Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x,y,z)=x2+y2+z2, (9,7,−2) maximum rate of change = direction vector =

∇f=⟨xx2+y2+z2,yx2+y2+z2,zx2+y2+z2⟩(9,7,−2) ∇f=⟨992+72+22,792+72+22,−292+72+22⟩=⟨9,7,−2⟩134 Maximum rate of change =|∇|f=92+72+22134=1 direction =∇f|∇|f=⟨9,7,−2⟩134

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maduregimc Answered2021-12-22

Find the maximum rate of change of f at the given point and the direction in which it occurs.

f (x, y, z) = \sqrt{x^{2} + y^{2} + z^{2}}, (9, 7, - 2)

maximum rate of change =
direction vector =

Answer & Explanation

Heather Fulton

Expert

2021-12-23Added 31 answers

\nabla f = {⟨ \frac{x}{\sqrt{x^{2} + y^{2} + z^{2}}}, \frac{y}{\sqrt{x^{2} + y^{2} + z^{2}}}, \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}} ⟩}_{(9, 7, - 2)}

\nabla f = ⟨ \frac{9}{\sqrt{9^{2} + 7^{2} + 2^{2}}}, \frac{7}{\sqrt{9^{2} + 7^{2} + 2^{2}}}, \frac{- 2}{\sqrt{9^{2} + 7^{2} + 2^{2}}} ⟩ = \frac{⟨ 9, 7, - 2 ⟩}{\sqrt{134}}

Maximum rate of change

= | \nabla | f = \frac{\sqrt{9^{2} + 7^{2} + 2^{2}}}{\sqrt{134}} = 1

direction

= \frac{\nabla f}{| \nabla | f} = \frac{⟨ 9, 7, - 2 ⟩}{\sqrt{134}}

Jordan Mitchell

Expert

2021-12-24Added 31 answers

The formula is just him finding the derivative with respect to whatever slot the equation is in (f gradient = fx',fy',fz') and so forth.

karton

Expert

2021-12-30Added 439 answers

where does that formula come from in?

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