Consider the function $f(x, y) = 5\sqrt{\frac{xy}{6}}$ which is differentiable at $(6, 4)$.\n\na) Find the gradient vector at the point $(6, 4)$ (Give a vector with exact components)\nYou cannot proceed with the rest of the problem until you correctly answer the above\n\nb) At the point $(6, 4)$, the total derivative is\nDot product form:\n$TD(h_1, h_2) = \text{________} \cdot \langle h_1, h_2 \rangle$\nExpanded form:\n$TD(h_1, h_2) = \text{________}$\n(Give an expression using $h_1$ and $h_2$ in your answer)\n\nc) Rewrite the total derivative action in the differential notation: Your answer should be an equation\ninvolving $dx$, $dy$, $df$\n\nd) The linear function $L_{(6, 4)}$ that well approximates the nonlinear function $f(x, y)$ nearby $(6, 4)$, aka\nthe local linearization of $f$ near $(6, 4)$ is given by\n$L_{(6, 4)}(x, y) = \text{________} + \text{________}(x - 6) + \text{________}(y - 4)$\n\ne) Estimate $f(6.43, 4.59)$ using local linearization near $(6, 4)$.\n$f(6.43, 4.59) \approx \text{________}$\nMake sure your answer is accurate to at least four decimal places, or give an exact answer. Do not\ngive the exact value of $f(6.43, 4.59)$ which is essentially $11.089353001866$.\n\ng) In the $xyz$ system, Find the equation of tangent plane to the surface $z = 5\sqrt{\frac{xy}{6}}$ at the point\n$(6, 4, z_0)$. (Make sure you calculate the numerical value of $z_0 = f(6, 4)$.)\n\nf) At the point $(6, 4)$, find the directional derivatives in the following directions:\ndirection of $\hat{i}$ \n$\text{________}$\ndirection of $\hat{j}$\n$\text{________}$\nsame direction as $\langle -3, 5 \rangle$\n$\text{________}$
