 ## MATH1901 Quizzes

Quiz 4: Curves and surfaces in 3-dimensional space
Question 1 Questions
What are suitable parametric equations for this plane curve? Exactly one option must be correct)
 a) $x=2cost+1,\phantom{\rule{1em}{0ex}}y=3sint+2$ b) $x=3cost+1,\phantom{\rule{1em}{0ex}}y=2sint+2$ c) $x=3cost-1,\phantom{\rule{1em}{0ex}}y=2sint-2$ d) $x=2cost-1,\phantom{\rule{1em}{0ex}}y=3sint-2$

Choice (a) is incorrect
This is an ellipse with centre $\left(1,2\right)$ and with axes of length $4$ in the $x$–direction and $6$ in the $y$–direction. Choice (b) is correct!
The curve is an ellipse centre (1,2) with axes length 6 in the $x$ direction and 4 in the $y$ direction.
Choice (c) is incorrect
This is an ellipse with centre $\left(-1,-2\right)$ and with axes of length $6$ in the $x$–direction and $4$ in the $y$–direction. Choice (d) is incorrect
This is an ellipse with centre $\left(-1,-2\right)$ and with axes of length $4$ in the $x$–direction and $6$ in the $y$–direction. Which of the following could be the parametric equations for the following space curve? (You may assume that $t\ge 0$ and that the scales are the same on the $x,\phantom{\rule{0.3em}{0ex}}y$ and $z$ axes.) Exactly one option must be correct)
 a) $x=cost,\phantom{\rule{1em}{0ex}}y=sint,\phantom{\rule{1em}{0ex}}z=t$ b) $x=tcost,\phantom{\rule{1em}{0ex}}y=tsint,\phantom{\rule{1em}{0ex}}z={t}^{2}$ c) $x=cost,\phantom{\rule{1em}{0ex}}y=sint,\phantom{\rule{1em}{0ex}}z={t}^{2}$ d) $x=tcost,\phantom{\rule{1em}{0ex}}y=tsint,\phantom{\rule{1em}{0ex}}z=t$

Choice (a) is incorrect
This curve has the following graph. Choice (b) is incorrect
In this curve the $z$ values increase as the square of $t$, making the spiral much ‘steeper’ than the curve shown in the question. The parametric equations given in this option correspond to the following graph. Choice (c) is incorrect
This curve has the following graph. Choice (d) is correct!
The curve circles around the $z$-axis and the distance of the curve from the $z$-axis increases linearly with $t$ (and hence $z$).
Which of the following could be the parametric equations for the following space curve? (Hint: look at the sign of $y$ as $t$ changes.) Exactly one option must be correct)
 a) $x={t}^{2}-1,\phantom{\rule{1em}{0ex}}y={t}^{3}-t,\phantom{\rule{1em}{0ex}}z={t}^{3}$ b) $x=cost,\phantom{\rule{1em}{0ex}}y=sint,\phantom{\rule{1em}{0ex}}z={t}^{3}$ c) $x={t}^{2}-1,\phantom{\rule{1em}{0ex}}y={t}^{2},\phantom{\rule{1em}{0ex}}z={t}^{2}$ d) $x={t}^{3}-t,\phantom{\rule{1em}{0ex}}y={t}^{2}-1,\phantom{\rule{1em}{0ex}}z={t}^{2}$

Choice (a) is correct!
As $t$ increases from large negative values through zero to large positive values, $y$ starts out negative then becomes positive, then negative and finally positive again. This fits the curve shown. It is actually quite hard to “see” where in space this curve lies, from the diagram provided, so don’t worry if you didn’t get this one correct.
Choice (b) is incorrect
The $cost$ and $sint$ terms make this curve wind around the $z$–axis, and the $z={t}^{3}$ term makes the height of the curve above the $xy$–plane increase rapidly. Choice (c) is incorrect
This is a “half” straight line because $z=y=x+1$, for all $t$. (It is only half a line because $x\ge -1$, $y\ge 0$ and $z\ge 0$ since ${t}^{2}\ge 0$ for all $t$.) (The box is there to help visualize the line).
Choice (d) is incorrect Let $f\left(x,y\right)={x}^{2}+{y}^{2}-1$. Which of the following could be the graph of $z=f\left(x,y\right)$? Exactly one option must be correct)
 a) b) c) d) Choice (a) is incorrect
This cannot be the graph of $f\left(x,y\right)$. Look again at the formula for $f\left(x,y\right)$! It tells us that $f\left(x,y\right)\ge -1$ for all $x,y$. In fact, this option is the graph of $z=1-\left({x}^{2}+{y}^{2}\right)$.
Choice (b) is incorrect
This is the graph of a regular cone (it has “straight sides”), whereas $f\left(x,y\right)$ is a paraboloid. In fact, this is the graph of $z=\sqrt{{x}^{2}+{y}^{2}-1}$.
Choice (c) is incorrect
Even though this is the graph of a paraboloid it cannot be the graph of $z=f\left(x,y\right)$ since $f\left(0,0\right)=-1$ whereas this graph goes through $\left(0,0,0\right)$. In fact, this is the graph of $z={x}^{2}+{y}^{2}$.
Choice (d) is correct!
This is the only graph listed which is paraboloid passing through $\left(0,0,-1\right)$.
Which option could be the equation of the following surface? Exactly one option must be correct)
 a) $z=4-{x}^{2}-{y}^{2}$ b) $z={y}^{2}-2$ c) $z=-{x}^{2}$ d) $z=4-{x}^{2}$

Choice (a) is incorrect
The function $f\left(x,y\right)=4-\left({x}^{2}+{y}^{2}\right)$ is an inverted paraboloid. It has the following graph. Choice (b) is incorrect
This function does not depend on $x$. Its graph is: Choice (c) is incorrect
This surface lies entirely below the $xy$–plane so it cannot be the function pictured. Its graph is Choice (d) is correct!
This graph has a parabolic cross-section $z=-{x}^{2}$ which does not depend on the value of $y$.
Which of the following equations could describe this surface? Exactly one option must be correct)
 a) $z=cosxcosy$ b) $z=sinxsiny$ c) $z=cosxsiny$ d) $z=cos\left({x}^{2}+{y}^{2}\right)$ e) $z=cos\sqrt{{x}^{2}+{y}^{2}}$ f) $z=sin\sqrt{{x}^{2}+{y}^{2}}$

Choice (a) is incorrect
This surface looks like an egg carton, as shown below. The surface in the question can’t be given by the equation $z=cosxcosy$ because in the plane $x=y$, for example, we have $z={cos}^{2}x$ and so $z\ge 0$, which is not the case. Choice (b) is incorrect
As $sinxsiny=cos\left(x+\frac{\pi }{2}\right)cos\left(y+\frac{\pi }{2}\right)$ this surface is just a shifted version of the surface $z=cosxcosy$. Choice (c) is incorrect
As $cosxsiny=cos\left(x\right)cos\left(y+\frac{\pi }{2}\right)$ this surface is just another shifted version of the surface $z=cosxcosy$. Choice (d) is incorrect
This function is a surface of revolution, so it could be the graph in the question. However, the ripples in this function get closer together as $\sqrt{{x}^{2}+{y}^{2}}$ gets large, so it cannot be the right answer. Choice (e) is correct!
This is a surface of revolution rotated about the $z$-axis and its equation is given by either $z=cos\sqrt{{x}^{2}+{y}^{2}}$ or $z=sin\sqrt{{x}^{2}+{y}^{2}}$. When $x=y=0$ we have $z=1$ (from the graph) so it is $z=cos\sqrt{{x}^{2}+{y}^{2}}$.
Choice (f) is incorrect
This is very similar to the graph in the question; however, this cannot be the right function because $z=0$ when $\left(x,y\right)=\left(0,0\right)$ which does not agree with the graph in the question. Which of the equations below could describe the following surface? Exactly one option must be correct)
 a) $z=log\left({x}^{2}+{y}^{2}\right)$ b) $z={e}^{{x}^{2}+{y}^{2}}$ c) $z=1-{e}^{{x}^{2}+{y}^{2}}$ d) $z=\frac{1}{{x}^{2}+{y}^{2}}$

Choice (a) is incorrect
This cannot be the right graph because $z\to -\infty$ as $\left(x,y\right)\to \left(0,0\right)$. In fact, the function $z=log\left({x}^{2}+{y}^{2}\right)$ has the graph: Choice (b) is incorrect
This function looks like a very “steep” paraboloid of revolution. It cannot be the function in the question because $z=0$ when $\left(x,y\right)=\left(0,0\right)$. In fact, $z={e}^{{x}^{2}+{y}^{2}}$ has the following graph. Choice (c) is incorrect
This function looks like a very “steep” inverted paraboloid of revolution. It cannot be the function in this option because here, $z$ is always less than 1 and becomes very large and negative as ${x}^{2}+{y}^{2}$ increases. In fact, $z=1-{e}^{{x}^{2}+{y}^{2}}$ has the following graph. Choice (d) is correct!
This is a surface of rotation about the $z$-axis. As ${x}^{2}+{y}^{2}\to \infty$ as $\left(x,y\right)\to \left(0,0\right)$ it is the only function listed which is consistent with the graph in the question.
Which one of the following shows some of the level curves for the function $z=x\left(x+y\right)$ ? Exactly one option must be correct)
 a) b) c) d) Choice (a) is incorrect
These level curves are ellipses with equations of the form $\frac{{\left(x-1\right)}^{2}}{{a}^{2}}+\frac{{\left(y-2\right)}^{2}}{{b}^{2}}=1$, which are not the same equations as the level curves for the function with equation $z=x\left(x+y\right)$.
Choice (b) is correct!
The level curve at height $z=c$ satisfies the equation $c=x\left(x+y\right)$; that is, $y=\frac{c}{x}-x$.
Choice (c) is incorrect
These level curves appear to have an asymptote at $x=2$ whereas the level curves of $z=f\left(x,y\right)$ have an asymptote at $x=0$.
Choice (d) is incorrect
These level curves are ellipses with equations of the form $\frac{{\left(x-1\right)}^{2}}{{a}^{2}}+\frac{{\left(y-2\right)}^{2}}{{b}^{2}}=1$, which are not the same equations as the level curves for the function with equation $z=x\left(x+y\right)$.
Some of the level curves of an unknown function $z=f\left(x,y\right)$ are given below. Which of the following functions could be $f\left(x,y\right)$ ? (Note: we do not know the height of these level curves.) Exactly one option must be correct)
 a) $f\left(x,y\right)={x}^{2}+{y}^{2}$. b) $f\left(x,y\right)=4{\left(x+1\right)}^{2}+9{\left(y+2\right)}^{2}$. c) $f\left(x,y\right)=9{\left(x-1\right)}^{2}+4{\left(y-2\right)}^{2}$. d) $f\left(x,y\right)=4{\left(x-1\right)}^{2}+9{\left(y-2\right)}^{2}$.

Choice (a) is incorrect
The level curves of this function are circles centred at the origin.
Choice (b) is incorrect
The level curves of this function are ellipses centered at $\left(-1,-2\right)$.
Choice (c) is incorrect
The level curves of this function are ellipses centered at $\left(1,2\right)$. The semi-major axis of each ellipse is vertical and the semi-minor axis is horizontal. That is, the ellipses are taller than they are wide, and so this option doesn’t match the given set of curves.
Choice (d) is correct!
The level curves of this function are ellipses centered at $\left(1,2\right)$. The semi-major axis of each ellipse is horizontal and the semi-minor axis is vertical. The ratio of these two axes is $3:2$, matching that shown on the set of curves.
Which of the following graphs could be part of the set of level curves for some surface $z=f\left(x,y\right)$ ? (Zero or more options can be correct)
 a) b) c) There is at least one mistake.
For example, choice (a) should be True.
For example, these could be some of the level curves of the surface given by
$f\left(x,y\right)=\sqrt{3{\left(x-1\right)}^{2}+2{\left(y-2\right)}^{2}}$
.
There is at least one mistake.
For example, choice (b) should be True.
For example, these could be some of the level curves of the surface given by
$f\left(x,y\right)={\left(x-1\right)}^{2}-{\left(y-2\right)}^{2}$
.
There is at least one mistake.
For example, choice (c) should be True.
For example, these could be some of the level curves of the surface given by
$f\left(x,y\right)=y-x$
.
Correct!
1. True For example, these could be some of the level curves of the surface given by
$f\left(x,y\right)=\sqrt{3{\left(x-1\right)}^{2}+2{\left(y-2\right)}^{2}}$
.
2. True For example, these could be some of the level curves of the surface given by
$f\left(x,y\right)={\left(x-1\right)}^{2}-{\left(y-2\right)}^{2}$
.
3. True For example, these could be some of the level curves of the surface given by
$f\left(x,y\right)=y-x$
.