Answer:
80.66
Step-by-step explanation:
[TeX] L\left( x \right) = f\left( a \right) + f'\left( a \right)\left( {x - a} \right) [/TeX]
[TeX] f(x, y, z) = x^2+y^2+z^2[/TeX]
Since we have three variables,
[TeX] L(x, y,z) = f(a, b,c) + f_x (a, b,c) (x - a) + f_y (a, b,c) (y - b) +f_z(a,b,c)(z-c)[/TeX]
[TeX] f(4, 4, 7) = 4^2+4^2+7^2=81[/TeX]
[TeX] f_x (a, b,c)=2x=2*4=8\\f_y (a, b,c)=2y=2*4=8\\ f_z(a,b,c)=2z=2*7=14[/TeX]
Therefore:
[TeX] L(x, y,z) = 81+ 8(x - 4) + 8 (y - 4) +14(z-7)[/TeX]
Using the above:
f(4.02, 3.99, 6.97) =81+ 8(4.02 - 4) + 8 (3.99 - 4) +14(6.97-7)=80.66.
The approximation of [TeX]4.02^2 + 3.99^2 + 6.97^2[/TeX] is 80.66000.
