## MATH1111 Quizzes

Surfaces Quiz
Web resources available Questions

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)
 a) $4z=\frac{{x}^{2}}{4}+{y}^{2}$ is a hyperbolic paraboloid. b) ${z}^{2}-{x}^{2}-{y}^{2}=0$ is a cone. c) $2{x}^{2}+3{y}^{2}-{z}^{2}=3$ is not in the catalogue. d) $4{x}^{2}+{y}^{2}+\frac{{z}^{2}}{4}=-4$ is not in the catalogue.

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!
1. False Making $z$ the subject makes $x=\frac{{x}^{2}}{16}+\frac{{y}^{2}}{4}$ which is an elliptical paraboloid.
2. True Rearranging into the form ${x}^{2}+{y}^{2}-{z}^{2}=0$ is a cone with $a=b=c=1\phantom{\rule{0.3em}{0ex}}.$
3. 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.
4. 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)
 a) The hyperboloid $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}-\frac{{z}^{2}}{16}=1\phantom{\rule{0.3em}{0ex}}.$ b) The top half of the ellipsoid $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}+\frac{{z}^{2}}{16}=1\phantom{\rule{0.3em}{0ex}}.$ c) The top half of the two sheet hyperboloid $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}-\frac{{z}^{2}}{16}=-1\phantom{\rule{0.3em}{0ex}}.$ d) The top half of the cone $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}-\frac{{z}^{2}}{16}=0\phantom{\rule{0.3em}{0ex}}.$

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.
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)
 a) A family of cylinders with radii 1,2 and 3. b) A family of ellipsoids centre (0,0,0). c) A family of spheres centre (-1,2,-3). d) A family of spheres centre (1,-2,3).

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)
 a) $f\left(x,y\right)=\sqrt{{x}^{2}+\frac{4{y}^{2}}{9}+4}$ and $g\left(x,y\right)=-\sqrt{{x}^{2}+\frac{4{y}^{2}}{9}+4}\phantom{\rule{0.3em}{0ex}}.$ b) $f\left(x,y\right)=\sqrt{4-{x}^{2}-\frac{4{y}^{2}}{9}}$ and $g\left(x,y\right)=-\sqrt{4-{x}^{2}-\frac{4{y}^{2}}{9}}\phantom{\rule{0.3em}{0ex}}.$ c) $f\left(x,y\right)=4\sqrt{1-\frac{{x}^{2}}{4}-\frac{{y}^{2}}{9}}$ and $g\left(x,y\right)=-4\sqrt{1-\frac{{x}^{2}}{4}-\frac{{y}^{2}}{9}}\phantom{\rule{0.3em}{0ex}}.$ d) $f\left(x,y\right)=4\sqrt{1+\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}}$ and $g\left(x,y\right)=-4\sqrt{1+\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}}\phantom{\rule{0.3em}{0ex}}.$

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.