
# 13.1E: Functions of Multiple Variables (Exercises)


For the following exercises, evaluate each function at the indicated values.

1) $$W(x,y)=4x^2+y^2.$$ Find $$W(2,−1), W(−3,6)$$.

$$W(2,−1) = 17,\quad W(−3,6) = 72$$

2) $$W(x,y)=4x^2+y^2$$. Find $$W(2+h,3+h).$$

3) The volume of a right circular cylinder is calculated by a function of two variables, $$V(x,y)=πx^2y,$$ where $$x$$ is the radius of the right circular cylinder and $$y$$ represents the height of the cylinder. Evaluate $$V(2,5)$$ and explain what this means.

$$V(2,5) = 20π\,\text{units}^3$$ This is the volume when the radius is $$2$$ and the height is $$5$$.

4) An oxygen tank is constructed of a right cylinder of height $$y$$ and radius $$x$$ with two hemispheres of radius $$x$$ mounted on the top and bottom of the cylinder. Express the volume of the cylinder as a function of two variables, $$x$$ and $$y$$, find $$V(10,2)$$, and explain what this means.

For exercises 5 - 10, find the domain and range of the given function. State the domain in set-builder notation and the range in interval notation.

5) $$V(x,y)=4x^2+y^2$$

Domain: $$\big\{(x, y) \, | \, x \in \rm I\!R, y \in \rm I\!R\big\}$$ That is, all points in the $$xy$$-plane
Range: $$[0, \infty)$$

6) $$f(x,y)=\sqrt{x^2+y^2−4}$$

Domain: $$\big\{(x, y) \, | \, x^2+y^2 \ge 4\big\}$$
Range: $$[0, \infty)$$

7) $$f(x,y)=4\ln(y^2−x)$$

Domain: $$\big\{(x, y) \, | \, x<y^2 \big\}$$
Range: $$(-\infty, \infty)$$

8) $$g(x,y)=\sqrt{16−4x^2−y^2}$$

Domain: $$\big\{(x, y) \, | \, \dfrac{x^2}{4} + \dfrac{y^2}{16} \le 1\big\}$$
Range: $$[0, 4]$$

9) $$z=\arccos(y−x)$$

Domain: $$\big\{(x, y) \, | \, x - 1 \le y \le x + 1\big\}$$ That is, all points between the graphs of $$y = x -1$$ and $$y = x +1$$.
Range: $$[0, \pi]$$

10) $$f(x,y)=\dfrac{y+2}{x^2}$$

Domain: $$\big\{(x, y) \, | \, x\neq 0 \big\}$$
Range: $$(-\infty, \infty)$$

Find the range of the functions.

11) $$g(x,y)=\sqrt{16−4x^2−y^2}$$

$$\big\{z \, | \, 0≤z≤4\big\}$$ or in interval notation: $$[0,4]$$

12) $$V(x,y)=4x^2+y^2$$

13) $$z=y^2−x^2$$

The set $$\rm I\!R$$

In exercises 14 - 29, find the level curves of each function at the indicated values of $$c$$ to visualize the given function. Sketch a contour plot for those exercises where you are asked for more than 3 values of $$c$$.

14) $$z(x,y)=y^2−x^2, \quad c=1$$

15) $$z(x,y)=y^2−x^2,\quad c=4$$

$$y^2−x^2=4,$$ a hyperbola

16) $$g(x,y)=x^2+y^2;\quad c=0, 1, 2, 3, 4, 9$$

17) $$g(x,y)=4−x−y;\quad c=0,1, 2, 3, 4$$

Level curves are lines with $$y = -x + (4 - c)$$.
For each value of $$c$$ these are:
$$c = 0: \, y = -x + 4$$,
$$c = 1: \, y = -x + 3$$,
$$c = 2: \, y = -x + 2$$,
$$c = 3: \, y = -x + 1$$,
$$c = 4: \, y = -x$$.
The contour plot consists of a series of parallel lines.

18) $$f(x,y)=xy;c=1;\quad c=−1$$

19) $$h(x,y)=2x−y;\quad c=-2,0,2$$

$$2x−y=0,2x−y=−2,2x−y=2;$$ three lines

20) $$f(x,y)=x^2−y;\quad c=1,2$$

21) $$g(x,y)=\dfrac{x}{x+y};c=−1,0,1,2$$

Level curves are lines with the form $$y = x \left( \dfrac{1-c}{c} \right)$$. At $$c = 0$$, we solve it directly from the equation $$\dfrac{x}{x+y}=0$$ to get $$x = 0$$.
For each value of $$c$$ these are:
$$c = -1: \, y = -2x$$,
$$c = 0: \, x = 0,\text{ with }y \ne 0$$,
$$c = 1: \, y = 0,\text{ with }x \ne 0$$,
$$c = 2: \, y = -\frac{1}{2}x$$.

22) $$g(x,y)=x^3−y;\quad c=−1,0,2$$

23) $$g(x,y)=e^{xy};\quad c=\frac{1}{2},3$$

The level curves have the form, $$y = \dfrac{\ln c}{x}$$.
For each value of $$c$$ these are:
$$c = \frac{1}{2}: \, y = \dfrac{\ln \frac{1}{2}}{x}$$ that can be rewritten as, $$y = -\dfrac{\ln 2}{x}$$

$$c = 3: \, y = \dfrac{\ln 3}{x}$$.

24) $$f(x,y)=x^2;\quad c=4,9$$

25) $$f(x,y)=xy−x;\quad c=−2,0,2$$

Level curves have the form: $$y = \dfrac{c}{x} + 1$$.
Here $$y = \dfrac{-2}{x} + 1,\quad y = 1,\quad y = \dfrac{2}{x} + 1$$ or $$xy−x=−2,\,xy−x=0,\,xy−x=2$$

26) $$h(x,y)=\ln(x^2+y^2);\quad c=−1,0,1$$

27) $$g(x,y)=\ln\left(\dfrac{y}{x^2}\right);\quad c=−2,0,2$$

The level curves have the form, $$y =e^c x^2$$.
For each value of $$c$$ these are:
$$c = -2: \, y = e^{-2} x^2$$,
$$c = 0: \, y = x^2$$,
$$c = 2: \, y = e^{2} x^2$$.

28) $$z=f(x,y)=\sqrt{x^2+y^2},\quad c=3$$

29) $$f(x,y)=\dfrac{y+2}{x^2},\quad c=$$ any constant

The level curves are parabolas of the form $$y=cx^2−2,\text{ with }x \ne 0$$.

In exercises 30-32, find the vertical traces of the functions at the indicated values of $$x$$ and $$y$$, and plot the traces.

30) $$z=4−x−y, \quad x=2$$

31) $$f(x,y)=3x+y^3, \quad x=1$$

$$z=3+y^3,$$ a curve in the $$zy$$-plane with rulings parallel to the $$x$$-axis

32) $$z=\cos\sqrt{x^2+y^2}, \quad x=1$$

In exercises 33 - 38, find the domain and range of each function.

33) $$z=\sqrt{100−4x^2−25y^2}$$

Domain: $$\big\{(x, y) \, | \, \dfrac{x^2}{25}+\dfrac{y^2}{4}≤1\big\}$$
Range: $$[0, 10]$$

34) $$z=\ln(x−y^2)$$

35) $$f(x,y,z)=\dfrac{1}{\sqrt{36−4x^2−9y^2−z^2}}$$

Domain: $$\big\{(x, y, z) \, | \, \dfrac{x^2}{9}+\dfrac{y^2}{4}+\dfrac{z^2}{36}<1\big\}$$
Range: $$\big[\frac{1}{6}, \infty\big)$$

36) $$f(x,y,z)=\sqrt{49−x^2−y^2−z^2}$$

37) $$f(x,y,z)=\sqrt[3]{16−x^2−y^2−z^2}$$

Domain: All points in $$xyz$$-space
Range: $$\big(-\infty, \sqrt[3]{16}\,\big]$$

38) $$f(x,y)=\cos\sqrt{x^2+y^2}$$

In exercises 39 - 40, plot a graph of the function.

39) $$z=f(x,y)=\sqrt{x^2+y^2}$$

40) $$z=x^2+y^2$$

41) Use technology to graph $$z=x^2y.$$

In exercises 42 - 46, sketch the function by finding its level curves. Verify the graph using technology, such as CalcPlot3D.

42) $$f(x,y)=\sqrt{4−x^2−y^2}$$

43) $$f(x,y)=2−\sqrt{x^2+y^2}$$

44) $$z=1+e^{−x^2−y^2}$$

45) $$z=\cos\sqrt{x^2+y^2}$$

46) $$z=y^2−x^2$$

47) Describe the contour lines for several values of $$c$$ for $$z=x^2+y^2−2x−2y.$$

The contour lines are concentric circles centered at the point, $$(1, 1)$$.
You can see this by completing the square after setting this function equal to $$c$$.
That is, we write $$x^2-2x+1+y^2−2y+1 = c + 2$$ which can be rewritten as, $$(x - 1)^2 + (y - 1)^2 = c + 2$$.
This gives us circles centered at the point, $$(1, 1)$$, each with a radius of $$\sqrt{c+2}$$.

In exercises, 48 - 52, find the level surface for the given value of $$c$$ for each function of three variables and describe it.

48) $$w(x,y,z)=x−2y+z,\quad c=4$$

49) $$w(x,y,z)=x^2+y^2+z^2,\quad c=9$$

$$x^2+y^2+z^2=9$$, a sphere of radius $$3$$

50) $$w(x,y,z)=x^2+y^2−z^2,\quad c=−4$$

51) $$w(x,y,z)=x^2+y^2−z^2,\quad c=4$$

$$x^2+y^2−z^2=4,$$ a hyperboloid of one sheet

52) $$w(x,y,z)=9x^2−4y^2+36z^2,\quad c=0$$

In exercises 53 - 55, find an equation of the level curve of $$f$$ that contains the point $$P$$.

53) $$f(x,y)=1−4x^2−y^2,\quad P(0,1)$$

$$4x^2+y^2=1,$$

54) $$g(x,y)=y^2\arctan x,\quad P(1,2)$$

55) $$g(x,y)=e^{xy}(x^2+y^2),\quad P(1,0)$$

$$1=e^{xy}(x^2+y^2)$$

56) The strength $$E$$ of an electric field at point $$(x,y,z)$$ resulting from an infinitely long charged wire lying along the $$y$$-axis is given by $$E(x,y,z)=k/\sqrt{x^2+y^2}$$, where $$k$$ is a positive constant. For simplicity, let $$k=1$$ and find the equations of the level surfaces for $$E=10$$ and $$E=100.$$

57) A thin plate made of iron is located in the $$xy$$-plane The temperature $$T$$ in degrees Celsius at a point $$P(x,y)$$ is inversely proportional to the square of its distance from the origin. Express $$T$$ as a function of $$x$$ and $$y$$.

$$T(x,y)=\dfrac{k}{x^2+y^2}$$

58) Refer to the preceding problem. Using the temperature function found there, determine the proportionality constant if the temperature at point $$P(1,2)$$ is $$50°C.$$ Use this constant to determine the temperature at point $$Q(3,4).$$

59) Refer to the preceding problem. Find the level curves for $$T=40°C$$ and $$T=100°C,$$ and describe what the level curves represent.

$$x^2+y^2=\dfrac{k}{40}, \quad x^2+y^2=\dfrac{k}{100}$$. The level curves represent circles of radii $$\sqrt{10k}/20$$ and $$\sqrt{k}/10$$