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MathQuiz 4.3
University of Sydney
School of Mathematics
and Statistics
© Copyright 2004-2006

Contour Diagrams Quiz

Question

Web resources available

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.

Question 1

Which of the following is the level curve for $z = f(x,y) = x2 +y2$ where $z = 9?$

Not correct. Choice (a) is false.

Try again, this could be the level curve for $z = x + y$ when $z = 3$ but is not what we are looking for.

Not correct. Choice (b) is false.

Try again, this could be the level curve for $z = x2 + y2$ where $z = 81$ but it is not what we are looking for.

Your answer is correct.

This is the curve $2 2 2 x + y = 3 = 9$ which is the circle centre zero with radius 3.

Not correct. Choice (d) is false.

Try again, this could be the level curve for $2 z = x + y$ where $z = 9$ but it is not what we are looking for.

Question 2

Suppose $2 2 z = f(x,y) = 4- (x + y )$
Which of the following are the level curves for $z = ±1 , z = ±2?$

Your answer is correct.

When $z = - 2$ the curve is $2 2 - 2 = 4 - (x + y )$ which simplifies to $2 2 x + y = 6$ which a circle centre $(0,0)$ radius $√ - 6.$
When $z = - 1$ the curve is $2 2 x + y = 5.$ When $z = 1$ the curve is $2 2 x + y = 3.$ When $z = 2$ the curve is $x2 + y2 = 2.$

Not correct. Choice (b) is false.

Try again, the curves are not labelled correctly.

Not correct. Choice (c) is false.

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 $√6-.$

Not correct. Choice (d) is false.

Try again, you have not calculated the curve correctly.
When $z = - 2$ the curve is $2 2 - 2 = 4 - (x + y )$ which simplifies to $2 2 x + y = 6$ which a circle centre $(0,0)$ radius $√ - 6.$

Question 3

Which of the following statements are correct?

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).

Your answers are correct

True. Suppose $z = c$ then the level curves are of the form $3x+ 2y+ 4 - c = 0$ which are straight lines.
True. Suppose $z = c$ then the level curves are of the form $y = 2x2 - c$ which are parabolas.
False. Suppose $z = c$ then the level curves are of the form $y = - 3x2 + c$ which are parabolas.
False. Suppose $z = c$ then the level curves are of the form $3x2 + 2y2 = c$ which are ellipses.
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).