5.10: Chapter 5 Review Exercises
- Page ID
- 84041
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
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.
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.