DAY 1: MATHEMATICAL METHODS PROBLEMS 2022
1. Given that the production function 𝑄(𝐿) = 𝑏𝐿2 , with 𝑏 > 0 (why is 𝑏 > 0) on ℝ+ . Derive
the cost function 𝐶(𝑄) and the profit function 𝜋(𝑄) for the perfectly competitive firm. Let
fixed costs be 𝑐0 and 𝑤 be the unit cost of 𝐿. Prove that the production function is
continuous and discuss the continuity of both the profit and cost functions.
2. Given that 𝑄𝑑(𝑝) = 50 − 2𝑝 and 𝑄𝑠(𝑝) = −10 + 𝑝. Using the mean value theorem
show that there is an existence of a positive equilibrium price (Don’t find the price). Hint:
Define new function as 𝑔(𝑝) = 𝑄𝑑(𝑝) – 𝑄𝑠(𝑝) and use the mean value theorem.
3. Given that 𝑥 = 𝑅𝐶𝑜𝑠𝜃 𝑎𝑛𝑑 𝑦 = 𝑅𝑆𝑖𝑛𝜃 𝑤ℎ𝑒𝑟𝑒 𝑅 ≥ 0 𝑎𝑛𝑑 𝑅 → 0 𝑎𝑠 (𝑥, 𝑦) → (0,0)
Evaluate
i.
𝑥3 + 𝑦3
(𝑥,𝑦)→(0,0) 𝑥 2 + 𝑦 2
lim
ii.
(𝑥 2 + 𝑦 2 )ln (𝑥 2 + 𝑦 2 )
lim
(𝑥,𝑦)→(0,0)
4. Find these limits if they exist.
i.
𝑥3𝑦
(𝑥,𝑦)→(0,0) 𝑥 6 + 𝑦 2
lim
ii.
𝑥2 − 𝑦2
lim
(𝑥,𝑦)→(0,0) 𝑥 2 + 𝑦 2
5. State Squeeze theorem and use it to find the values of these limits.
i.
𝑥𝑦
lim
(𝑥,𝑦)→(0,0) √𝑥 2
+ 𝑦2
ii.
𝑥2𝑦
lim
(𝑥,𝑦)→(0,0) 𝑥 2 + 𝑦 2
6. Discuss the continuity of:
𝑥 2 −𝑦 2
𝑖𝑓 (𝑥, 𝑦) ≠ (0,0)
{𝑥 2+𝑦 2
0 𝑖𝑓 (𝑥, 𝑦) = (0,0)
7. Let 𝑓(𝑥, 𝑦) = 𝑥 2 + 3𝑥𝑦 − 𝑦 2 = 𝑘
i.
Find the total derivative of 𝑓(𝑥, 𝑦) with respect to 𝑦.
ii.
Use the above to find 𝑥 ′ (𝑦), and verify using the implicit function theorem.
iii.
If x changes from 2 to 2.05 and y changes from 3 to 2.96, use (i) to estimate the change
in 𝑓(𝑥, 𝑦). Moreover, calculate the actual change in 𝑓(𝑥, 𝑦) and compare the two values.