menuicon

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 = x2 4 + y2 is a hyperbolic paraboloid.
b)
z2 x2 y2 = 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 x = x2 16 + y2 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 3 2 2 + y2 z2 32 = 1 which is a hyperboloid of one sheet.
There is at least one mistake.
For example, choice (d) should be True.
4x2 + y2 + z2 4 > 0 for all real x,y,z so this cannot represent a surface and is not in the catalogue.
Correct!
  1. False Making z the subject makes x = x2 16 + y2 4 which is an elliptical paraboloid.
  2. True Rearranging into the form x2 + y2 z2 = 0 is a cone with a = b = c = 1.
  3. False This can be rearranged to x2 3 2 2 + y2 z2 32 = 1 which is a hyperboloid of one sheet.
  4. True 4x2 + y2 + z2 4 > 0 for all real x,y,z so this cannot represent a surface and is not in the catalogue.
Consider the function f(x,y) = 16 4x2 16y2 9 .
Which of the following is the corresponding level surface g(x,y,z) = f(x,y) z = 0? Exactly one option must be correct)
a)
The hyperboloid x2 4 + y2 9 z2 16 = 1.
b)
The top half of the ellipsoid x2 4 + y2 9 + z2 16 = 1.
c)
The top half of the two sheet hyperboloid x2 4 + y2 9 z2 16 = 1.
d)
The top half of the cone x2 4 + y2 9 z2 16 = 0.

Choice (a) is incorrect
Try again, this would correspond to the function f(x,y) = 4x2 + 16y2 9 16.
Choice (b) is correct!
Rearranging 16 4x2 16y2 9 z = 0 gives z = 16 4x2 16y2 9 .
Squaring both sides gives z2 = 16 4x2 16y2 9 . 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(x,y) = 4x2 + 16y2 9 + 16.
Choice (d) is incorrect
Try again, this would correspond to the function f(x,y) = 4x2 + 16y2 9 .
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? 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(x,y,z) = (x a)2 + (y b)2 + (z c)2 gives a family of spheres centre (a,b,c).
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 x2 4 + y2 9 z2 4 = 1? Exactly one option must be correct)
a)
f(x,y) = x2 + 4y2 9 + 4 and g(x,y) = x2 + 4y2 9 + 4.
b)
f(x,y) = 4 x2 4y2 9 and g(x,y) = 4 x2 4y2 9 .
c)
f(x,y) = 41 x2 4 y2 9 and g(x,y) = 41 x2 4 y2 9 .
d)
f(x,y) = 41 + x2 4 + y2 9 and g(x,y) = 41 + x2 4 + y2 9 .

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.