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# Determine the second order Taylor formula for the given function about (xo,yo)?

f(x,y) = cosye^(x-1)^(2), where xo = 1, yo = 0

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- kbLv 79 years agoFavorite Answer
Assuming that we have f(x,y) = e^((x-1)^2) * cos y:

f(1, 0) = 1

f_x = 2(x-1) e^((x-1)^2) * cos y ==> f_x(1, 0) = 0

f_y = -e^((x-1)^2) * sin y ==> f_y(1, 0) = 0.

f_xx = [2 * e^((x-1)^2) + (2(x-1))^2 e^((x-1)^2)] cos y ==> f_xx (1, 0) = 2

f_xy = -2(x-1) e^((x-1)^2) * sin y ==> f_xy(1, 0) = 0

f_yy = -e^((x-1)^2) * cos y ==> f_yy (1, 0) = -1.

Hence,

f(x, y) = 1 + 0(x - 1) + 0(y - 0) + (1/2!) [2(x - 1)^2 + 2 * 0(x - 1)(y - 0) + (-1)(y - 0)^2] + ...

.........= 1 + (x - 1)^2 - (1/2)y^2 + ...

I hope this helps!

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