The temperature at a point (x, y) on a flat metal plate is given by T(x, y) = 88/(2 + x2 + y2), where T is measured in °C and x, y in meters. Find the rate of change of temperature with respect to distance at the point (3, 1) in the x-direction and the y-direction.

Step-by-step explanation:

To find the rate of change of temperature with respect to distance at the point (3, 1) in the x-direction and the y-direction we need to find the Directional Derivative of T(x,y). The definition of the directional derivative is given by:

Where i and j are the rectangular components of a unit vector. In this case, the problem don't give us additional information, so let's asume:

So, we need to find the partial derivative with respect to x and y:

In order to do the things easier let's make the next substitution:

and express T(x,y) as:

The partial derivative with respect to x is:

Using the chain rule:

Hence:

Symplying the expression and replacing the value of u:

The partial derivative with respect to y is:

Using the chain rule:

Hence:

Symplying the expression and replacing the value of u:

Therefore:

Evaluating the point (3,1)

it has the same slope and a different y-intercept

hope this : )

a 270° rotation about point p says on e d g e n u it y (had to space it out because it says it's a misspelling so i couldn't submit it