Business Calculus: Marginal Analysis, Exponential & Logarithmic Functions, and Elasticity of Demand

Marginal Analysis in Business & Economics

Key Definitions

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

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

• Profit (P(x)): The difference between revenue and cost, given by $P(x) = R(x) - C(x)$.

• Break-even Point: The production level where revenue equals cost ($R(x) = C(x)$), resulting in zero profit.

Marginal Analysis

Marginal analysis examines the additional benefit or cost associated with producing one more unit of a good or service. It is a fundamental tool for optimizing business decisions.

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

• Marginal Revenue (MR): The derivative of the revenue function, $R'(x)$, estimates the additional revenue from selling one more unit.

• Marginal Profit (MP): The derivative of the profit function, $P'(x)$, gives the additional profit from one more unit.

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

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

Similar approximations hold for revenue and profit.

Example: Marginal Cost Calculation

• Given $C(x) = 0.0005x^3 - 0.38x^2 + 120x$, estimate the marginal cost at $x = 150$ by computing $C'(150)$.

• To find the exact cost of the 151st passenger, calculate $C(151) - C(150)$.

Demand and Revenue Functions

• Demand Function: Relates price p to quantity demanded x, often written as $p = f(x)$.

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

Example: Marginal Revenue

• If $p = 50 - \frac{x}{25}$, then $R(x) = x \left(50 - \frac{x}{25}\right)$.

• Marginal revenue at $x = 10$ is $R'(10)$.

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

Exponential Functions

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

• Natural Exponential Function: $f(x) = e^x$, where $e \approx 2.71828$.

Exponential Growth & Decay

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

• $y_0$: Initial quantity

• $k$: Relative growth rate ($k > 0$ for growth, $k < 0$ for decay)

• $t$: Time

Example: Bacterial Growth

• If $N = N_0 e^{1.386t}$ and $N_0 = 25$, find $N$ after $t = 3.5$ hours.

Example: Carbon Dating

• Radioactive decay: $A = A_0 e^{-0.000124t}$

• Given $A_0 = 500$ mg, find $A$ after $t = 15,000$ years.

Compound and Continuous Interest

• Compound Interest Formula:

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

  • $P$: Principal

  • $r$: Annual interest rate (decimal)

  • $m$: Number of compounding periods per year

  • $t$: Time in years

• Continuous Compound Interest:

  $A = Pe^{rt}$

Example: Continuous Interest

• If $P = 100$, $r = 0.06$, $t = 2$, then $A = 100e^{0.12}$.

Logarithmic Functions

• Logarithmic Function: $y = \log_b x$ is equivalent to $x = b^y$.

• Natural Logarithm: $\ln x = \log_e x$

• Common Logarithm: $\log x = \log_{10} x$

• Domain: $(0, \infty)$; Range: $(-\infty, \infty)$

Properties of Logarithms

• $\log_b 1 = 0$

• $\log_b b = 1$

• $\log_b (MN) = \log_b M + \log_b N$

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

• $\log_b (M^p) = p \log_b M$

Example: Logarithmic Equations

• Solve $\log_2 x - \log_2(x-1) = 1$

• Solve $\log x + \log(x-1) = \log(3x+12)$

Derivatives of Exponential & Logarithmic Functions

Basic Derivative Rules

• $\frac{d}{dx}[e^x] = e^x$

• $\frac{d}{dx}[e^{kx}] = k e^{kx}$

• $\frac{d}{dx}[\ln x] = \frac{1}{x}$

• $\frac{d}{dx}[b^x] = b^x \ln b$

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

Example: Differentiation

• Find $\frac{d}{dx}[3e^x + 5\ln x]$

• Find $\frac{d}{dx}[x^7 \ln x]$

Product and Quotient Rules

Product Rule

If $f(x) = n(x) d(x)$, then:

$f'(x) = n'(x) d(x) + d'(x) n(x)$

Quotient Rule

If $f(x) = \frac{n(x)}{d(x)}$, $d(x) \neq 0$, then:

$f'(x) = \frac{n'(x)d(x) - d'(x)n(x)}{[d(x)]^2}$

Example: Quotient Rule

• Differentiate $y = \frac{2x-1}{4x+3}$

The Chain Rule

Definition and Application

• Composite Function: $(f \circ g)(x) = f(g(x))$

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

Generalized Power Rule

$\frac{d}{dx}[g(x)^n] = n[g(x)]^{n-1} \cdot g'(x)$

Generalized Exponential Rule

$\frac{d}{dx}[e^{g(x)}] = g'(x) e^{g(x)}$

Generalized Logarithm Rule

$\frac{d}{dx}[\ln g(x)] = \frac{g'(x)}{g(x)}$

Example: Chain Rule

• If $h(x) = (2x-1)^4$, then $h'(x) = 4(2x-1)^3 \cdot 2$

• If $f(x) = e^{4x^2+3x+2}$, then $f'(x) = (8x+3) e^{4x^2+3x+2}$

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: $100 \times \frac{f'(x)}{f(x)}$

Example: Investment Growth

• If $F(t) = 5000e^{0.07t}$, the relative rate of change is $0.07$ (i.e., 7% per year).

Elasticity of Demand

• Elasticity of Demand: For price-demand equation $x = f(p)$, elasticity at price $p$ is:

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

• Interpretation:

  • If $E(p) > 1$: Demand is elastic (demand is sensitive to price; price increase decreases revenue).

  • If $E(p) < 1$: Demand is inelastic (demand is not sensitive; price increase increases revenue).

  • If $E(p) = 1$: Unit elastic (revenue is maximized).

Example: Elasticity Calculation

• Given $x + 500p = 10,000$, find $E(p)$ at $p = 4, 10, 16$.

• Given $x = 2400 - 2p^2$, determine elasticity at $p = 8$.

Summary Table: Elasticity Types

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

Additional info:

• All formulas and rules are essential for Business Calculus, especially in applications to economics and business decision-making.

• Examples provided in the notes illustrate practical applications, such as cost estimation, revenue optimization, and investment growth.