ECON7150 S1 2022
Assignment 1
Due May 6 2022 at 4 PM
Instructions:
• All answers to the assignment must be neatly hand written.
• Scan your assignment and save it as one pdf document. If you do not have access to a
flatbed scanner you can use a phone app such as “Adobe Scan” or “Microsoft Office Lens”.
• Submit your assignment on Blackboard through Turnitin.
• Make sure you show all steps, key formulae, and workings clearly. Final solutions should
be simplified as much as possible and either highlighted or circled. Round to the nearest
hundredth if necessary.
• 100 Marks - 30% of overall assessment
1. (8 marks) Find the domain and range of the each of these functions.
a) (4 marks) f(x) =
√
x2 − x+ 3− 2
b) (4 marks) g(x) = e
√
x−1
2. (8 marks) Consider the function f(x) = 3x+ 2x2 + ex.
a) (4 marks) Find the second order Taylor polynomial for f(x) about x = −1.
b) (2 marks) Use your answer from part a) to calculate approximate values for f(−1.01) and
f(−5). Round to the nearest hundredth.
c) (2 marks) Compare your answers from part b) to the true values of f(−2) and f(−5).
How far off the true value was each approximation?
1
23. (8 marks) The graph function f(x), defined on the interval [−2, 2] is as follows:
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
3.5
4
a) (4 marks) Using set notation, report the values of x for which f ′(x) exists and f ′(x) ≥ 0.
b) (4 marks) Using set notation, report the values of x for which f ′′(x) exists and f ′′(x) < 0.
4. (8 marks) The graph function g(x), defined on the interval [−2, 2] is as follows:
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-2
-1
0
1
2
3
4
5
6
7
a) (4 marks) Using set notation, report the values of x for which g′(x) exists and g′(x) ≤ 0.
b) (4 marks) Using set notation, report the values of x for which g′′(x) exists and g′′(x) ≤ 0.
5. (24 marks) Find the first and second derivatives of each of these functions. Simplify as much
as possible.
a) (8 marks) f(x) = e2x ln(3x+ 1)
b) (8 marks) g(x) = (x5 + 3x2 + 43x)(12x+ 3)
3c) (8 marks) h(x) = (ln(x+ 3)− ln(x+ 2))2
6. (16 marks) Use implicit differentiation to find dydx and
d2y
dx2
for each of the following cases:
a) (8 marks) y4 + y = x
b) (8 marks) y = e2x+y
7. (12 marks) Suppose that a firm can sell a quantity q at a price of 100− 2q. The firm’s costs
are given by C(q) = q3 + 2q2 + F for some F > 0. The constant F is a fixed cost that only
needs to be paid if the firm opens its doors (think of this as registering the firm, buying land
for production, etc.). If the firm does not open its doors, it earns 0 profits.
a) (6 marks) Suppose that the firm has chosen to open its doors. What is the level of q which
maximizes profits?
b) (6 marks) For what values of F should the firm open its doors? In other words, for what
values of F will the firm earn positive profits given the solution to part a)?
8. (16 marks) Suppose that there is a worker who is deciding how much to work. They have
a job that pays them at wage w and they can choose how many hours ` that they want to
work.
a) (4 marks) The decision maker values dollars of consumption as c
2
3 and has a linear dislike
of labor provision. Thus, the decision maker chooses ` to maximize
(w`)
2
3 − `
Find the ` which solves the decision maker’s problem, treating w as a constant.
b) (6 marks) Suppose that now the decision maker is now faced with a labor tax of τ (where
0 ≤ τ < 1), so their take-home pay is only (1− τ)w`. Find the value of ` that maximizes
((1− τ)w`) 23 − `,
treating both w and τ as constants in the maximization problem. Does a higher value of
τ lead to higher or lower `?
c) (6 marks) Suppose now that the decision maker faces a “poll tax” of T that does not
depend on their earnings, so their take-home pay is only w`−T .1 Find the value of ` that
maximizes
(w`− T ) 23 − `,
1Assume that T is “small enough” so that w`− T is always strictly positive. In other words, do not worry about
the possibility that w`− T is negative.
4treating both w and T as constants in the maximization problem. Does a higher value of
T lead to higher or lower `?