Chapter 13

Find the critical points, if any, of \(F\). a. \(\quad F(x, y)=2 x+5 y+7\) b. \(\quad F(x, y)=x^{2}+4 x y+3 y^{2}\) c. \(\quad F(x, y)=x^{3}(1-x)+y\) d. \(\quad F(x, y)=x y(1-x y)\) e. \(\quad F(x, y)=\left(x-x^{2}\right)\left(y-y^{2}\right) \quad\) f. \(\quad F(x, y)=\frac{x}{y}\) g. \(\quad F(x, y)=e^{x+y}\) h. \(\quad F(x, y)=\sin (x+y)\) \(\begin{array}{lll}\text { i. } & F(x, y)=\frac{x^{2}}{1+y^{2}} & \text { j. } \quad F(x, y)\end{array}=\cos x \sin y\)

Draw three dimensional graphs of a. $$ F(x, y)=2 \quad \text { b. } \quad F(x, y)=x $$ c. $$ F(x, y)=x^{2} $$ $$ \text { d. } \quad F(x, y)=(x+y) / 2 $$ e. \(F(x, y)=0.2 x+0.3 y \quad\) f. \(\quad F(x, y)=\left(x^{2}+y^{2}\right) / 4\) $$ F(x, y)=0.5 x e^{-y} $$ g. h. \(\quad F(x, y)=\sin y\) i. \(\quad F(x, y)=0.5 x+\sin y \quad\) j. \(\quad F(x, y)=x \sin y\) k. \(\quad F(x, y)=\sqrt{x^{2}+y^{2}} \quad\) 1. \(\quad F(x, y)=x y\) m. \(\quad F(x, y)=\frac{1}{0.4+x^{2}+y^{2}}\) n. \(\quad F(x, y)=e^{\left(-x^{2}-y^{2}\right)}\) o. \(F(x, y)=|x y| \quad\) p. \(\quad F(x, y)=\sin \left(x^{2}+y^{2}\right)\)

For each of the following functions, find the critical points and use Theorem 13.2 .1 to determine whether they are local maxima, local minima, or saddle points or none of these. a. $$ F(x, y)=-x^{2}+x y-y^{2} $$ b. $$ F(x, y)=x^{2}+x y-y^{2} $$ c. $$ F(x, y)=x^{2}+y^{2}-2 x y+2 x-2 y $$ d. $$ F(x, y)=-x^{2}-5 y^{2}+2 x y-10 x+6 y+20 $$

Find the partial derivatives, \(F_{1}, F_{2}, F_{1,1}, F_{1,2}, F_{2,1}\) and \(F_{2,2}\) of the following functions. a. \(\quad F(x, y)=3 x-5 y+7\) b. \(\quad F(x, y)=x^{2}+4 x y+3 y^{2}\) c. \(\quad F(x, y)=x^{3} y^{5}\) d. \(\quad F(x, y)=\sqrt{x y}\) e. \(\quad F(x, y)=\ln (x \times y) \quad\) f. \(\quad F(x, y)=\frac{x}{y}\) g. \(\quad F(x, y)=e^{x+y}\) h. \(\quad F(x, y)=x^{2} e^{-y}\) i. \(\quad F(x, y)=\sin (2 x+3 y)\) j. \(F(x, y)=e^{-x} \cos y\)

Find \(C\) and \(b\) so that \(C e^{0 x}\) closely approximates the data \begin{tabular}{|r|r|r|r|r|r|} \hline\(x\) & 0 & 1 & 2 & 3 & 4 \\ \hline\(y\) & 2.18 & 5.98 & 16.1 & 43.6 & 129.7 \\ \hline \end{tabular} Observe that for \(y=C e^{b x}, \ln y=\ln C+b x\). Therefore, fit \(a+b x\) to the number pairs, \((x, \ln y)\) using linear least squares. Then \(\ln y_{k} \doteq a+b x_{k},\) and $$ y_{k} \doteq e^{a+b x_{k}}=e^{a} \times e^{b x_{k}}=C e^{b x_{k}}, \quad \text { where } \quad C=e^{a} $$

a. Show that $$ u(x, t)=20 e^{-t} \sin \pi x, \quad 0 \leq x \leq 1, \quad 0 \leq t $$ solves $$ u_{t}(x, t)=\frac{1}{\pi^{2}} u_{x x}(x, t), \quad u(x, 0)=20 \sin \pi x, \quad \text { and } \quad u(0, t)=u(1, t)=0 $$ b. Describe a physical problem for which this is a solution. c. What is the 'eventual' value of \(u(x, t)\) (what is \(\left.\lim _{t \rightarrow \infty} u(x, t)\right) ?\) d. At what time, \(t,\) will the maximum value of \(u(x, t)\) for \(0 \leq x \leq 1\) be \(20 ?\)

For \(P=n R T / V\), find \(\frac{\partial}{\partial V} P\) and \(\frac{\partial}{\partial T} P\). For fixed \(T\), how does \(P\) change as \(V\) increases? For fixed \(V\), how does \(P\) change as \(T\) increases?

Find the largest box that will fit in the positive octant \((x \geq 0, y \geq 0,\) and \(z \geq 0)\) and underneath the plane \(z=12-2 x-3 y\).

Find the largest box that will fit in the positive octant \((x \geq 0, y \geq 0,\) and \(z \geq 0)\) and underneath the plane \(z=12-2 x-3 y\).

Let \(F\) be defined by $$ \begin{aligned} F(x, y) &=x^{2} & \text { for } & y>0 \\ &=0 & \text { for } \quad y \leq 0 \end{aligned} $$ 1\. Sketch a graph of \(F\) in three dimensional space. 2\. Is \(F_{1}(x, y)\) continuous on the interior of a circle with center (0,0)\(?\) 3\. Let \(L(x, y)=0\) for all \((x, y) .\) Is it true that $$ \lim _{(x, y) \rightarrow(0,0)} \frac{F(x, y)-L(x, y)}{\sqrt{x^{2}+y^{2}}}=0 \text { ? } $$ 4\. Are you willing to call the plane \(z=0\) a tangent plane to the graph of \(F ?\)

Find the point of the plane \(z=2 x+3 y-12\) that is 1\. closest to the origin. 2\. closest to (4,5,6)

Suppose there is an infinitely long tube containing water lying along the \(X\) -axis from \(-\infty\) to \(\infty\) and at time \(t=0\) a bolus injection of one gram of salt is made at the origin. Let \(u(x, t)\) be the concentration of salt at position \(x\) in the tube at time \(t\). Considering \(t=0\) is a bit of stressful: \(u(x, 0)=0\) for \(x \neq 0 ;\) but the bolus injection of one \(\mathrm{gm}\) at the origin causes the concentration at \(x=0\) and \(t=0\) to be rather large; \(u(0,0)=\infty\). Moving on, for \(t>0\) we may assume that $$ u_{t}(x, t)=k u_{x x}(x, t) $$ where the diffusion coefficient, \(k,\) describes the rate at which salt diffuses in water. a. Show that $$ u(x, t)=\frac{1}{\sqrt{4 \pi k t}} e^{-x^{2} /(4 k t)} $$ is a solution to Equation 13.38 . b. Suppose \(k=1 / 4\). Sketch the graphs of \(u(x, 1), u(x, 4),\) and \(u(x, 8)\). c. Suppose \(k=1 / 4\). Sketch the graphs of \(u(x, 1), u(x, 1 / 2),\) and \(u(x, 1 / 4)\). d. Estimate the areas under the previous curves. For any time, \(t_{0},\) what do you expect to be the area under the curve of \(u\left(x, t_{0}\right), \infty

Find the point of the sphere \(x^{2}+y^{2}+z^{2}=25\) that is closest to (3,4,5) .