Business Mathematics #1

Derive inverse and direct demand functions from linear relations and identify slope as demand. Solve a system to obtain Qd = 600 - 5P and P = 50 - 0.25Q.

defines market equilibrium as where demand equals supply, with price and quantity at their intersection; algebraic solving gives equilibrium price about 13.3 and quantity about 7.3.

Explore how market disequilibrium occurs when prices create surpluses or shortages, illustrating with price floors and ceilings and their effects on producers and consumers.

Distinguish supply function and demand function by slope in price as a function of quantity. Then set demand equal to supply to find market equilibrium quantity and price.

Derive market equilibrium for sneakers with a hyperbolic demand and linear supply by solving a quadratic equation analytically, yielding price ~39.4 and quantity ~11.7 (thousand pairs), with a graphical check.

This lecture explains cost functions by separating fixed costs from variable costs, showing total costs as fixed plus variable, with linear and quadratic examples illustrating how costs rise with output.

Explore the revenue function as revenue equals price times quantity. Derive revenue from price functions tied to a demand function, e.g., R = 2Q and R = 4Q − 0.5Q^2.

Compute average revenue and average costs by dividing total revenue and total cost by quantity. Examine how average fixed costs and average variable costs shape with production.

Master differentiation rules, including the power rule d/dx x^n = n x^{n-1} and the constant multiple rule for a·x^n. Apply to revenue function c(q) = q - 0.5 q^2.

Explore marginal functions by differentiating revenue, cost, and profit functions to determine the extra revenue and cost of each additional unit of production.

Explore the profit function as total revenue minus total costs, with a quadratic revenue and linear cost; expand brackets and analyze profit regions via the graph.

Identify the break even point as where total revenue equals total costs. Use fixed costs divided by price minus variable cost per unit to compute it.

Identify break-even points at q=4 and q=20 and the maximum profit at q=12, value 128, using first and second derivatives on a downward-opening quadratic.

Derive the marginal cost from the total cost function, locate Q = 10 by MC' = 0, and confirm a minimum with the second derivative, MC = 25.

Compute the minimum average cost from a quadratic total cost function by deriving the average cost and its first and second derivatives, showing q = 50 yields AC = 250.