Suppose the f(x,y) is a smooth function and that its partial derivatives have th

Question

Suppose the f(x,y) is a smooth function and that its partial derivatives have the values, fx(9,6)=1 and fy(9,6)=4. Given that f(9,6)=0, use this information to estimate the value of f(8,5). Note this is analogous to finding the tangent line approximation to a function of one variable. In fancy terms, it is the first Taylor approximation. Estimate of (integer value) f(8,5)

Suppose the f(x,y) is a smooth function and that its partial derivatives have the values, fx(??9,??6)=??1 and fy(??9,??6)=??4. Given that f(??9,??6)=0, use this information to estimate the value of f(??8,??5). Note this is analogous to finding the tangent line approximation to a function of one variable. In fancy terms, it is the first Taylor approximation. Estimate of (integer value) f(??8,??5)

Explanation / Answer

Solution:

The partial derivatives give you the tangent plane. You end up with a multivariable version of Taylor's series.

F is approximated by

F f(x0,y0) + [fx(x0,y0)](x-x0) + [fy(x0,y0)](y-y0)

F = 0 + (-1)(x+9) + (-4)(y+6)

F = - (x + 9) - 4 (y + 6)

Substitute the various values for (x,y) into this linear approximation to obtain

f(-8, -5) = - (-8 +9) - 4 ( -5 + 6) = - 5

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