Consider the function f(x, y) = 5√((xy)/(6)) which is differentiable at (6, 4).) Find the gradie

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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$\text{}$

          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$\text{________}$

Consider the function f(x, y) = 5√((xy)/(6)) which is differentiable at (6, 4).) Find the gradient vector at the point (6, 4) (Give a vector with exact components)cannot proceed with the rest of the problem until you correctly answer the above) At the point (6, 4), the total derivative isproduct form:TD(h1, h2) = ·⟨ h1, h2 ⟩form:TD(h1, h2) =(Give an expression using h1 and h2 in your answer)) Rewrite the total derivative action in the differential notation: Your answer should be an equationdx, dy, df) The linear function L(6, 4) that well approximates the nonlinear function f(x, y) nearby (6, 4), akalocal linearization of f near (6, 4) is given byL(6, 4)(x, y) = + (x - 6) + (y - 4)) Estimate f(6.43, 4.59) using local linearization near (6, 4).f(6.43, 4.59) ≈sure your answer is accurate to at least four decimal places, or give an exact answer. Do notthe exact value of f(6.43, 4.59) which is essentially 11.089353001866.ŋ) In the xyz system, Find the equation of tangent plane to the surface z = 5√((xy)/(6)) at the point(6, 4, z0). (Make sure you calculate the numerical value of z0 = f(6, 4).)) At the point (6, 4), find the directional derivatives in the following directions:of î of ĵdirection as ⟨ -3, 5 ⟩

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