Business Calculus: Marginal Analysis, Exponential & Logarithmic Functions, and Derivative Applications

Marginal Analysis in Business & Economics

Key Definitions

• Revenue (R(x)): The total money received from selling x units of a good or service.

• Cost (C(x)): The total money required to produce x units, including fixed and variable costs.

• Profit (P(x)): The difference between revenue and cost, P(x) = R(x) - C(x). If R(x) < C(x), the company incurs a loss.

Break-Even Point

• The production level where R(x) = C(x). At this point, profit is zero.

Marginal Analysis

Marginal analysis examines the additional benefit or cost of producing one more unit. It is a key tool for maximizing profit.

• Marginal Cost (MC): The derivative of the cost function, C'(x), approximates the cost of producing one additional unit after x units.

• Marginal Revenue (MR): The derivative of the revenue function, R'(x).

• Marginal Profit (MP): The derivative of the profit function, P'(x).

Theorem: The marginal cost at x approximates the exact cost of producing the (x+1)st item:

$C'(x) \approx C(x+1) - C(x)$

Similar approximations hold for revenue and profit.

• Example: If C(x) = 10,000 + 150x - 0.2x^2, then C'(x) gives the approximate cost of the next item.

Demand and Revenue Functions

• Demand Function: Relates price (p) and quantity demanded (x): p = f(x).

• Total Revenue: R(x) = x \cdot p = x \cdot f(x).

Average and Marginal Average Functions

• Average Cost: $\overline{C}(x) = \frac{C(x)}{x}$

• Average Revenue: $\overline{R}(x) = \frac{R(x)}{x}$

• Average Profit: $\overline{P}(x) = \frac{P(x)}{x}$

• Marginal Average: The derivative of the average function, e.g., $\overline{C}'(x)$.

Exponential & Logarithmic Functions

The Number e and Exponential Functions

• e is a mathematical constant, approximately 2.71828.

• Exponential Function: $f(x) = b^x$, where $b > 0$, $b \neq 1$.

Exponential Growth and Decay

Many real-world processes grow or decay exponentially:

$f(t) = y_0 e^{kt}$

• y0: Initial amount

• k: Growth rate (positive for growth, negative for decay)

• t: Time

• Example: Bacterial growth: $N = N_0 e^{1.386t}$

• Example: Carbon-14 decay: $A = A_0 e^{-0.000124t}$

Compound and Continuous Interest

• Compound Interest: $A = P \left(1 + \frac{r}{m}\right)^{mt}$

• Continuous Compound Interest: $A = Pe^{rt}$

• P: Principal, r: annual rate (decimal), m: compounding periods per year, t: years

Logarithmic Functions

• Logarithm: The inverse of an exponential function. $y = \log_b x$ is equivalent to $x = b^y$.

• Common Logarithms: $\log x = \log_{10} x$, $\ln x = \log_e x$

• Domain: $x > 0$

Properties of Logarithms

• $\log_b 1 = 0$

• $\log_b b = 1$

• $\log_b (MN) = \log_b M + \log_b N$

• $\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N$

• $\log_b (M^p) = p \log_b M$

Derivatives of Exponential & Logarithmic Functions

Basic Derivative Rules

• $\frac{d}{dx} [e^x] = e^x$

• $\frac{d}{dx} [e^{kx}] = k e^{kx}$

• $\frac{d}{dx} [\ln x] = \frac{1}{x}$

• $\frac{d}{dx} [b^x] = b^x \ln b$

• $\frac{d}{dx} [\log_b x] = \frac{1}{x \ln b}$

Product and Quotient Rules

• Product Rule: $[f(x)g(x)]' = f'(x)g(x) + f(x)g'(x)$

• Quotient Rule: $\left[\frac{f(x)}{g(x)}\right]' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

Chain Rule

• If $h(x) = f(g(x))$, then $h'(x) = f'(g(x)) \cdot g'(x)$

• Generalized Power Rule: $\frac{d}{dx} [g(x)]^n = n [g(x)]^{n-1} g'(x)$

• Generalized Exponential Rule: $\frac{d}{dx} [e^{g(x)}] = g'(x) e^{g(x)}$

• Generalized Logarithm Rule: $\frac{d}{dx} [\ln g(x)] = \frac{g'(x)}{g(x)}$

Elasticity of Demand

Relative and Percentage Rate of Change

• Relative Rate of Change: $\frac{f'(x)}{f(x)}$ or $\frac{d}{dx} \ln f(x)$

• Percentage Rate of Change: $100 \times \frac{f'(x)}{f(x)}$

Elasticity of Demand

• Measures how sensitive demand is to changes in price.

• Formula: $E(p) = -\frac{p f'(p)}{f(p)}$

• If $E(p) > 1$: Demand is elastic (demand changes more than price).

• If $E(p) < 1$: Demand is inelastic (demand changes less than price).

• If $E(p) = 1$: Unit elastic (demand changes proportionally to price).

Elasticity	Interpretation	Revenue Effect of Price Increase
0 < E(p) < 1	Inelastic	Revenue increases
E(p) > 1	Elastic	Revenue decreases
E(p) = 1	Unit Elastic	Revenue unchanged

Summary Table: Key Formulas

Concept	Formula
Marginal Cost	$C'(x)$
Marginal Revenue	$R'(x)$
Marginal Profit	$P'(x)$
Average Cost	$\overline{C}(x) = \frac{C(x)}{x}$
Exponential Growth/Decay	$f(t) = y_0 e^{kt}$
Compound Interest	$A = P \left(1 + \frac{r}{m}\right)^{mt}$
Continuous Compound Interest	$A = Pe^{rt}$
Elasticity of Demand	$E(p) = -\frac{p f'(p)}{f(p)}$

Examples and Applications

• Marginal Cost Example: If $C(x) = 0.0005x^3 - 0.38x^2 + 120x$, then $C'(x)$ gives the marginal cost at x passengers.

• Exponential Growth Example: Starting with 25 bacteria, after 3.5 hours: $N = 25e^{1.386 \times 3.5}$

• Compound Interest Example: $1000 at 10\%$ compounded monthly for 10 years: $A = 1000 \left(1 + \frac{0.10}{12}\right)^{120}$

• Elasticity Example: If $x = 2400 - 2p^2$, at $p = 8$, compute $E(8)$ to determine elasticity.

Additional info: Some context and explanations have been expanded for clarity and completeness, including the summary tables and explicit formula listings.