This quiz tests the work covered in the lecture on Surfaces and corresponds to
Section 12.5 of the textbook Calculus: Single and Multivariable (Hughes-Hallett,
Gleason, McCallum et al.).

There are more web quizzes at Wiley, select Sections 4 and 5. This quiz has 14 questions on both this topic and the last.

Use the catalogue on page 636 of Calculus: Single and Multivariable
(Hughes-Hallett, Gleason, McCallum et al.) to identify which of the
statements below are correct. (Zero or more options can be correct)

For example, choice (a) should be False.

For example, choice (b) should be True.

For example, choice (c) should be False.

For example, choice (d) should be True.

*There is at least one mistake.*

For example, choice (a) should be False.

Making $z$ the subject makes
$x=\frac{{x}^{2}}{16}+\frac{{y}^{2}}{4}$ which is an elliptical
paraboloid.

*There is at least one mistake.*

For example, choice (b) should be True.

Rearranging into the form ${x}^{2}+{y}^{2}-{z}^{2}=0$
is a cone with $a=b=c=1\phantom{\rule{0.3em}{0ex}}.$

*There is at least one mistake.*

For example, choice (c) should be False.

This can be rearranged to
$\frac{{x}^{2}}{{\left(\sqrt{\frac{3}{2}}\right)}^{2}}+{y}^{2}-\frac{{z}^{2}}{{\left(\sqrt{3}\right)}^{2}}=1$ which is a hyperboloid
of one sheet.

*There is at least one mistake.*

For example, choice (d) should be True.

$4{x}^{2}+{y}^{2}+\frac{{z}^{2}}{4}>0$
for all real $x\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}z$
so this cannot represent a surface and is not in the catalogue.

*Correct!*

*False*Making $z$ the subject makes $x=\frac{{x}^{2}}{16}+\frac{{y}^{2}}{4}$ which is an elliptical paraboloid.*True*Rearranging into the form ${x}^{2}+{y}^{2}-{z}^{2}=0$ is a cone with $a=b=c=1\phantom{\rule{0.3em}{0ex}}.$*False*This can be rearranged to $\frac{{x}^{2}}{{\left(\sqrt{\frac{3}{2}}\right)}^{2}}+{y}^{2}-\frac{{z}^{2}}{{\left(\sqrt{3}\right)}^{2}}=1$ which is a hyperboloid of one sheet.*True*$4{x}^{2}+{y}^{2}+\frac{{z}^{2}}{4}>0$ for all real $x\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}z$ so this cannot represent a surface and is not in the catalogue.

Consider the function $f\left(x,y\right)=\sqrt{16-4{x}^{2}-\frac{16{y}^{2}}{9}}\phantom{\rule{0.3em}{0ex}}.$

Which of the following is the corresponding level surface $g\left(x,y,z\right)=f\left(x,y\right)-z=0\phantom{\rule{0.3em}{0ex}}?$ Exactly one option must be correct)

Which of the following is the corresponding level surface $g\left(x,y,z\right)=f\left(x,y\right)-z=0\phantom{\rule{0.3em}{0ex}}?$ Exactly one option must be correct)

*Choice (a) is incorrect*

Try again, this would correspond to the function
$f\left(x,y\right)=\sqrt{4{x}^{2}+\frac{16{y}^{2}}{9}-16}\phantom{\rule{0.3em}{0ex}}.$

*Choice (b) is correct!*

Rearranging $\sqrt{16-4{x}^{2}-\frac{16{y}^{2}}{9}}-z=0$
gives $z=\sqrt{16-4{x}^{2}-\frac{16{y}^{2}}{9}}\phantom{\rule{0.3em}{0ex}}.$

Squaring both sides gives ${z}^{2}=16-4{x}^{2}-\frac{16{y}^{2}}{9}\phantom{\rule{0.3em}{0ex}}.$ Divide both sides by 16 and rearrange and we have the required level surface.

Squaring both sides gives ${z}^{2}=16-4{x}^{2}-\frac{16{y}^{2}}{9}\phantom{\rule{0.3em}{0ex}}.$ Divide both sides by 16 and rearrange and we have the required level surface.

*Choice (c) is incorrect*

Try again, this would correspond to the function
$f\left(x,y\right)=\sqrt{4{x}^{2}+\frac{16{y}^{2}}{9}+16}\phantom{\rule{0.3em}{0ex}}.$

*Choice (d) is incorrect*

Try again, this would correspond to the function
$f\left(x,y\right)=\sqrt{4{x}^{2}+\frac{16{y}^{2}}{9}}\phantom{\rule{0.3em}{0ex}}.$

Consider the function $f\left(x,y,z\right)={\left(x+1\right)}^{2}+{\left(y-2\right)}^{2}+{\left(z+3\right)}^{2}\phantom{\rule{0.3em}{0ex}}.$

Which of the following most accurately describes the set of level surfaces? Exactly one option must be correct)

Which of the following most accurately describes the set of level surfaces? Exactly one option must be correct)

*Choice (a) is incorrect*

Try again,
check the catalogue of surfaces.

*Choice (b) is incorrect*

Try again, it is a
special type of ellipsoid but check the centre.

*Choice (c) is correct!*

$f\left(x,y,z\right)={\left(x-a\right)}^{2}+{\left(y-b\right)}^{2}+{\left(z-c\right)}^{2}$ gives a family of
spheres centre $\left(a,b,c\right)\phantom{\rule{0.3em}{0ex}}.$

*Choice (d) is incorrect*

Try again, you do not have the correct centre.

Which of the following are the two functions which together form the two sheet hyperboloid
$\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}-\frac{{z}^{2}}{4}=-1\phantom{\rule{0.3em}{0ex}}?$ Exactly one option
must be correct)

*Choice (a) is correct!*

You have
successfully made $z$
the subject of the equation and taken both the positive and negative square root.

*Choice (b) is incorrect*

Make
$z$ the
subject of the equation and take both the positive and negative square root.

*Choice (c) is incorrect*

Make
$z$ the
subject of the equation and take both the positive and negative square root.

*Choice (d) is incorrect*

Make
$z$ the
subject of the equation and take both the positive and negative square root.