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

Linear Functions Quiz

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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.

Question 1

Consider the linear function $z = 3y - 2x- 1$ . Which of the following statements are true?

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.

Your answers are 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.

Question 2

The following table contains the values of a linear function.

$x\y$	1	2
1	4	2
2	7	?

Which of the following is the missing value and the equation of the plane, respectively?

Not correct. Choice (a) is false.

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

Not correct. Choice (b) is false.

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.

Not correct. Choice (c) is false.

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

Your answer 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.
$4 = 3- 2+ c ⇒ c = 3$ so the equation is $z = 3x- 2y+ 3 .$ It is always a good idea to check another point satisfies this equation as well. ( $x = 1 , y = 2$ and $z = 2$ from the table which gives 2= 3-4+3 in the equation, which is true)

Question 3

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?

Your answer is correct.

$z = 2x- 2y +c$ and we substitute $x = - 1, y = 2$ and $z = 5$ into the equation to find $c .$
$5 = - 2 - 4+ c ⇒ c = 11$
We could also solve $z - 5 = 2(x+ 1)- 2(y- 2)$ and we will get the same result.

Not correct. Choice (b) is false.

Try again, you may have used $x = 1$ instead of $x = - 1.$

Not correct. Choice (c) is false.

Try again, the slope in the $x$ direction is the coefficient of $x$ and the slope in the $y$ direction is the coefficient of $y.$

Not correct. Choice (d) is false.

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

Question 4

Consider the table below.

$x\y$	1	2	4
1	1	3	7
2	4	6	10
4	10	12	16

Which one of the following statements is correct.

Not correct. Choice (a) is false.

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

Not correct. Choice (b) is false.

Try again, you have not calculated the slopes correctly.

Your answer 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 $z = 3x + 2y- 4.$

Not correct. Choice (d) is false.

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