f(7) > f(x) + Vf(x)"(j – x) Vĩ,jeR" | 1. Use Taylor's theorem to prove (*).

Q: find the linearization L(x) of ƒ(x) at x = a. ƒ(x) = tan x, a = π

A: The linearization of a function f(x) at x = a is, Now differentiating  given function with respect…

Q: Let f(x, y, z) = In(x² + 2) + y²e== – cos(2yz). Find fzyz- %3D

A: Given that: f(x, y, z)=ln(x2+z)+y2exz-cos(2yz)

Q: Compute fx + fxy + fyyzz, where f(x, y, z) = z sin(xy)

A: Given f(x,y,z)= z sin(xy)

Q: Let  f(x, y) = sin(15x2 + 3y2) Calculate the quadratic approximation of f at the point (0,0)

A: Given that,  fx, y=sin15x2+3y2 The quadratic approximation of the function fx, y at the point x0, y0…

Q: (c(2z+y), %3D ue of c in the joi F of W.

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Q: Let y = 2x + 2x +3. If Ar = 0.2 at a = 2, use linear approximation to estimate Ay %3D Ay Submit…

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Q: Let f(x,y, z) = xtan (Vyz)- sin(e*) , evaluate zy %3D дудхда

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Q: Let f (x, y) = 2x3 + x2y - y2. Compute d2f/dx2, d2f/dy2, and d2f/dxdy at (x, y) = (1, 2).

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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…

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Q: Sx, ifx 1 f(x) = Show that f is R- integrable.

A: R- integrable function

Q: 2. Let f(x, y) = In(x² + y²). Show that fre + fyy = 0. %3D

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

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Q: Suppose that z = f(x, y) is differentiable at the point (1024, 256) withf(1024, 256) = 2 and f,…

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Q: Prove that af +y. -yof=nf (x, y). əx (Hint: Differentiate both sides of the definition of a…

A: Given  a) x∂f∂x+y∂f∂y=nf(x,y)b) x2∂2f∂x2+2xy∂2f∂x∂y+y2∂2f∂y2=n(n-1)f(x,y)

Q: find a potential function for F or determine that F is not conservative. F=(z,x,y)

A: F is conservative when curl F = 0 Given that f=(z, x, y)=zi+xj+yk

Q: Find the local linearization of f(z) =z² at 9. lo(z) = %3D

A: Given : f(x) = x2   x0 =  9

Q: Suppose that the only Införmation we have ab8ut a fuRcEiOH FIS y= f'(x) (a) Use a linear…

A: Given: f(1)=-7

Q: Let H(x) integral ,for -2 ≤ x ≤2.Use geometry to evaluate H(2).

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

Q: 2. For f(z, y) = z-y, prove that f is differentiable at the point (1, 1).

A: Given: f(x, y) = x2 - y2

Q: Let y = 2x + 5x +. 2. If Ax = 0.4 at a = 4, use linear approximation to estimate Ay Ay =

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Q: W= Pcxiy,Z) = In (x+y+z) ,x= cost, 2. %3D 2X = Cost, y- Sint and z= 4VE >use chain rule to Rind dw…

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Q: Find the linear approximation of f(x) = In r at a = 1 and use it to estimate In(1.14). L(2) = %3D…

A: Please refer the attached image for complete solution.

Q: Let F(x, y, z) = z* +x+y sin x +x In z 0, where z =f(x, y), then at (0,0,1) and %3D are equal to:

A: Given,          Fx,y,z=z4+x+ysinx+xlnz=0 , where z=fx,y

Q: Let F(x, y) = G , y 1 )- Calculate the work against F required to move an object from (1, 1) to (3,…

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Q: Suppose f(z) = r + sin z and F is an antiderivative of f that satisfies F(#) = 0. Then F(z) =

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Q: Calculate of of and for the function f(x, y) defined as : ax' dy sin (xy) 1 + y f(x, y) =

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Q: Let f(x, y) = xcosy- y e*y. fxy at (1,0)is equal to:

A: For finding the Partial differentiation we need to take one variable constant and differentiate the…

Q: Find the linearization of the function f (x)= x³ at x = 1. qb |a| sin (a) L (x) = Using this…

A: Given, fx=x5 We know that, If we have a differentiable function f defined near "a" the linearization…

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

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Q: -2 X Given f(x,y,2) sin (z²y), show that fxyuy = fyxy %3D %3D

A: Given:- f(x, y, z) =e-2x sin(z2y)  To prove:- fxyy = fyxy

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: Find g: R R, continuous in [0. ) and position in (0, ) satisfying g1) %3D )dt

A: Here we have,  g:R→R, continuous in [0,∞) and position in (0,∞) Also given g(1)=1 and…

Q: Let f(x, y) = xcosy- y'e*y. fxy at (1,0)is equal to: %3D -2/3 -2 1 0.

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Q: Let y = 5x + 4c +3. If Ar 0.1 at z 5, use linear approximation to estimate Ay

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Q: (a) Let f(r, y) = x - 3ry + 18y, where 3r?y- y- 6x = 0. Prove that f(r, y) has extrema at the points…

A: Calculate partial derivative of f(x,y) with respect to x and y ,

Q: Find the lineanzation LlXiy) of f (iy) =x*-5Xy at (2,1). Then use the linearization to approximate f…

A: Here in this question we need to linearize the given function and then approximate the value of…

Q: Let F(x,y,z) = z* + x+ y sin x + x In z 0, where z fx.y), then at (0,0,1) and are dy equal to: ax

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Q: Find the surfaces on which the function u = 2 (x2 + y²)/z is constar

A: Since u is a constant so, x2+y2z=4a , where a is an arbitrary constant.

Q: Show that the function u = p(xy) + VrY v(y/x) provides the uation x2 = 0. dy?

A: First find the partial derivative of the given function with  respect to x :

Q: ) Compute f-(, ") and fy(§, T) of f(r, y) = e9 cos(ry)

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

Q: Let y 4x + 3x + 4. If Ax 0.3 at a = 3, use linear approximation to estimate A %3D Ay =

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Q: is function oF x $ y xz + y In x = ze-sin (xy) D. Ausumo that z Find zx iF

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Q: Suppose that f(0) = 1, ƒ(0.5) = 2.5, f(1) = 2, and f(0.25) = f (0.75) = a. Using trapezoidal rule…

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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: Determine the linearization of f(x, y) = arctan(x² + y²) at

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

A: f(h,k)-f(0,0) =hfx(0,0) +kfy(0,0) +h2+k2∅(h,k) h2kh2+k2-0 =hA+kB+h2+k2ϕh,k ...(1) A =fx(0,0)…

Q: Let y = 3x + 2x + 4. If Aæ = 0.4 at x = 1, use linear approximation to estimate Ay %3D Ay 2 Submit…

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Q: Let Juce) 2ntv () %3D A=O n!F (I+u+n) Bessel function Of the first Kind be a Of Order V Show that…

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