Answer:
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)