Business and Financial Math: Cost, Revenue & Interest Formulas

Cost, Revenue, and Break-Even

Cost = VariableCost + FixedCost     Revenue = X * price     Break-even condition: P(x) = 0

C(x) = 8x + 100             R(x) = 10x             R(x) = C(x)         Profit = Revenue - Cost

Profit Function and Example

P(x) = R(x) - C(x) = 10x - (8x + 100) = 2x - 100

Demand and Supply Equilibrium

Demand: demand as a function of unit price P: Q_d = a - bP. Equilibrium when D = S.

Supply: q (# items) as a function of unit price P. Example (demand): q = -20p + 800.

Example supply: q = 10p - 100 (supply). Solve equilibrium: -20p + 800 = 10p - 100 → -30p = -900 → p = $30 (equilibrium price). Then q = -20(30) + 800 → q = 200 (equilibrium quantity).

Compound Interest and Future Value

Variables: P = present value, r = annual nominal rate, t = time (years), A = future value.

• Compounded annually: A(t) = P(1 + r)^t.
• Compounded m times per year: A(t) = P(1 + r/m)^{m t}.
• Example: A(t) = P(1 + r/4)^{4 t} (compounded quarterly).

To find r when A, P, m, and t are known:

r = m( (A/P)^{1/(m t)} - 1 ). For annual compounding (m = 1): r = (A/P)^1/t - 1.

Future value formula (n periods): FV = PV(1 + i)ⁿ. When using r and m: FV = PV(1 + r/m)^{m t} (t in years).

Calculate Present Value

PV = FV(1 + i)^-n or equivalently PV = FV / (1 + r/m)^{m t}.

Annuities

Present value of an annuity:

PV = PMT · [1 - (1 + i)^-n] / i

Quadratic Features: Vertex and Intercepts

Vertex (x-coordinate): x = -b / (2a), and value f(-b/(2a)).

x-intercept(s): Solve f(x) = 0 (factor or use the quadratic formula).

y-intercept: f(0) (occurs when x = 0).

Maximum Annual Revenue

Revenue R = p q (substitute q from the demand equation).

Demand: Q_d (quantity demanded) = a (intercept) - b (slope) · P (price).

Example: If q = -2000p + 150000, then R(p) = p(-2000p + 150000) = -2000p² + 150000p. Use vertex formula p = -b/(2a) to find the price that maximizes revenue: p = -150000 / (2 · -2000) = 37.5. Then evaluate f(37.5) to find the maximum revenue.

Exponential Functions and Growth

General form: F(x) = A b^x.

Exponential growth: y = A b^t with b > 1.

Exponential decay: y = A b^t with 0 < b < 1.

Example: A(t) = 2000(1.015)^{12 t} is exponential growth because b > 1.

Continuous Compounding and Logarithms

Continuous compounding: A(t) = P e^{r t}.

If r > 0, this is growth; if r < 0, this is decay. Use natural logarithms to solve for variables in exponents.

For example, if A = P(1.003)^{12 t}, take ln of both sides: ln(A/P) = 12t · ln(1.003).

Note: ln(e) = 1.

Logistic Function

Form: f(x) = N / (1 + A b^-x), where N is the limiting value (carrying capacity).

If b > 1 and A > 0, the graph rises toward N as x increases; if 0 < b < 1 or A changes sign, the behavior differs (it may fall or shift).

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