Question 13.4

Are the following functions vector-valued or scalar-valued?

• (a) \(f(x, y, z)=e^x z^x \sin y\)
• (b) \(g(x, y)= (x^2y^2, 2x-1)\)
• (c) \(h(t)=(\cos t, \sin t, t^2, t^3)\)

Question 13.5

Are the following functions vector-valued or scalar-valued?

• (a) \(f(u, v, w)=(u^2v, we^u, 5v)\)
• (b) \(g(x)=\log \sqrt{x}\)
• (c) \(h(x, y)=x^5y^{-3}\)

Question 13.6

• (a) \(f(x,y) = x^2 - y^2 = c, \quad c=0, 1, -1\)
• (b) \(f(x,y) = 2x^2 + 3y^2 = c, \quad c=6, 12\)

86

Question 13.7

• (a) \(f(x,y) = (x - y)^2 = c, \quad c=0, 1, 4\)
• (b) \(f(x,y) = (x + y)^2 = c, \quad c=0,1,4\)

Question 13.8

Draw the level curves for \(f\) of values \(c\).

• (a) \(f(x, y)=x^3-y, \ c=-1, 0, 1\)
• (b) \(f(x, y)=y-2\log x, \ c=-3, 0, 3\)
• (c) \(f(x, y)=y\csc x, \ c=0, 1, 2\)
• (d) \(f(x, y)=x/(x^2+y^2), \ c=-2, 0, 4\)

Question 13.9

Let \(f(x, y)=9x^2+y^2\). Sketch the following.

• (a) The level curves for \(f\) of values \(c=0, 1, 9\)
• (b) The sections of the graph of \(f\) in the planes \(x=-1, x=0, x=1\)
• (c) The sections of the graph of \(f\) in the planes \(y=-1, y=0, y=1\)
• (d) The graph of \(f\)

Question 13.10

Sketch the level curves and graphs of the following functions:

• (a) \(f\colon\,{\mathbb R}^2 \rightarrow {\mathbb R},\ (x,y)\mapsto x-y +2 \)
• (b) \(f\colon\, {\mathbb R}^2 \rightarrow {\mathbb R},\ (x,y)\mapsto x^2 +4y^2 \)
• (c) \(f\colon\, {\mathbb R}^2 \rightarrow {\mathbb R},\ (x,y)\mapsto -xy\)

Question 13.11

Sketch level sets of values \(c=0, 1, 4, 9\) for both \(f(x,y)=x^2+y^2\) and \(g(x, y)=\sqrt{x^2+y^2}\). How are the graphs of \(f\) and \(g\) different? How are their sections different?

Question 13.12

Let \(S\) be the surface in \(\mathbb{R}^3\) defined by the equation \(x^2y^6-2z=3\).

• (a) Find a real-valued function \(f(x, y, z)\) of three variables and a constant \(c\) such that \(S\) is the level set of \(f\) of value \(c\).
• (b) Find a real-valued function \(g(x, y)\) of two variables such that \(S\) is the graph of \(g\).

Question 13.13

Describe the behavior, as \(c\) varies, of the level curve \(f(x,y)=c\) for each of these functions:

• (a) \(f(x,y)=x^2+y^2 +1 \)
• (b) \(f(x,y)= 1- x^2 - y^2 \)
• (c) \(f(x,y)=x^3 - x\)

Question 13.14

For the functions in Examples 2, 3, and 4, compute the section of the graph defined by the plane \[ S_\theta = \{ (x,y,z) \mid y = x \tan \theta \} \] for a given constant \(\theta\). Do this by expressing \(z\) as a function of \(r\), where \(x=r \cos \theta, y = r \sin\, \theta\). Determine which of these functions \(f\) have the property that the shape of the section \(S_{\theta} \cap \hbox{graph } f\) is independent of \(\theta\). (The solution for Example 3 only is in the Study Guide.)

Question 13.15

\(f(x,y)=4 -3x + 2y, c=0,1,2,3,-1,-2,-3\)

Question 13.16

\(f(x,y)= (100 -x^2- y^2)^{1/2}, c=0,2,4,6,8,10\)

Question 13.17

\(f(x,y)= (x^2 + y^2)^{1/2}, c=0,1,2,3,4,5\)

Question 13.18

\(f(x,y)= x^2 + y^2, c=0,1,2,3,4,5\)

Question 13.19

\(f(x,y)= 3x - 7y, c=0,1,2,3,-1,-2,-3\)

Question 13.20

\(f(x,y)= x^2 +xy , c=0,1,2,3,-1,-2,-3\)

Question 13.21

\(f(x,y)= x/y, c=0,1,2,3,-1,-2,-3\)

Question 13.22

\(f\colon\, {\mathbb R}^3 \rightarrow {\mathbb R}, (x,y,z) \mapsto -x^2 - y^2 - z^2\)

Question 13.23

\(f\colon\, {\mathbb R}^3 \rightarrow {\mathbb R}, (x,y,z) \mapsto 4x^2 + y^2 + 9z^2\)

Question 13.24

\(f\colon\, {\mathbb R}^3 \rightarrow {\mathbb R}, (x,y,z) \mapsto x^2 + y^2\)

Question 13.25

\(f\colon\, {\mathbb R}^3 \rightarrow {\mathbb R}, (x,y,z) \mapsto xy\)

Question 13.26

\(f\colon\, {\mathbb R}^3 \rightarrow {\mathbb R}, (x,y,z) \mapsto xy +yz\)

Question 13.27

\(f\colon\, {\mathbb R}^3 \rightarrow {\mathbb R}, (x,y,z) \mapsto xy + z^2\)

Question 13.28

\(f\colon\, {\mathbb R}^2 \rightarrow {\mathbb R}, (x,y) \mapsto |y|\)

Question 13.29

\(f\colon\, {\mathbb R}^2 \rightarrow {\mathbb R}, (x,y) \mapsto \max\, (|x|, |y|)\)

Question 13.30

\(4x^2 + y^2 = 16\)

Question 13.31

\(x + 2z = 4\)

Question 13.32

\(z^2 = y^2 + 4\)

Question 13.33

\(x^2 + y^2 -2x = 0\)

Question 13.34

\(\displaystyle \frac{x}{4} = \frac{y^2}{4} + \frac{z^2}{9}\)

Question 13.35

\(\displaystyle \frac{y^2}{9} + \frac{z^2}{4} = 1 + \frac{x^2}{16}\)

Question 13.36

\(z=x^2\)

Question 13.37

\(y^2 + z^2 = 4\)

Question 13.38

\(z = \displaystyle \frac{y^2}{4} - \frac{x^2}{9}\)

Question 13.39

\(y^2 = x^2 + z^2\)

Question 13.40

\(4x^2 - 3y^2 + 2z^2 = 0\)

Question 13.41

\(\displaystyle \frac{x^2}{9} + \frac{y^2}{12} + \frac{z^2}{9} = 1\)

Question 13.42

\(x^2 + y^2 + z^2 + 4x -by + 9z - b = 0\), where \(b\) is a constant

Question 13.43

Using polar coordinates, describe the level curves of the function defined by \[ f(x,y)=2xy/(x^2+y^2)\ \hbox{ if }\ (x,y)\neq (0,0)\hbox{ and } f(0,0)=0. \]

Question 13.44

Let \(f\colon\, {\mathbb R}^2\backslash \{{\bf 0}\} \rightarrow {\mathbb R}\) be given in polar coordinates by \(f(r,\theta) = (\cos 2\theta)/r^2\). Sketch a few level curves in the \(xy\) plane. Here, \({\mathbb R}^2\backslash \{{\bf 0}\} = \{ {\bf x} \in {\mathbb R}^2 \mid {\bf x} \neq {\bf 0}\}.\)

Question 13.45

Show that in Figure 13.13, the level “curve” \(z=3\) consists of two points.