Chapter 13

Approximate the volume of the region between the graph of \(F\) and the \(x, y\) -plane using six or more subregions of its domain and a point selected in each subregion. a. \(\quad F(x, y)=x \times y \quad 0 \leq x \leq 3 \quad 0 \leq y \leq 2\) b. \(\quad F(x, y)=x+y \quad 1 \leq x \leq 3 \quad 2 \leq y \leq 5\) c. \(\quad F(x, y)=x \times \ln y \quad 0 \leq x \leq 3 \quad 1 \leq y \leq 3\) d. \(\quad F(x, y)=e^{-x-y} \quad 0 \leq x \leq 1 \quad 0 \leq y \leq 1\)

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. \(F(x, y)=x^{3}(1-x)+y \quad\) 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. \(\quad F(x, y)=\sin (x+y)\) i. \(\quad F(x, y)=\frac{x^{2}}{1+y^{2}} \quad\) j. \(\quad F(x, y)=\cos x \sin y\)

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

Sketch the domains over which the integrals are defined. a \(\quad \int_{1}^{5} \int_{x}^{x^{2}} F(x, y) d y d x \quad \mathrm{~b} \quad \int_{1}^{5} \int_{y}^{y^{2}} F(x, y) d x d y \quad\) c \(\quad \int_{1}^{5} \int_{3}^{4} F(x, y) d y d x\) d \(\int_{0}^{\pi} \int_{-\sin x}^{\sin x} F(x, y) d y d x\) e \(\int_{\pi}^{2 \pi} \int_{\sin y}^{-\sin y} F(x, y) d x d y\) \(\int_{0}^{1} \int_{0}^{1-x^{2}} F(x, y) d y d x\)

Cyclic AMP is released by a slime mold amoeba at the center of a \(6 \mathrm{~mm}\) by 4 \(\mathrm{mm}\) flat plate. The concentration at position \((x, y)\) is \(e^{-x^{2}+y^{2}}\) molecules \(/ \mathrm{mm}^{2}\), where the \(x\) -axis runs through the center of the plate in the \(6 \mathrm{~mm}\) direction, the \(y\) -axis runs through the center of the plate and is perpendicular to the \(x\) -axis. a. Write an integral that is the amount of cyclic AMP released by the amoeba. b. Compute an approximate value of the integral.

Find \(C\) and \(b\) so that \(C e^{b x}\) closely approximates the data $$\begin{array}{|r|r|r|r|r|r|}\hline x & 0 & 1 & 2 & 3 & 4 \\ \hline y & 2.18 & 5.98 & 16.1 & 43.6 & 129.7 \\\\\hline\end{array}$$ 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 $$y_{k} \doteq e^{a+b x_{k}}=e^{a} \cdot e^{b x_{k}}=C e^{b x_{k}}, \quad \text { where } \quad C=e^{a} .$$

Is the plane \(z=0\) a tangent plane to the graph of \(F(x, y)=\sqrt{x^{2}+y^{2}}\) shown in Figure 13.2D.

Write but do not compute the iterated form of the integral \(\int_{R} \int F(P) d A\) for the functions \(F\) and domains indicated. In i. and j. write the integral as the sum of two iterated integrals. a. \(\quad F(x, y)=x \times y \quad 0\) \(\leq x \quad \leq 3 \quad 0 \leq y \leq 2\) b. \(\quad F(x, y)=x+y \quad 1 \quad \leq x \quad \leq 3 \quad 2 \leq y \leq 5\) c. \(\quad F(x, y)=x \times \ln y \quad 0 \quad \leq x \quad \leq 3 \quad 1 \leq y \leq 3\) d. \(\quad F(x, y)=x^{2} y\) 0\(\leq x \quad \leq \pi \quad 0 \leq y \leq \sin x\) e. \(\quad F(x, y)=x+y \quad 1 \quad \leq x\) \(x \leq y \leq x^{2}\) f. \(\quad F(x, y)=x \times y^{2} \quad 1 \quad \leq y \quad \leq 2 \quad y \leq x \leq y^{2}\) g. \(\quad F(x, y)=x \times y \quad 0 \quad \leq x+y \quad \leq 2 \quad 0 \leq x, \quad 0 \leq y\) h. \(\quad F(x, y)=x+y \quad 0 \leq x^{2}+y^{2} \leq 1\) i. \(\quad F(x, y)=x \times \ln y \quad 1 \quad \leq x+y \quad \leq 3 \quad 0 \leq x, \quad 0 \leq y\) j. \(F(x, y)=x \times \ln y \quad 1 \leq x^{2}+y^{2} \leq 4 \quad 0 \leq x, \quad 0 \leq y\)

a. Find \(a, b,\) and \(c\) so that \(y=a+b x+c x^{2}\) is the least squares approximation to data, \(\left.\left(x_{1}, y_{1}\right), x_{2}, y_{2}\right), \cdots,\left(x_{n}, y_{n}\right) .\) To do so you will need to minimize $$S S=\sum_{k=1}^{n}\left(y_{k}-\left(a+b x_{k}+c x_{k}^{2}\right)\right)^{2}$$ This is a three-variable minimization problem. The solution will be similar to the least squares line approximation to data of Example 13.2.2. b. In Exercise Table 13.2 .4 are data showing the height of a ball falling in air above a Texas Instruments CBL motion detector. Find the parabola that is the least squares fit to the data. c. Check your answer on a computer or calculator.

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 \(20 ?\)

Evaluate the integrals. a. \(\int_{0}^{1} \int_{2}^{4} x y^{2} d y d x\) b. \(\int_{2}^{4} \int_{0}^{1} x y^{2} d x d y\) c. \(\quad \int_{0}^{1} \int_{2}^{4} x y^{2} d x d y\) d. \(\int_{0}^{1} \int_{0}^{y} x y^{2} d x d y\) e. \(\int_{0}^{1} \int_{x^{2}}^{x} x y d y d x\) f. \(\int_{1}^{4} \int_{y}^{y^{2}} x^{2}+y^{2} d x d y\) g. \(\quad \int_{1}^{2} \int_{e^{-x}}^{e^{x}} \frac{x}{y} d y d x\) h. \(\int_{0}^{\sqrt{3}} \int_{1}^{4-x^{2}} x+y d y d x\) i. \(\int_{0}^{1} \int_{1-x^{2}}^{4-x^{2}} x+y d y d x\)

Find \(a\) and \(b\) so that \(\sin (a x+b)\) closely approximates the data $$\begin{array}{|r|r|r|r|r|r|}\hline \mathrm{x} & 0 & 1 & 2 & 3 & 4 \\ \hline \mathrm{y} & 0.97 & 0.70 & 0.26 & -0.26 & -0.5 \\\\\hline\end{array}$$ Observe that for \(y=\sin (a x+b), \arcsin y=a x+b\). Therefore, fit \(a x+b\) to the number pairs, \((x, \arcsin y)\) using linear least squares.

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 and underneath the hemisphere \(z=\sqrt{25-x^{2}-y^{2}}\)

Let \(F\) be defined by $$\begin{aligned}F(x, y) &=x^{2} \quad \text { for } \quad y>0 \\ &=0 \quad \text { for } \quad y \leq 0\end{aligned}$$ 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 $$\lim _{(x, y) \rightarrow(0,0)} \frac{F(x, y)-L(x, y)}{\sqrt{x^{2}+y^{2}}}=0 \quad ?$$ 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 stressful. The bolus injection of one \(\mathrm{gm}\) at the origin causes the concentration at \(x=0\) and \(t=0\) to be rather large; \(u(0,0)=\infty ;\) but \(u(x, 0)=0\) for \(x \neq 0\) Moving on, we assume that for \(t>0\) $$u_{t}(x, t)=k u_{x x}(x, t)$$ 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 13.42 . 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) .