Part 1: Calculus exercises

Find the derivative (dY/dX) of the following functions: (Please provide step by step)

1. Y=8X⁴ - 20X³
2. Y=X³-8X²+57X+2
3. Y=4X³(2X- 5)
4. Y=3X³ / X²
5. Y=U² and U = 2X³+3
6. Y=(2X³+5)²

Part 2: Optimization Questions

1. Quilted Products, Ltd., offers mail-order storage containers for fine china to newlyweds. The company is the low-cost provider of these quilted boxes with fixed costs of $480,000 per year, plus variable costs of $30 for each box. Annual demand relations for the company are:

P= $70 - $0.0005Q

(Hint: Please calculate the TR by taking P x Q = TR = ($70 - $0.0005Q) Q = $70Q - 0.0005Q²

a. Calculate the profit-maximizing activity level.

b. Calculate the company's optimal profit and return-on-sales levels. (Hint: Return on sales is profit divided by total revenue.)

2. Given the following total revenue and total cost functions of a firm:

TR=22Q-0.5Q²  ; 

TC=(1/3)Q³-8.5Q²+50Q+90

a. Determine the level of output at which the firm maximizes it total profit. (Hint: Calculate the MR and MC function. Let MC = MR)

1. Determine the maximum profit that the firm could earn.