9.6: Directional Derivatives and the Gradient
- Page ID
- 144346
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 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?