full working out please Consider the function f:R2 R, defined by f(x,y) =Py+3y-2

Question

full working out please

Consider the function f:R2 R, defined by f(x,y) =Py+3y-2 (a) Find the first-order Taylor approximation at the point xo?(1,-2) and use it to find an approximate value for f(1.1,-2.1 (b) Calculate the Hessian (X-X0)t (Hf(Xy)) (X-Xy) at X-(1,-2). (c) Find the second-order Taylor approximation at xo?(1,-2) and use it to find an approximate value for f(1.1, -2.1 Use the calculator to compute the exact value of the function f(1.1,-2.1) Com- pare the values of the found approximations with the exact value of the function

Explanation / Answer

f = x^2y + 3y - 2

Deriing partially :
fx = 2xy ---> 2*1*-2 --> -4
fy = x^2 + 3 ---> 1 + 3 --> 4

So, we have
f(x,y) = f(x0,y0) + fx(x-x0) + fy(y-y0)

At (1,-2), f(x0,y0) = -2 - 6 - 2 ---> -10

So, we have
-10 - 4(x - 1) + 4(y-(-2))

-10 - 4x + 4 + 4y + 8

f(x,y) = -4x + 4y + 2

Now, plug in (1.1,-2.1) :
-4(1.1) + 4(-2.1) + 2
-10.8 ---> ANS

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b)
H = fxx*fyy - fxy^2

Now, fx = 2xy and fy = x^2 + 3

Deriving :
fxx = 2y , fyy = 0 , fxy = 2x

Values become :
fxx = -4 , fyy = 0 , fxy = 2*1 = 2 , fyx = 2 also

Now, H is given by (2x2 matrix)

[fxx   fxy]
[fyx   fyy]

= [-4   2]
[2    0]

----------------------------------------------------------------

c)
f = -4x + 4y + 2 + fxx/2*(x-x0)^2 + fxy(x-x0)(y-y0) + fyy/2(y-y0)^2

Plug in all values :
f = -4x + 4y + 2 + -2(x-1)^2 + 2(x-1)(y+2) + 0

f = -4x + 4y + 2 - 2(x-1)^2 + 2(x-1)(y+2)

Now, plug in 1.1 ,- 2.1 :
f = -4(1.1) + 4(-2.1) + 2 - 2(1.1-1)^2 + 2(1.1-1)(-2.1+2)

f = -10.84 ----> ANS

Exact value is : -10.841 , which is mightily close.

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