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.

*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is correct!*

*Choice (d) is incorrect*

Which of the following are the level curves for $z=\pm 1\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}z=\pm 2\phantom{\rule{0.3em}{0ex}}?$ Exactly one option must be correct)

*Choice (a) is correct!*

When $z=-1$ the curve is ${x}^{2}+{y}^{2}=5\phantom{\rule{0.3em}{0ex}}.$ When $z=1$ the curve is ${x}^{2}+{y}^{2}=3\phantom{\rule{0.3em}{0ex}}.$ When $z=2$ the curve is ${x}^{2}+{y}^{2}=2\phantom{\rule{0.3em}{0ex}}.$

*Choice (b) is incorrect*

*Choice (c) is incorrect*

When $z=-2$ the curve is $-2=4-\left({x}^{2}+{y}^{2}\right)$ which simplifies to ${x}^{2}+{y}^{2}=6$ which a circle centre $\left(0,0\right)$ radius $\sqrt{6}\phantom{\rule{0.3em}{0ex}}.$

*Choice (d) is incorrect*

When $z=-2$ the curve is $-2=4-\left({x}^{2}+{y}^{2}\right)$ which simplifies to ${x}^{2}+{y}^{2}=6$ which a circle centre $\left(0,0\right)$ radius $\sqrt{6}\phantom{\rule{0.3em}{0ex}}.$

*There is at least one mistake.*

For example, choice (a) should be True.

*There is at least one mistake.*

For example, choice (b) should be True.

*There is at least one mistake.*

For example, choice (c) should be False.

*There is at least one mistake.*

For example, choice (d) should be False.

*There is at least one mistake.*

For example, choice (e) should be True.

*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 $y=2{x}^{2}-c$ which are parabolas.*False*Suppose $z=c$ then the level curves are of the form $y=-3{x}^{2}+c$ which are parabolas.*False*Suppose $z=c$ then the level curves are of the form $3{x}^{2}+2{y}^{2}=c$ which are ellipses.*True*Suppose $z=c$ then the level curves are of the form ${\left(x-1\right)}^{2}+{\left(y-2\right)}^{2}=c$ which are circles, centre (1,2).

*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is correct!*

*Choice (d) is incorrect*