In this activity, we will use Equation (11.7.2) to calculate some directional derivatives and make sense of these results for a few cases. For all parts of this activity, let \(f(x,y) = 3xy-x^2y^3\text{.}\)
Use Equation (11.7.2) to determine \(D_{\vi} f(x,y)\) and \(D_{\vj} f(x,y)\text{.}\) Write a couple of sentences to describe what familiar functions \(D_{\vi} f\) and \(D_{\vj} f\) are. Remember that \(\vi\) is the unit vector in the positive \(x\)-direction and \(\vj\) is the unit vector in the positive \(y\)-direction.
Use Equation (11.7.2) to find the derivative of \(f\) in the direction of the vector \(\vv = \langle 2, 3 \rangle\) at the point \((1,-1)\text{.}\) Remember that a unit direction vector is needed.
Use Equation (11.7.2) to find the derivative of \(f\) in the direction of the vector \(\vv = \langle -2, -3 \rangle\) at the point \((1,-1)\text{.}\) Write a couple of sentences to explain why this result is different than your answer to the previous two tasks, even though the direction vectors are parallel.
