9.6: Directional Derivatives and the Gradient
- Page ID
- 144346
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- Find the directional derivative of \(f(x, y) = xy\) at point \(P(-2, 0)\) in the direction of \(\mathbf{v} = \mathbf{i} + \sqrt{3}\,\mathbf{j}\).
- Find the directional derivative of \(g(x, y) = e^x\sin y\) at point \(Q\left(1, \dfrac{\pi}{4}\right)\) in the direction of \(\mathbf{u} = 4\mathbf{i} - 3\mathbf{j}\).
- Find the directional derivative of \(h(x, y) = \ln(x^2 + y^2)\) at point \(P(1, 2)\) in the direction of \(\mathbf{v} = (2, -5)\).
- Find the directional derivative of \(f(x, y, z) = y^2 + xz\) at point \(P(1, 2, 2)\) in the direction of \(\mathbf{u} = (2, -1, 2)\).
- The temperature of a thin plate in the \(xy\)-plane is \(T = x^2 + y^2\). How fast does the temperature change at the point \((1, 5)\) moving in the direction of \(\mathbf{v} = (\sqrt{3}, 1)\)?
- Suppose the density of a thin plate at the point \((x, y)\) is \(\rho(x, y) = \dfrac{1}{\sqrt{1 + x^2 + y^2}}\). Find the rate of change of the density at \((2, 1)\) in the direction \(\mathbf{u} = -\mathbf{j}\).
- In what direction does \(f(x, y) = x^2 + xy + y^2\) increase most rapidly at the point \(P(-5, -4)\)? In what direction does \(f(x, y)\) decrease most rapidly at \(P\)?
- In what direction does \(f(x, y) = \arctan\left(\dfrac{y}{x}\right)\) increase most rapidly at the point \(P(-9, 9)\)? In what direction does \(f(x, y)\) decrease most rapidly at \(P\)?
- Suppose the temperature at \((x, y, z)\) is given by \(T = xy + \sin(yz)\). In what direction should you go from point \((1, 1, 1)\) to decrease the temperature as quickly as possible?
- A bug is crawling on the surface of a hot plate. The temperature of the plate at point \((x, y)\) is given by \(T(x, y) = 100 - x^2 - 3y^3\).
- If the bug is at the point \((2, 1)\), in what direction should it move to cool off the fastest? How fast will the temperature drop in this direction?
- If the bug is at the point \((1, 3)\), in what direction should it move in order to maintain its temperature?
- If the bug is at the point \((2, 1)\), in what direction should it move to cool off the fastest? How fast will the temperature drop in this direction?
- The National Oceanic and Atmospheric Administration (NOAA) measures the atmospheric pressure inside a hurricane. The NOAA finds that the atmospheric pressure of the hurricane at point \((x, y, z)\) can be modeled by the function \(\displaystyle P(x, y, z) = 2^{2(55-z)/11}\left(1 - \dfrac{1}{10(1 + x^2 + y^2)}\right)\). At point \((20, 80, 10)\), in what direction does the atmospheric pressure increase the fastest?