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Surfaces Quiz

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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.

Question 1

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.

a)		$x2 2 4z = 4 + y$ is a hyperbolic paraboloid.
b)		$2 2 2 z - x - y = 0$ is a cone.
c)		$2x2 + 3y2 - z2 = 3$ is not in the catalogue.
d)		$4x2 +y2 + z2= - 4 4$ is not in the catalogue.

There is at least one mistake.
For example, choice (a) should be false.

Making $z$ the subject makes $x2 y2 x = 16 + 4$ which is an elliptical paraboloid.

There is at least one mistake.
For example, choice (b) should be true.

Rearranging into the form $x2 + y2 - z2 = 0$ is a cone with $a = b = c = 1.$

There is at least one mistake.
For example, choice (c) should be false.

This can be rearranged to $--x2---+ y2 - -z2--= 1 (∘ -3)2 (√3-)2 2$ which is a hyperboloid of one sheet.

There is at least one mistake.
For example, choice (d) should be true.

$4x2 +y2 + z2> 0 4$ for all real $x, y, z$ so this cannot represent a surface and is not in the catalogue.

Your answers are correct

False. Making $z$ the subject makes $x2 y2 x = 16 + 4$ which is an elliptical paraboloid.
True. Rearranging into the form $x2 + y2 - z2 = 0$ is a cone with $a = b = c = 1.$
False. This can be rearranged to $--x2---+ y2 - -z2--= 1 (∘ -3)2 (√3-)2 2$ which is a hyperboloid of one sheet.
True. $4x2 +y2 + z2> 0 4$ for all real $x, y, z$ so this cannot represent a surface and is not in the catalogue.

Question 2

Consider the function $∘ ---------16y2- f(x,y) = 16- 4x2 -----. 9$
Which of the following is the corresponding level surface $g(x,y,z) = f(x,y)- z = 0?$

a)	The hyperboloid $x2 y2 z2 4-+ 9-- 16 = 1.$	b)	The top half of the ellipsoid $x2 y2 z2 -4 + 9-+ 16 = 1.$
c)	The top half of the two sheet hyperboloid $x2 y2 z2 -- + --- -- = - 1. 4 9 16$	d)	The top half of the cone $2 2 2 x- + y-- -z = 0. 4 9 16$

Not correct. Choice (a) is false.

Try again, this would correspond to the function $∘ ------16y2---- f(x,y) = 4x2 + -9--- 16.$

Your answer is correct.

Rearranging $∘ ----------16y2 16- 4x2 - ----- z = 0 9$ gives $∘ ----------16y2 z = 16 - 4x2 -----. 9$
Squaring both sides gives $2 z2 = 16 - 4x2 - 16y-. 9$ Divide both sides by 16 and rearrange and we have the required level surface.

Not correct. Choice (c) is false.

Try again, this would correspond to the function $∘ ---------2---- f(x,y) = 4x2 + 16y-+ 16. 9$

Not correct. Choice (d) is false.

Try again, this would correspond to the function $∘ --------2- f(x,y) = 4x2 + 16y-. 9$

Question 3

Consider the function $f(x,y,z) = (x + 1)2 + (y - 2)2 +(z + 3)2.$
Which of the following most accurately describes the set of level surfaces?

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).

Not correct. Choice (a) is false.

Try again, check the catalogue of surfaces.

Not correct. Choice (b) is false.

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

Your answer is correct.

$f(x,y,z) = (x- a)2 + (y- b)2 + (z - c)2$ gives a family of spheres centre $(a,b,c).$

Not correct. Choice (d) is false.

Try again, you do not have the correct centre.

Question 4

Which of the following are the two functions which together form the two sheet hyperboloid $x2 y2 z2 4-+ 9-- -4 = - 1?$

a)		$∘ ----------- 4y2 f(x,y) = x2 + 9--+ 4$ and $∘ ----4y2---- g(x,y) = - x2 +--- +4. 9$
b)		$∘ --------4y2 f(x,y) = 4 - x2 --9-$ and $∘ -------4y2- g(x,y) = - 4- x2 --9-.$
c)		$∘ ----------- x2 y2 f(x,y) = 4 1 - 4-- -9$ and $∘ ----------- x2 y2 g(x,y) = - 4 1- 4-- -9 .$
d)		$∘ ----------- x2 y2 f(x,y) = 4 1 + 4-+ -9$ and $∘ ----------- x2 y2 g(x,y) = - 4 1+ 4-+ -9 .$