14.R: Chapter 14 Review Exercises

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Gilbert Strang & Edwin “Jed” Herman
OpenStax

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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.