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Mathematics LibreTexts

6.6E:

  • Page ID
    25952
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     Directional Derivatives and the Gradient

    For the following exercises, find the directional derivative using the limit definition only.

    1) \(\displaystyle f(x,y)=5−2x^2−\frac{1}{2}y^2\) at point \(\displaystyle P(3,4)\) in the direction of \(\displaystyle u=(cos\frac{π}{4})i+(sin\frac{π}{4})j\)

    2) \(\displaystyle f(x,y)=y^2cos(2x)\) at point \(\displaystyle P(\frac{π}{3},2)\) in the direction of \(\displaystyle u=(cos\frac{π}{4})i+(sin\frac{π}{4})j\)

    Solution:\(\displaystyle −3\sqrt{3}\)

    3) Find the directional derivative of \(\displaystyle f(x,y)=y^2sin(2x)\) at point \(\displaystyle P(\frac{π}{4},2)\) in the direction of \(\displaystyle u=5i+12j.\)

     

    For the following exercises, find the directional derivative of the function at point P in the direction of v.

    4) \(\displaystyle f(x,y)=xy, P(0,−2), v=\frac{1}{2}i+\frac{\sqrt{3}}{2}j\)

    Solution:\(\displaystyle −1\)

    5) \(\displaystyle h(x,y)=e^xsiny,P(1,\frac{π}{2}),v=−i\)

    6) \(\displaystyle h(x,y,z)=xyz,P(2,1,1),v=2i+j−k\)

    Solution:\(\displaystyle \frac{2}{\sqrt{6}}\)

    7) \(\displaystyle f(x,y)=xy,P(1,1),u=⟨\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}⟩\)

    8) \(\displaystyle f(x,y)=x^2−y^2,u=⟨\frac{\sqrt{3}}{2},\frac{1}{2}⟩,P(1,0)\)

    Solution:\(\displaystyle \sqrt{3}\)

    9) \(\displaystyle f(x,y)=3x+4y+7,u=⟨\frac{3}{5},\frac{4}{5}⟩,P(0,\frac{π}{2})\)

    10) \(\displaystyle f(x,y)=e^xcosy,u=⟨0,1⟩,P=(0,\frac{π}{2})\)

    Solution:\(\displaystyle −1.0\)

    11) \(\displaystyle f(x,y)=y^{10},u=⟨0,−1⟩,P=(1,−1)\)

    12) \(\displaystyle f(x,y)=ln(x^2+y^2),u=⟨\frac{3}{5},\frac{4}{5}⟩,P(1,2)\)

    Solution:\(\displaystyle \frac{22}{25}\)

    13) \(\displaystyle f(x,y)=x^2y,P(−5,5),v=3i−4j\)

    14) \(\displaystyle f(x,y)=y^2+xz,P(1,2,2),v=⟨2,−1,2⟩\)

    Solution:\(\displaystyle \frac{2}{3}\)

     

    For the following exercises, find the directional derivative of the function in the direction of the unit vector \(\displaystyle u=cosθi+sinθj.\)

    15) \(\displaystyle f(x,y)=x^2+2y^2,θ=\frac{π}{6}\)

    16) \(\displaystyle f(x,y)=\frac{y}{x+2y},θ=−\frac{π}{4}\)

    Solution:\(\displaystyle \frac{−\sqrt{2}(x+y)}{2(x+2y)^2}\)

    17) \(\displaystyle f(x,y)=cos(3x+y),θ=\frac{π}{4}\)

    18) \(\displaystyle w(x,y)=ye^x,θ=\frac{π}{3}\)

    Solution:\(\displaystyle \frac{e^x(y+\sqrt{3})}{2}\)

    19) \(\displaystyle f(x,y)=xarctan(y),θ=\frac{π}{2}\)

    20) \(\displaystyle f(x,y)=ln(x+2y),θ=\frac{π}{3}\)

    Solution:\(\displaystyle \frac{1+2\sqrt{3}}{2(x+2y)}\)

     

    For the following exercises, find the gradient.

    21) Find the gradient of \(\displaystyle f(x,y)=\frac{14−x^2−y^2}{3}\). Then, find the gradient at point \(\displaystyle P(1,2).\)

    22) Find the gradient of \(\displaystyle f(x,y,z)=xy+yz+xz\) at point \(\displaystyle P(1,2,3).\)

    Solution:\(\displaystyle ⟨5,4,3⟩\)

    23) Find the gradient of \(\displaystyle f(x,y,z))\) at \(\displaystyle P\) and in the direction of \(\displaystyle u: f(x,y,z)=ln(x^2+2y^2+3z^2),P(2,1,4),u=\frac{−3}{13}i−\frac{4}{13}j−\frac{12}{13}k.\)

    24) \(\displaystyle f(x,y,z)=4x^5y^2z^3,P(2,−1,1),u=\frac{1}{3}i+\frac{2}{3}j−\frac{2}{3}k\)

    Solution:\(\displaystyle −320\)

     

    For the following exercises, find the directional derivative of the function at point \(\displaystyle P\) in the direction of \(\displaystyle Q\).

    25) \(\displaystyle f(x,y)=x^2+3y^2,P(1,1),Q(4,5)\)

    26) \(\displaystyle f(x,y,z)=\frac{y}{x+z},P(2,1,−1),Q(−1,2,0)\)

    Solution:\(\displaystyle \frac{3}{\sqrt{11}}\)

     

    For the following exercises, find the derivative of the function at \(\displaystyle P\) in the direction of \(\displaystyle u\).

    27) \(\displaystyle f(x,y)=−7x+2y,P(2,−4),u=4i−3j\)

    28) \(\displaystyle f(x,y)=ln(5x+4y),P(3,9),u=6i+8j\)

    Solution:\(\displaystyle \frac{31}{255}\)

    29) [T] Use technology to sketch the level curve of \(\displaystyle f(x,y)=4x−2y+3\) that passes through \(\displaystyle P(1,2)\) and draw the gradient vector at \(\displaystyle P\).

    30) [T] Use technology to sketch the level curve of \(\displaystyle f(x,y)=x^2+4y^2\) that passes through \(\displaystyle P(−2,0)\) and draw the gradient vector at P.

    Solution:

    The top of half of an ellipse centered at the origin with major axis horizontal and of length 4 and minor axis 2. The point (–2, 0) is marked, and there is an arrow pointing out from it to the left marked –4i.

     

    For the following exercises, find the gradient vector at the indicated point.

    31) \(\displaystyle f(x,y)=xy^2−yx^2,P(−1,1)\)

    32) \(\displaystyle f(x,y)=xe^y−ln(x),P(−3,0)\)

    Solution:\(\displaystyle \frac{4}{3}i−3j\)

    33) \(\displaystyle f(x,y,z)=xy−ln(z),P(2,−2,2)\)

    34) \(\displaystyle f(x,y,z)=x\sqrt{y^2+z^2}, P(−2,−1,−1)\)

    Solution:\(\displaystyle \sqrt{2}i+\sqrt{2}j+\sqrt{2}k\)

     

    For the following exercises, find the derivative of the function.

    35) \(\displaystyle f(x,y)=x^2+xy+y^2\) at point \(\displaystyle (−5,−4)\) in the direction the function increases most rapidly

    36) \(\displaystyle f(x,y)=e^{xy}\) at point \(\displaystyle (6,7)\) in the direction the function increases most rapidly

    Solution:\(\displaystyle 1.6(10^{19})\)

    37) \(\displaystyle f(x,y)=arctan(\frac{y}{x})\) at point \(\displaystyle (−9,9)\) in the direction the function increases most rapidly

    38) \(\displaystyle f(x,y,z)=ln(xy+yz+zx)\) at point \(\displaystyle (−9,−18,−27)\) in the direction the function increases most rapidly

    Solution:\(\displaystyle \frac{5\sqrt{2}}{99}\)

    39) \(\displaystyle f(x,y,z)=\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\) at point \(\displaystyle (5,−5,5)\) in the direction the function increases most rapidly

     

    For the following exercises, find the maximum rate of change of \(\displaystyle f\) at the given point and the direction in which it occurs.

    40) \(\displaystyle f(x,y)=xe^{−y}, (1,0)\)

    Solution:\(\displaystyle \sqrt{5},⟨1,2⟩\)

    41) \(\displaystyle f(x,y)=\sqrt{x^2+2y}, (4,10)\)

    42) \(\displaystyle f(x,y)=cos(3x+2y),(\frac{π}{6},−\frac{π}{8})\)

    Solution:\(\displaystyle \sqrt{\frac{13}{2}},⟨−3,−2⟩\)

     

    For the following exercises, find equations of

    a. the tangent plane and

    b. the normal line to the given surface at the given point.

    43) The level curve \(\displaystyle f(x,y,z)=12\) for \(\displaystyle f(x,y,z)=4x^2−2y^2+z^2\) at point \(\displaystyle (2,2,2).\)

    44) \(\displaystyle f(x,y,z)=xy+yz+xz=3\) at point \(\displaystyle (1,1,1)\)

    Solution:\(\displaystyle a. x+y+z=3, b. x−1=y−1=z−1\)

    45) \(\displaystyle f(x,y,z)=xyz=6\) at point \(\displaystyle (1,2,3)\)

    46) \(\displaystyle f(x,y,z)=xe^ycosz−z=1\) at point \(\displaystyle (1,0,0)\)

    Solution:\(\displaystyle a. x+y−z=1, b. x−1=y=−z\)

     

    For the following exercises, solve the problem.

    47) The temperature \(\displaystyle T\) in a metal sphere is inversely proportional to the distance from the center of the sphere (the origin: \(\displaystyle (0,0,0))\). The temperature at point \(\displaystyle (1,2,2)\) is \(\displaystyle 120°C.\)

    a. Find the rate of change of the temperature at point \(\displaystyle (1,2,2)\) in the direction toward point \(\displaystyle (2,1,3).\)

    b. Show that, at any point in the sphere, the direction of greatest increase in temperature is given by a vector that points toward the origin.

    48) The electrical potential (voltage) in a certain region of space is given by the function \(\displaystyle V(x,y,z)=5x^2−3xy+xyz.\)

    a. Find the rate of change of the voltage at point \(\displaystyle (3,4,5)\) in the direction of the vector \(\displaystyle ⟨1,1,−1⟩.\)

    b. In which direction does the voltage change most rapidly at point \(\displaystyle (3,4,5)\)?

    c. What is the maximum rate of change of the voltage at point \(\displaystyle (3,4,5)\)?

    Solution:\(\displaystyle a. \frac{32}{\sqrt{3}}, b. ⟨38,6,12⟩, c. 2\sqrt{406}\)

    49) If the electric potential at a point \(\displaystyle (x,y)\) in the xy-plane is \(\displaystyle V(x,y)=e^{−2x}cos(2y)\), then the electric intensity vector at \(\displaystyle (x,y)\) is \(\displaystyle E=−∇V(x,y).\)

    a. Find the electric intensity vector at \(\displaystyle (\frac{π}{4},0).\)

    b. Show that, at each point in the plane, the electric potential decreases most rapidly in the direction of the vector \(\displaystyle E.\)

    50) In two dimensions, the motion of an ideal fluid is governed by a velocity potential \(\displaystyle φ\). The velocity components of the fluid u in the x-direction and v in the y-direction, are given by \(\displaystyle ⟨u,v⟩=∇φ\). Find the velocity components associated with the velocity potential \(\displaystyle φ(x,y)=sinπxsin2πy.\)

    Solution:\(\displaystyle ⟨u,v⟩=⟨πcos(πx)sin(2πy),2πsin(πx)cos(2πy)⟩\)