 ## 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 $z=f\left(x,y\right)={x}^{2}+{y}^{2}$ where $z=9\phantom{\rule{0.3em}{0ex}}?$ Exactly one option must be correct)
 a) b) c) d) 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 $z={x}^{2}+{y}^{2}$ where $z=81$ but it is not what we are looking for.
Choice (c) is correct!
This is the curve ${x}^{2}+{y}^{2}={3}^{2}=9$ which is the circle centre zero with radius 3.
Choice (d) is incorrect
Try again, this could be the level curve for $z={x}^{2}+y$ where $z=9$ but it is not what we are looking for.
Suppose $z=f\left(x,y\right)=4-\left({x}^{2}+{y}^{2}\right)$
Which of the following are the level curves for $z=±1\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}z=±2\phantom{\rule{0.3em}{0ex}}?$ Exactly one option must be correct)
 a) b) c) d) Choice (a) is correct!
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 $\sqrt{6}\phantom{\rule{0.3em}{0ex}}.$
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
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 $-2=4-\left({x}^{2}+{y}^{2}\right)$ 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
Try again, you have not calculated the curve correctly.
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 $\sqrt{6}\phantom{\rule{0.3em}{0ex}}.$
Which of the following statements are correct? (Zero or more options can be correct)
 a) The contour diagram for $f\left(x,y\right)=3x+2y+4$ has straight lines for its level curves. b) The contour diagram for $f\left(x,y\right)=2{x}^{2}-y$ has parabolas for its level curves. c) The contour diagram for $f\left(x,y\right)=3{x}^{2}+y$ has hyperbolas for its level curves. d) The contour diagram for $f\left(x,y\right)=3{x}^{2}+2{y}^{2}$ has circles for its level curves. e) The contour diagram for $f\left(x,y\right)={\left(x-1\right)}^{2}+{\left(y-2\right)}^{2}$ 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 $y=2{x}^{2}-c$ 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 $y=-3{x}^{2}+c$ 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 $3{x}^{2}+2{y}^{2}=c$ 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 ${\left(x-1\right)}^{2}+{\left(y-2\right)}^{2}=c$ 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 $y=2{x}^{2}-c$ which are parabolas.
3. False Suppose $z=c$ then the level curves are of the form $y=-3{x}^{2}+c$ which are parabolas.
4. False Suppose $z=c$ then the level curves are of the form $3{x}^{2}+2{y}^{2}=c$ which are ellipses.
5. 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).
Which one of the following could be the contour diagram for $z=f\left(x,y\right)=x-{y}^{2}\phantom{\rule{0.3em}{0ex}}?$ Exactly one option must be correct)
 a) b) c) d) Choice (a) is incorrect
Try again, this could be the contour diagram for $z=f\left(x,y\right)={x}^{2}-y\phantom{\rule{0.3em}{0ex}}.$
Choice (b) is incorrect
Try again, this could be the contour diagram for $z=f\left(x,y\right)=x-{y}^{3}\phantom{\rule{0.3em}{0ex}}.$
Choice (c) is correct!
Let $c=x-{y}^{2}$ then $y=\sqrt{x-c}$ so these are the correct level curves for the contour map.
Choice (d) is incorrect
Try again, this could be the contour diagram for $z=f\left(x,y\right)=y-{x}^{3}\phantom{\rule{0.3em}{0ex}}.$