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Mathematics LibreTexts

13R: Chapter 13 Review Exercises

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    67642
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    For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.

    1.  The domain of \(f(x,y)=x^3\arcsin(y)\) is  \( \big\{ (x,y) \, | \, x \in \mathbb R\text{ and }−\pi≤y≤\pi \big\}.\)

    2.  If the function \(f(x,y)\) is continuous everywhere, then \(f_{xy}(x,y) =f_{yx}(x,y).\)

    Answer:
    True, by Clairaut’s theorem

    3.  The linear approximation to the function of \(f(x,y)=5x^2+x\tan y\) at the point \((2,π)\) is given by \(L(x,y)=22+21(x−2)+(y−π).\)

    4.  \((34,916)\) is a critical point of \(g(x,y)=4x^3−2x^2y+y^2−2.\)

    Answer:
    False

     

    For the following exercises, sketch the function in one graph and, in a second, sketch several level curves.

    5.  \(f(x,y)=e^{−\left(x^2+2y^2\right)}\)

    6.  \(f(x,y)=x+4y^2\)

    Answer:
    Contour Plot for function z = x + 4y^2

    For the following exercises, evaluate the following limits, if they exist. If they do not exist, prove it.

    7.  \(\displaystyle \lim_{(x,y)→(1,1)}\frac{4xy}{x−2y^2}\)

    8.  \(\displaystyle \lim_{(x,y)→(0,0)}\frac{4xy}{x−2y^2}\)

    Answer:
    Does not exist.

     

    For the following exercises, find the largest interval of continuity for the function.

    9.  \(f(x,y)=x^3\arcsin y\)

    10.  \(g(x,y)=\ln(4−x^2−y^2)\)

    Answer:
    Continuous at all points on the \(xy\)-plane, except where \(x^2 + y^2 > 4.\)

     

    For the following exercises, find all first partial derivatives.

    11.  \(f(x,y)=x^2−y^2\)

    12.  \(u(x,y)=x^4−3xy+1,\) with \(x=2t\) and \(y=t^3\)

    Answer:
    \(\dfrac{∂u}{∂x}=4x^3−3y,\)

    \( \dfrac{∂u}{∂y}=−3x,\)

    \(\dfrac{dx}{dt} = 2\)  and \(\dfrac{dy}{dt} = 3t^2\)

    \(\begin{align*} \dfrac{du}{dt}  &= \dfrac{∂u}{∂x}\cdot\dfrac{dx}{dt} + \dfrac{∂u}{∂y}\cdot\dfrac{dy}{dt}\\[4pt]
    &= 8x^3 -6y -9xt^2\\[4pt]
    &= 8\big(2t\big)^3 - 6(t^3) - 9(2t)t^2 \\[4pt]
    &= 64t^3 - 6t^3 - 18t^3 \\[4pt]
    &= 40t^3 \end{align*}\)

     

    For the following exercises, find all second partial derivatives.

    13.  \(g(t,x)=3t^2−\sin(x+t)\)

    14.  \(h(x,y,z)=\dfrac{x^3e^{2y}}{z}\)

    Answer:
    \(h_{xx}(x,y,z) = \dfrac{6xe^{2y}}{z},\)
    \(h_{xy}(x,y,z) = \dfrac{6x^2e^{2y}}{z},\)
    \(h_{xz}(x,y,z) = −\dfrac{3x^2e^{2y}}{z^2},\)
    \(h_{yx}(x,y,z) = \dfrac{6x^2e^{2y}}{z},\)
    \(h_{yy}(x,y,z) = \dfrac{4x^3e^{2y}}{z},\)
    \(h_{yz}(x,y,z) = −\dfrac{2x^3e^{2y}}{z^2},\)
    \(h_{zx}(x,y,z) = −\dfrac{3x^2e^{2y}}{z^2},\)
    \(h_{zy}(x,y,z) = −\dfrac{2x^3e^{2y}}{z^2},\)
    \(h_{zz}(x,y,z) = \dfrac{2x^3e^{2y}}{z^3}\)

     

    For the following exercises, find the equation of the tangent plane to the specified surface at the given point.

    15.  \(z=x^3−2y^2+y−1\) at point \((1,1,−1)\)

    16.  \(z=e^x+\dfrac{2}{y}\) at point \((0,1,3)\)

    Answer:
    \(z = x - 2y + 5\)

     

    17.  Approximate \(f(x,y)=e^{x^2}+\sqrt{y}\) at \((0.1,9.1).\) Write down your linear approximation function \(L(x,y).\) How accurate is the approximation to the exact answer, rounded to four digits?

    18.  Find the differential \(dz\) of \(h(x,y)=4x^2+2xy−3y\) and approximate \(Δz\) at the point \((1,−2).\) Let \(Δx=0.1\) and \(Δy=0.01.\)

    Answer:
    \(dz=4\,dx−dy, \; dz(0.1,0.01)=0.39, \; Δz = 0.432\)

    19.  Find the directional derivative of \(f(x,y)=x^2+6xy−y^2\) in the direction \(\vecs v=\mathbf{\hat i}+4\,\mathbf{\hat j}.\)

    20.  Find the maximal directional derivative magnitude and direction for the function \(f(x,y)=x^3+2xy−\cos(πy)\) at point \((3,0).\)

    Answer:
    \(3\sqrt{85}\langle 27, 6\rangle\)

     

    For the following exercises, find the gradient.

    21.  \(c(x,t)=e(t−x)^2+3\cos t\)

    22.  \(f(x,y)=\dfrac{\sqrt{x}+y^2}{xy}\)

    Answer:
    \(\vecs \nabla f(x, y) = -\dfrac{\sqrt{x}+2y^2}{2x^2y}\,\mathbf{\hat i} + \left( \dfrac{1}{x} + \dfrac{1}{\sqrt{x}y^2} \right) \,\mathbf{\hat j}\)

     

    For the following exercise, find and classify the critical points.

    23.  \(z=x^3−xy+y^2−1\)

     

    For the following exercises, use Lagrange multipliers to find the maximum and minimum values for the functions with the given constraints.

    24.  \(f(x,y)=x^2y,\)  subject to the constraint:  \(x^2+y^2=4\)

    Answer:
    maximum: \(\dfrac{16}{3\sqrt{3}},\) minimum: \(-\dfrac{16}{3\sqrt{3}},\)

    25.  \(f(x,y)=x^2−y^2,\)  subject to the constraint:  \(x+6y=4\)

     

    26.  A machinist is constructing a right circular cone out of a block of aluminum. The machine gives an error of \(5\%\) in height and \(2\%\) in radius. Find the maximum error in the volume of the cone if the machinist creates a cone of height \(6\) cm and radius \(2\) cm.

    Answer:
    \(2.3228\) cm3

    27.  A trash compactor is in the shape of a cuboid. Assume the trash compactor is filled with incompressible liquid. The length and width are decreasing at rates of \(2\) ft/sec and \(3\) ft/sec, respectively. Find the rate at which the liquid level is rising when the length is \(14\) ft, the width is \(10\) ft, and the height is \(4\) ft.

     

    Contributors

    • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

     

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