The temperature of any given point on a plate is given by:
T(x,y)=180e^(−(x^2/4)−(y^2/3))
I need to find the rate of change in temperature along the direction from (2,1) to (-1,-3). I have tried taking the dot product of the gradient and unit vector, but this has not worked. I have correctly used the same technique to find the direction and magnitude of greatest increase in temperature. I have found the gradient of T to be <-180e^(-4/3), -120e^(-4/3). How do I find the rate of change from (2,1) to (-1, -3)?
[Math] Find the rate of change in a given direction of two variable function
multivariable-calculusslopevector analysis
Best Answer
Let $A = (2,1), B = (-1,-3)\implies \vec{u} = \dfrac{1}{|\vec{AB}|}\vec{AB}= \left(\dfrac{-4}{5}, \dfrac{-3}{5}\right)$ is the unit vector which is also the direction in which the derivative being taken along. Thus the directional derivative you are looking for is: $\nabla_{\vec{u}} f= \nabla f\cdot \vec{u}= (f_x,f_y)\cdot \vec{u}= \dfrac{-4}{5}f_x - \dfrac{3}{5}f_y$. You can find the partial derivatives, can't you ....? Also, you need to specify the point that the directional derivative is calculated at...