Help with study questions

[allhalf425]

I have a big exam coming up in a few days, and I just couldn't get this problem on the study guide. I was hoping you guys might be able to help me out! This is the question:

Find the directional derivative of f(x, y, z) = 4(x^2)(y^2) + 9x(z^2) - 5xyz at the point P(-1, 2, -1) in the direction of the vector v = i - 2j + 2k

Any help on this would be GREATLY appreciated! And please don't just give me the answer, it would be much more beneficial if you actually described what was happening. Thank you!

 

[BigGlenntheHeavy]

\(\displaystyle Directional \ derivative \ of \ f: \ D_u f(x,y,z) \ = \ af_x (x,y,z)+bf_y (x,y,z)+cf_z (x,y,z), \ where\)

\(\displaystyle u \ = \ ai+bj+ck \ (unit \ vector).\)

\(\displaystyle Given: \ v \ = \ <1,-2,2> \ \implies \ u \ = \ \bigg<\frac{1}{3},\frac{-2}{3},\frac{2}{3}\bigg>, \ f(x,y,z) \ = \ 4x^{2}y^{2}+9xz^{2}-5xyz, \ and\)
\(\displaystyle P \ = \ (-1,2,-1).\)

\(\displaystyle Ergo, \ D_u f(x,y,z) \ = \ \frac{1}{3}f_x (x,y,z)-\frac{2}{3}f_y (x,y,z)+\frac{2}{3}f_z (x,y,z).\)

\(\displaystyle D_u f(x,y,z) \ = \ \frac{8xy^{2}+9z^{2}-5yz}{3}-\frac{16x^{2}y-10xz}{3}+\frac{36xz-10xy}{3}\)

\(\displaystyle Therefore, \ D_u f(-1,2,-1) \ = \ -\frac{13}{3}-\frac{22}{3}+\frac{56}{3} \ = \ 7.\)

 

[allhalf425]

Okay, all of that makes sense except for the part where v = <1, -2, 2> ==> u = <1/3, -2/3, 2/3>.

Why are they all divided by 3? Thanks for the reply!

 

[BigGlenntheHeavy]

\(\displaystyle u \ is \ the \ unit \ vector \ of \ v.\)

\(\displaystyle Note: \ That \ should \ be \ obvious.\)

 

[allhalf425]

Okay, thank you.