Homework 1
1. Given the sets 𝑆1 = {2, 4, 6}, 𝑆2 = {7, 2, 6}, 𝑆3 = {4, 2, 6}, and 𝑆4 = {2, 4}, which of the
following statements are true?
(a) 𝑆1 = 𝑆3
(d) 3 ∉ 𝑆2
(g) 𝑆1 ⊃ 𝑆4
(b) 𝑆1 = 𝑹
(e) 4 ∉ 𝑆3
(h) 𝜙 ⊂ 𝑆2
(c) 8 ∈ 𝑆2
(f) 𝑆4 ⊂ 𝑹
(i) 𝑆3 ⊃ {1,2}
2. Enumerate all the subsets of the set 𝑆 = {𝑎, 𝑏, 𝑐, 𝑑}. How many subsets are there altogether?
3. Does any of the following, drawn in a rectangular coordinate plane, represent a function?
(a) A circle
(c) A rectangle
(b) A triangle
(d) A downward-sloping straight line
4. If the domain of the function 𝑦 = 5 + 3𝑥 is the set {𝑥 ∈ 𝑹|1 ≤ 𝑥 ≤ 9}, find the range of the
function and express it as a set.
5. Given the demand and supply functions:
(a) 𝑞𝐷 = 200 − 2𝑝; 𝑞𝑆 = 100 + 𝑝,
(b) 𝑞𝐷 = 3𝑝−1/2; 𝑞𝑆 = 𝑝,
(c) 𝑞𝐷 = 10 + 𝑝; 𝑞𝑆 = 20 + 𝑝/2,
(d) 𝑞𝐷 = 20 − 𝑝2 ; 𝑞𝑆 = 12 + 𝑝,
find equilibrium price and quantity in each case. Sketch the supply and demand curves. Calculate
consumer surplus and producer surplus in case (d).
6. Given a demand function 𝑞𝐷 = 15 − 2𝑝, and a supply function
𝑞𝑆 = 0 for 𝑝 < 5,
𝑞𝑆 = 10 + 𝑝 for 𝑝 ≥ 5,
analyze the problem of finding an equilibrium (hint: draw a graph). Can you give a plausible
economic explanation for this supply function?
7. Find the marginal and average functions for the following total functions and graph the results:
Total-cost function: (a) 𝐶 = 3𝑄 2 + 7𝑄 + 12
Total-revenue function: (b) 𝑅 = 100𝑄 − 𝑄 2
Total-product function: (c) 𝑄 = 𝑎𝐿 + 𝑏𝐿2 − 𝑐𝐿3 (a,b,c > 0)
8. Given 𝑦 = (16𝑥 + 3)−2,
(a) use the chain rule to find dy/dx,
(b) sketch the function,
(c) comment on the geometric properties of the function y (continuity, monotonicity and
differentiability).
(d) comment on the geometric properties of the function y on the domain 𝑹+ (continuity,
monotonicity, differentiability and convexity/concavity)