Question

Find the linear approximation of the function f(x, y, z) = x2 + y2 + z2 at (2, 6, 3) and use it to approximate the number 2.012 + 5.972 + 2.982 . (Round your answer to five decimal places.) f(2.01, 5.97, 2.98) ≈ 

          Find the linear approximation of the function f(x, y, z) = x2 +
y2 + z2 at (2, 6, 3) and use it to approximate the number 2.012 +
5.972 + 2.982 . (Round your answer to five decimal places.) f(2.01,
5.97, 2.98) ≈

Added by Robert S.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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Find the linear approximation of the function f(x, y, z) = x2 + y2 + z2 at (2, 6, 3) and use it to approximate the number 2.012 + 5.972 + 2.982 . (Round your answer to five decimal places.) f(2.01, 5.97, 2.98) ≈
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Transcript

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00:01 Okay, so first of all, let's keep in mind that the gradient of f is given by 2x, comma, 2y, comma 2 z.
00:13 Now let's evaluate the gradient at p, where p is our point to 6 .3.
00:24 Okay, so this guy is gonna be 4, 12, 6, 6.
00:30 So, at this point, what do we know? well, we know that the value of f at 2 .012, comma 5 .972, comma, 2 .982.
00:51 Okay so well this guy is approximated by the value of f at p which is okay so this one is gonna be four plus 36 which is 40 plus 9 which is 49 so 49 here plus the gradient of f evaluated at p dot product what dot product this point here minus our point p so here we are going to have 0 .012 okay this one minus 6 which is 0 negative 0 point 0 point okay so let's use a calculator so 5 .972 minus 6 .0 is negative 0 .0288 and finally, okay finally we have this guy here 2 .982 minus 3 so 2 .982 minus 3 which is negative 0 .018 perfect okay so we just need to compute this dot product now well okay let's compute this dot product this is pretty easy to compute this dot product this dot product is gonna be four multiplied by this guy so four multiplied by zero 12 minus 12 multiplied by 0 .028 plus 6 multiplied by 0 .028 plus 6 multiplied by this guy so negative 6 multiplied by 0 .0 18...
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