Section 1
Functions of Two or More Variables
No question.
Match the functions(a) $-(d)$ with their contour maps (A) $-(D)$ in Figure $21 .$(a) $f(x, y)=3 x+4 y$(b) $g(x, y)=x^{3}-y$(c) $h(x, y)=4 x-3 y$(d) $k(x, y)=x^{2}-y$
Sketch the graph and draw several vertical and horizontal traces.$$f(x, y)=12-3 x-4 y$$
Sketch the graph and draw several vertical and horizontal traces.$$f(x, y)=\sqrt{4-x^{2}-y^{2}}$$
Sketch the graph and draw several vertical and horizontal traces.$$f(x, y)=x^{2}+4 y^{2}$$
Sketch the graph and draw several vertical and horizontal traces.$$f(x, y)=y^{2}$$
Sketch the graph and draw several vertical and horizontal traces.$$f(x, y)=\sin (x-y)$$
Sketch the graph and draw several vertical and horizontal traces.$$f(x, y)=\frac{1}{x^{2}+y^{2}+1}$$
Sketch contour maps of $f(x, y)=x+y$ with contour intervals $m=1$ and 2 .
Sketch the contour map of $f(x, y)=x^{2}+y^{2}$ with level curves $c=0$, 4,8,12,16
Draw a contour map of $f(x, y)$ with an appropriate contour interval, showing at least six level curves.$$f(x, y)=x^{2}-y$$
Draw a contour map of $f(x, y)$ with an appropriate contour interval, showing at least six level curves.$$f(x, y)=\frac{y}{x^{2}}$$
Draw a contour map of $f(x, y)$ with an appropriate contour interval, showing at least six level curves.$$f(x, y)=\frac{y}{x}$$
Draw a contour map of $f(x, y)$ with an appropriate contour interval, showing at least six level curves.$$f(x, y)=x y$$
Draw a contour map of $f(x, y)$ with an appropriate contour interval, showing at least six level curves.$$f(x, y)=x^{2}+4 y^{2}$$
Draw a contour map of $f(x, y)$ with an appropriate contour interval, showing at least six level curves.$$f(x, y)=x+2 y-1$$
Draw a contour map of $f(x, y)$ with an appropriate contour interval, showing at least six level curves.$$f(x, y)=x^{2}$$
Draw a contour map of $f(x, y)$ with an appropriate contour interval, showing at least six level curves.$$f(x, y)=3 x^{2}-y^{2}$$
Find the linear function whose contour map (with contour interval $m=6$ ) is shown in Figure 22. What is the linear function if $m=3$ (and the curve labeled $c=6$ is relabeled $c=3$ )?
Use the contour map in Figure 23 to calculate the average rate of change:(a) from $A$ to $B$.(b) from $A$ to $C$.
Refer to the map in Figure $24 .$(a) At which of $A-C$ is pressure increasing in the northern direction?(b) At which of $A-C$ is pressure increasing in the westerly direction?
Refer to the map in Figure $24 .$For each of $\mathrm{A}-\mathrm{C}$ indicate in which of the four cardinal directions, $\mathrm{N}, \mathrm{S}, \mathrm{E},$ or $\mathrm{W},$ pressure is increasing the greatest.
Refer to the map in Figure $24 .$Rank the following states in order from greatest change in pressure across the state to least: Arkansas, Colorado, North Dakota, Wisconsin.
Let $T(x, y, z)$ denote temperature at each point in space. Draw level surfaces (also called isotherms) corresponding to the fixed temperatures given.$$T(x, y, z)=x-y+2 z, T=0,1,2$$
Let $T(x, y, z)$ denote temperature at each point in space. Draw level surfaces (also called isotherms) corresponding to the fixed temperatures given.$$T(x, y, z)=x^{2}+y^{2}-z, T=0,1,2$$
Let $T(x, y, z)$ denote temperature at each point in space. Draw level surfaces (also called isotherms) corresponding to the fixed temperatures given.$$T(x, y, z)=x^{2}-y^{2}+z^{2}, T=0,1,2,-1,-2$$
$ \rho(S, T)$ is seawater density (kilograms per cubic meter) as a function of salinity $S$ (parts per thousand) and temperature $T$ (degrees Celsius). Refer to the contour map in Figure $25 .$Calculate the average rate of change of $\rho$ with respect to $T$ from $B$ to $A$.
$ \rho(S, T)$ is seawater density (kilograms per cubic meter) as a function of salinity $S$ (parts per thousand) and temperature $T$ (degrees Celsius). Refer to the contour map in Figure $25 .$Calculate the average rate of change of $\rho$ with respect to $S$ from $B$ to $C$.
$ \rho(S, T)$ is seawater density (kilograms per cubic meter) as a function of salinity $S$ (parts per thousand) and temperature $T$ (degrees Celsius). Refer to the contour map in Figure $25 .$At a fixed level of salinity, is seawater density an increasing or a decreasing function of temperature?
$ \rho(S, T)$ is seawater density (kilograms per cubic meter) as a function of salinity $S$ (parts per thousand) and temperature $T$ (degrees Celsius). Refer to the contour map in Figure $25 .$Does water density appear to be more sensitive to a change in temperature at point $A$ or point $B$ ?
Refer to Figure $26 .$Find the change in seawater density from $A$ to $B$.
Refer to Figure $26 .$Estimate the average rate of change from $A$ to $B$ and from $A$ to C.
Refer to Figure $26 .$Estimate the average rate of change from $A$ to points i, ii, and iii.
Refer to Figure $26 .$Sketch the path of steepest ascent beginning at $D$.
Let temperature in 3-space be given by $T(x, y, z)=x^{2}+y^{2}-z$. Draw isotherms corresponding to temperatures $T=-2,-1,0,1,2 .$
Let temperature in 3 -space be given by $T(x, y, z)=\frac{x^{2}}{4}+\frac{y^{2}}{9}+z^{2}$ Draw isotherms corresponding to temperatures $T=0,1,2$.
Let temperature in 3 -space be given by $T(x, y, z)=x^{2}-y^{2}-z$ Draw isotherms corresponding to temperatures $T=-1,0,1 .$
Let temperature in 3-space be given by $T(x, y, z)=x^{2}-y^{2}-z^{2}$. Draw isotherms corresponding to temperatures $T=-2,-1,0,1,2$.
The function $f(x, t)=t^{-1 / 2} e^{-x^{2} / t},$ whose graph is shown in Figure $27,$ models the temperature along a metal bar after an intense burst of heat is applied at its center point.(a) Sketch the vertical traces at times $t=1,2,3 .$ What do these traces tell us about the way heat diffuses through the bar?(b) Sketch the vertical traces $x=c$ for $c=\pm 0.2,\pm 0.4$. Describe how temperature varies in time at points near the center.
Let$$f(x, y)=\frac{x}{\sqrt{x^{2}+y^{2}}} \quad \text { for }(x, y) \neq(0,0)$$Write $f$ as a function $f(r, \theta)$ in polar coordinates, and use this to find the level curves of $f$