# Thread: Rate of change / Directional Derivatives

1. ## Rate of change / Directional Derivatives

I can't quite figure out where to begin / find an equation for this problem.... I need a good push in the right direction, thanks ahead of time.
The temperature at a point (x, y, z) is given by the following equation where T is measured in °C and x, y, z in meters. T(x, y, z) = 200e^((-x^2)(-3y^2)(-9z^2))

(a) Find the rate of change of temperature at the point P(2, -1, 2) in the direction towards the point (3, -3, 3).

(b) In which direction does the temperature increase fastest at P?

(c) Find the maximum rate of increase at P.
What equation do I need to use, in terms of the given variables in my problem??

2. Originally Posted by piyourface166
[INDENT]The temperature at a point (x, y, z) is given by the following equation where T is measured in °C and x, y, z in meters. T(x, y, z) = 200e^((-x^2)(-3y^2)(-9z^2))

(a) Find the rate of change of temperature at the point P(2, -1, 2) in the direction towards the point (3, -3, 3).
Let $\displaystyle u=<3,-3,3>-<2,-1,2>$.

For a) you want $\displaystyle \dfrac{{\nabla T(2, - 1,2) \cdot u}}{{\left\| u \right\|}}$.

3. when you have ||u|| does that equal - abs of root(6) - because vector is <1,-2,1>? The ||vector|| always confused me. I know what |vector| asks, but the second '||' throws me off.

4. Some authors (older ones particularity) do use $\displaystyle |u|$ for length of a vector.
More modern notation is $\displaystyle ||u||$ to distinguish from absolute value as in $\displaystyle ||\alpha u||=(|\alpha|)||u||$.

5. thank you so much that has been bothering me. Ok so I got [-518400e^(-432)] / root (6) - for part a, which I'm fairly confident is correct. So for part b - which direction does the temperature increase the fastest at P - we only have two points, which can only give us one vector and two directions, to P from the second point and away from P towards the second point correct? But that doesn't seem correct to me.

6. I got the problem. Thank you for all your help!