This quiz tests the work covered in the lecture on linear functions and corresponds
to Section 12.4 of the textbook Calculus: Single and Multivariable (Hughes-Hallett,
Gleason, McCallum et al.).

There is a great applet at http://www.univie.ac.at/future.media/moe/galerie/geom2/geom2.html
which gets you to work out the equations of 5 planes. Make sure you rotate the
planes so that you can see where they cut the axes. You need to do some calculations
to find the equations.

There are more web quizzes at Wiley, select Sections 4 and 5. This quiz has 14
questions on both this topic and the next.

Consider the linear function $z=3y-2x-1$.
Which of the following statements are true? (Zero or more options
can be correct)

There is at least one mistake. For example, choice (a) should be False.

Substitute
$x=3$ and
$y=-2$ into the equation
and we get $z=-6-6-1=-13$
so (3,-2,-1) is not on the plane.

There is at least one mistake. For example, choice (b) should be False.

The slope of
the plane in the $x$ direction
is the coefficient of $x$
when the equation of the plane is written in this form, so the slope is -2.

There is at least one mistake. For example, choice (c) should be True.

The slope of
the plane in the $y$ direction
is the coefficient of $y$
when the equation of the plane is written in this form, so the slope is 3.

There is at least one mistake. For example, choice (d) should be True.

The slope of
the plane in the $x$ direction
is the coefficient of $x$
when the equation of the plane is written in this form, so the slope is -2.

There is at least one mistake. For example, choice (e) should be False.

The slope of
the plane in the $y$ direction
is the coefficient of $y$
when the equation of the plane is written in this form, so the slope is 3.

Correct!

False Substitute
$x=3$ and
$y=-2$ into the equation
and we get $z=-6-6-1=-13$
so (3,-2,-1) is not on the plane.

False The slope of
the plane in the $x$ direction
is the coefficient of $x$
when the equation of the plane is written in this form, so the slope is -2.

True The slope of
the plane in the $y$ direction
is the coefficient of $y$
when the equation of the plane is written in this form, so the slope is 3.

True The slope of
the plane in the $x$ direction
is the coefficient of $x$
when the equation of the plane is written in this form, so the slope is -2.

False The slope of
the plane in the $y$ direction
is the coefficient of $y$
when the equation of the plane is written in this form, so the slope is 3.

The following table contains the values of a linear function.

Which of the following is the missing value and the equation of the
plane, respectively? Exactly one option must be correct)

Choice (a) is incorrect

Try again, the
$y$ values are decreasing
by 2 and the $x$ values are
increasing by 3.

Choice (b) is incorrect

Try again, you have the correct values for the slopes in the
$x$ and
$y$
directions but you have not calculated the missing value correctly.

Choice (c) is incorrect

Try again, you
have calculated the missing value correctly but you do not have the correct equation of the plane.

Choice (d) is correct!

The
$x$ values are increasing
by 3 and the $y$
values are decreasing by 2 so the missing number is 5 and the equation is
$z=3x-2y+c.$ We
find $c$
by substituting one of the points into the equation. $4=3-2+c\Rightarrow c=3$ so the
equation is $z=3x-2y+3\phantom{\rule{0.3em}{0ex}}.$
It is always a good idea to check another point satisfies this equation as well.
($x=1\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y=2$ and
$z=2$ from
the table which gives 2= 3-4+3 in the equation, which is true)

We are given that the slope of a plane in the
$x$ direction is 2 and
the slope in the $y$
direction is -2 and (-1,2,5) is a point on the plane. Which of the following is the equation of that plane? Exactly one option must be correct)

Choice (a) is correct!

$z=2x-2y+c$ and we
substitute $x=-1\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}y=2$
and $z=5$ into the
equation to find $c\phantom{\rule{0.3em}{0ex}}.$ $5=-2-4+c\Rightarrow c=11$ We could also solve $z-5=2\left(x+1\right)-2\left(y-2\right)$ and we
will get the same result.

Choice (b) is incorrect

Try
again, you may have used $x=1$
instead of $x=-1\phantom{\rule{0.3em}{0ex}}.$

Choice (c) is incorrect

Try again, the slope
in the $x$ direction is the
coefficient of $x$ and the
slope in the $y$ direction
is the coefficient of $y\phantom{\rule{0.3em}{0ex}}.$

Choice (d) is incorrect

Try
again, you have not used the correct form for the equation of the plane.

Which one of the following statements is correct. Exactly one option must be correct)

Choice (a) is incorrect

Try again, it does. Note that
the $x$
and $y$
values are not evenly spaced.

Choice (b) is incorrect

Try again,
you have not calculated the slopes correctly.

Choice (c) is correct!

You have
noted that the $x$
and $y$
values are not evenly spaced and that the slope in the
$x$ direction is 3 and the
slope in the $y$ direction
is 2 so the equation is $z=3x+2y-4\phantom{\rule{0.3em}{0ex}}.$

Choice (d) is incorrect

Try
again, look carefully at the order of the variables.