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Note: The page numbers are the page numbers of the original Wisniewski text.

1. Co-ordinate system. Determine which quadrant the following points are in: a. b. c. d. (x, y) = (-2, 2) (x, y) = (2, -2) (x, y) = ((-1)2, 2) = (1, 2) (x, y) = (2-1, 2) = (1/2, 2)

y

2.5

Coordinates (x, y)

(2-1, 2) ((-1)2, 2)

(-2, 2)

2 1.5 1 0.5

x

-2.5 -2 -1.5 -1 -0.5 -0.5 -1 -1.5 0.5 1 1.5 2 2.5

(2, -2)

-2 -2.5

2. Functions a. Graph y = |x|. (The bar symbols mean take the absolute value of x ie |-3| = 3). If you are not sure what it looks like, plug in a couple values for x and find the corresponding y. ECO2IQA 2010 Tutorial Chapter 2 Solutions Page 1 of 17

10 9 8 7 6 5 4 3 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5

y

x

1 2 3 4 5 6 7 8 9 10

y = |x|

b. Is y a function of x? Explain.

Yes. For each x value there is only one y value.

c. Re-do a) and b) using y2 = x – 1 (where x ≥ 1).

ECO2IQA 2010 Tutorial Chapter 2 Solutions

Page 2 of 17

y

4

y2 = x - 1

3 2 1

x

1 -1 -2 -3 -4 2 3 4 5 6 7 8 9 10

y is not a function of x: one x value (say x = 5) has two y values (y = 2 and y = -2). d. Let f(x) = x2 + 1. Compute f(0), f(-1), f(1/2) and f ( 2) .

f(0) = 02 + 1 = 1. f(-1) = (-1)2 + 1 = 2. f(1/2) = (1/2)2 + 1 = 1.25. f( 2 ) = ( 2 )2 + 1 = 3.

3. Linear functions. Do Exercise 2.2 from the text. (Note: Exercise questions are located at the end of a chapter. Student Activity questions are found within the chapter.)

TC = 100 + 5Q i. ii. Change in TC is 5. Change in TC is 5.

Page 3 of 17

ECO2IQA 2010 Tutorial Chapter 2 Solutions

iii. iv.

Change in Total Cost from a unit change in output is called Marginal Cost (MC). Not realistic to expect constant MC in most industries. With a fixed factor of production (say capital) we would expect (eventually) diminishing marginal productivity of the variable factor (say labour) since more workers are sharing the same amount of capital – eg think of more and more chefs using the same kitchen. Since marginal productivity of the next worker is lower than the previous worker, to produce an extra unit of output we need to increase labour by more than the increase in labour to produce the previous unit of output. Thus marginal cost would (eventually) increase as Q increases. However, for digital goods eg mp3s the cost of producing another unit is essentially zero (there are no variable costs) so here MC is constant and in fact MC = 0.

4. Let z = 2x – y. a. Is this an explicit or implicit function of z? This is an explicit function of z ie we have z by itself on one side of the equation and no z on the other side. b. Is y a dependent or independent variable in this equation? y is an independent variable here. (At least that is what the equation implies as we normally try to put the dependent variable on the left-hand side (lhs) of the equation. In general, whether a variable is dependent or independent depends on the model we are using). c. Draw the relationship between z and x assuming y = 1. ECO2IQA 2010 Tutorial Chapter 2 Solutions Page 4 of 17

When y = 1, z = 2x – 1.

5 4 3 2 1 x -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 1 2 3 4 5 z

z = 2x - y y = 1

d. How would the curve in c) change if y increases to 2? y is not on the axis and when y increases z decreases, so the curve will shift down when y increases. When y = 2, z = 2x – 2.

ECO2IQA 2010 Tutorial Chapter 2 Solutions

Page 5 of 17

5 4 3 2 1

z

x -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 1 2 3 4 5

z = 2x - y y = 1 z = 2x - y y = 2

e. Draw the relationship between z and y assuming x = 1. When x = 1, z = 2 – y z

5 4 3 2 1

z = 2x - y x = 1

y -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 1 2 3 4 5

ECO2IQA 2010 Tutorial Chapter 2 Solutions

Page 6 of 17

f. How would the curve in e) change if y increases from 1 to 2? It wouldn’t change: Since y is on the axis, when y changes we are moving along the curve.

5. Polynomial functions. Do Exercise...

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