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MATH1111 Quizzes

Contour Diagrams Quiz
Web resources available Questions

This quiz tests the work covered in the lecture on contour diagrams and corresponds to Section 12.3 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.).

There is an animation at http://www.math.ou.edu/ tjmurphy/Teaching/2443/LevelCurves/levelCurves.html which demonstrates level curves. This is about half-way down the page.

If you want some harder questions on level curves try questions 8, 9 and 10 from the MATH1001 quiz 4 at http://www.maths.usyd.edu.au/u/UG/JM/MATH1001/Quizzes/quiz4.html.

There are more web quizzes at Wiley, select Section 3. This only has 2 questions.

Which of the following is the level curve for z = f(x,y) = x2 + y2 where z = 9? Exactly one option must be correct)
a)
PIC
b)
PIC
c)
PIC
d)
PIC

Choice (a) is incorrect
Try again, this could be the level curve for z = x + y when z = 3 but is not what we are looking for.
Choice (b) is incorrect
Try again, this could be the level curve for z = x2 + y2 where z = 81 but it is not what we are looking for.
Choice (c) is correct!
This is the curve x2 + y2 = 32 = 9 which is the circle centre zero with radius 3.
Choice (d) is incorrect
Try again, this could be the level curve for z = x2 + y where z = 9 but it is not what we are looking for.
Suppose z = f(x,y) = 4 (x2 + y2)
Which of the following are the level curves for z = ±1,z = ±2? Exactly one option must be correct)
a)
PIC
b)
PIC
c)
PIC
d)
PIC

Choice (a) is correct!
When z = 2 the curve is 2 = 4 (x2 + y2) which simplifies to x2 + y2 = 6 which a circle centre (0,0) radius 6.
When z = 1 the curve is x2 + y2 = 5. When z = 1 the curve is x2 + y2 = 3. When z = 2 the curve is x2 + y2 = 2.
Choice (b) is incorrect
Try again, the curves are not labelled correctly.
Choice (c) is incorrect
Try again, you have not calculated the curve correctly.
When z = 2 the curve is 2 = 4 (x2 + y2) which simplifies to x2 + y2 = 6 which a circle centre (0,0) radius 6.
Choice (d) is incorrect
Try again, you have not calculated the curve correctly.
When z = 2 the curve is 2 = 4 (x2 + y2) which simplifies to x2 + y2 = 6 which a circle centre (0,0) radius 6.
Which of the following statements are correct? (Zero or more options can be correct)
a)
The contour diagram for f(x,y) = 3x + 2y + 4 has straight lines for its level curves.
b)
The contour diagram for f(x,y) = 2x2 y has parabolas for its level curves.
c)
The contour diagram for f(x,y) = 3x2 + y has hyperbolas for its level curves.
d)
The contour diagram for f(x,y) = 3x2 + 2y2 has circles for its level curves.
e)
The contour diagram for f(x,y) = (x 1)2 + (y 2)2 has circles, centre (1,2) for its level curves.

There is at least one mistake.
For example, choice (a) should be True.
Suppose z = c then the level curves are of the form 3x + 2y + 4 c = 0 which are straight lines.
There is at least one mistake.
For example, choice (b) should be True.
Suppose z = c then the level curves are of the form y = 2x2 c which are parabolas.
There is at least one mistake.
For example, choice (c) should be False.
Suppose z = c then the level curves are of the form y = 3x2 + c which are parabolas.
There is at least one mistake.
For example, choice (d) should be False.
Suppose z = c then the level curves are of the form 3x2 + 2y2 = c which are ellipses.
There is at least one mistake.
For example, choice (e) should be True.
Suppose z = c then the level curves are of the form (x 1)2 + (y 2)2 = c which are circles, centre (1,2).
Correct!
  1. True Suppose z = c then the level curves are of the form 3x + 2y + 4 c = 0 which are straight lines.
  2. True Suppose z = c then the level curves are of the form y = 2x2 c which are parabolas.
  3. False Suppose z = c then the level curves are of the form y = 3x2 + c which are parabolas.
  4. False Suppose z = c then the level curves are of the form 3x2 + 2y2 = c which are ellipses.
  5. True Suppose z = c then the level curves are of the form (x 1)2 + (y 2)2 = c which are circles, centre (1,2).
Which one of the following could be the contour diagram for z = f(x,y) = x y2? Exactly one option must be correct)
a)
PIC
b)
PIC
c)
PIC
d)
PIC

Choice (a) is incorrect
Try again, this could be the contour diagram for z = f(x,y) = x2 y.
Choice (b) is incorrect
Try again, this could be the contour diagram for z = f(x,y) = x y3.
Choice (c) is correct!
Let c = x y2 then y = x c so these are the correct level curves for the contour map.
Choice (d) is incorrect
Try again, this could be the contour diagram for z = f(x,y) = y x3.