The temperature at a point x y z is given by tx y z 100ex2...

The temperature at a point (x, y, z) is given by T(x, y, z) = 100e^(-x^2 - 3y^2 - 7z^2) where T is measured in °C and x, y, z in meters. (a) Find the rate of change of temperature at the point P(2, -1, 5) in the direction towards the point (3, -2, 6). _______ °C/m (b) In which direction does the temperature increase fastest at P? (c) Find the maximum rate of increase at P.

Transcript

00:02 Here in this question it is given that t x comma y comma z is equals to 100 e negative x square negative 3 y square negative negative 9 z square so here in a part we need to find rate of change of temperature the point 2 comma negative 1 .3 in the direction of 4.

01:02 Negative 3 .6.

01:04 So here for that we first evaluate the gradient of t so it will be equals to negative 2x i cap negative 6 y j cap negative 18 z k cap so here gradient of t at 2 comma negative 1 comma 3 will be equals to negative 2 here we have 1 i cap negative 6 here y is negative 1 j cap negative 80 z is 3 k cap so here on solving it here we have negative 2 i cap plus 6 j cap negative 54 k cap now we need to find the unit vector that is here we have 4 i cap negative 3 j cap plus 6 k cap which is divided by square root 4 square plus negative 3 and a square of it plus 6 and a square of it.

02:23 So here it will be equals to 4 i cap negative 3 j cap plus 6 k cap which is divided by square root 16 plus 9 plus 36.

02:37 So here on solving it we have the unit vector as 4 icap negative 3 j cap plus 6 k cap divided by square root 61.

02:57 So now here we have d u if that will be equals to gradient of t and here we have a unit vector so it will be equal to negative.

03:22 Sorry, here note that the value of x is 2, so here we have 2 and here we have 4.

03:36 Negative 4 i cap plus 6 j cap negative 54 k cap and here we have 4 i cap negative 5 i cap square root 61.

04:01 So here on solving it we have negative 16 negative 18 negative 3 to 9 which is divided by square root 61 so here we have negative 358 divided by square root 61 so this is the required rate of change of temperature at point p in the direction of 4, negative 3 and 6.

04:40 B part here we need to find the direction where the temperature increases faster...

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