Gradient Problem: Find Rate of Change, Direction, & Maximum Increase

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SUMMARY

The discussion focuses on solving a gradient problem involving the temperature function T(x,y,z)=8x^2-7xy+7xyz. The rate of change of T at point P(-1,1,-1) in the direction of the vector u=<8,10,-8> was calculated to be -2.914. The gradient at point P was determined to be <-30,14,-7>, indicating the direction of fastest temperature increase. The maximum rate of increase at point P was found to be 33.84.

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Homework Statement


T(x,y,z)=8x^2-7xy+7xyz
a. find the rate of change of t at point p(-1,1,-1) in the direction u=<8,10,-8>
b. which direction does the temperature increase fastest
c. find the maximum rate of increase at P.

Homework Equations


gradient of T=(16x-7y+7yz, -7x+7xz, 7xy)


The Attempt at a Solution


a. i multiplied gradient by unit vector <8,10,-8>/square root of 228 and i got -2.914
please tell if this is right.
b. i just found gradient T(-1,1,-1)=<-30,14,-7>
c. i found the magnitude of b which was 33.84
 
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yep, all good. Nice work.
 

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