Business Calculus – Complete Reference Guide

Functions Review

Linear Functions

$f(x) = mx + b$

$m$ = slope, $b$ = y-intercept

Domain: All real numbers

Quadratic Functions

$f(x) = ax^2 + bx + c$

Vertex: $x = -\frac{b}{2a}$

Opens up if $a > 0$, down if $a < 0$

Polynomial Functions

$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$

Degree determines behavior at infinity

Exponential Functions

$f(x) = a \cdot b^x$ where $b > 0, b \neq 1$

Growth: $b > 1$, Decay: $0 < b < 1$

$f(x) = Pe^{rx}$ (natural base)

Logarithmic Functions

$f(x) = \log_b(x)$ or $f(x) = \ln(x)$

Inverse of exponential: $b^{\log_b(x)} = x$

Limits & Continuity

Limit Definition

$\lim_{x \to a} f(x) = L$ means $f(x)$ approaches $L$ as $x$ approaches $a$

Limit Laws

$\lim(f + g) = \lim f + \lim g$

$\lim(f \cdot g) = \lim f \cdot \lim g$

$\lim \frac{f}{g} = \frac{\lim f}{\lim g}$ (if $\lim g \neq 0$)

Continuity at $x = a$

1. $f(a)$ is defined

2. $\lim_{x \to a} f(x)$ exists

3. $\lim_{x \to a} f(x) = f(a)$

One-Sided Limits

Left limit: $\lim_{x \to a^-} f(x)$

Right limit: $\lim_{x \to a^+} f(x)$

If both equal $L$, then $\lim_{x \to a} f(x) = L$

Derivative Definition

Formal Definition

$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

Slope of tangent line at point $(a, f(a))$

Alternative Form

$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

Interpretation

Instantaneous rate of change

Slope of tangent line

Velocity (if $f$ is position)

Differentiability

$f$ is differentiable at $x = a$ if $f'(a)$ exists

Requires continuity at $a$

No sharp corners or cusps

Basic Differentiation Rules

Power Rule

$\frac{d}{dx}[x^n] = nx^{n-1}$

Constant Rule

$\frac{d}{dx}[c] = 0$

Sum/Difference Rule

$\frac{d}{dx}[f \pm g] = f' \pm g'$

Constant Multiple

$\frac{d}{dx}[cf] = c \cdot f'$

Exponential & Log

$\frac{d}{dx}[e^x] = e^x$

$\frac{d}{dx}[a^x] = a^x \ln(a)$

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

$\frac{d}{dx}[\log_a(x)] = \frac{1}{x \ln(a)}$

Trig Functions

$\frac{d}{dx}[\sin x] = \cos x$

$\frac{d}{dx}[\cos x] = -\sin x$

Product, Quotient, Chain

Product Rule

$(fg)' = f'g + fg'$

"First times derivative of second, plus second times derivative of first"

Quotient Rule

$\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$

"Low d-high minus high d-low, over low squared"

Chain Rule

$(f \circ g)' = f'(g(x)) \cdot g'(x)$

Or: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

Use chain rule for composite functions. Look for function within function.

Common Chain Rule Cases

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

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

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

Marginal Analysis

Marginal Cost

$MC = C'(x)$

Approx. cost of producing one more unit

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

Marginal Revenue

$MR = R'(x)$

Revenue gained from selling one more unit

Marginal Profit

$MP = P'(x) = R'(x) - C'(x)$

Optimization

Max Profit: Solve $P'(x) = 0$

Or: $R'(x) = C'(x)$ (MR = MC)

Check: $P''(x) < 0$ for maximum

Average Cost

$\overline{C}(x) = \frac{C(x)}{x}$

Min occurs when $\overline{C}'(x) = 0$

At minimum: $\overline{C}(x) = C'(x)$

Elasticity of Demand

Definition

$E = -\frac{p}{q} \cdot \frac{dq}{dp}$

Percentage change in quantity / percentage change in price

Interpretation

$|E| > 1$: Elastic (sensitive to price)

$|E| < 1$: Inelastic (not sensitive)

$|E| = 1$: Unit elastic

Revenue Relationship

Elastic: Price ↑ ⟹ Revenue ↓

Inelastic: Price ↑ ⟹ Revenue ↑

Unit elastic: Revenue maximized

From Demand Function

If demand is $q = f(p)$:

$E = \frac{p}{q} \cdot \frac{dq}{dp}$

Optimization Techniques

First Derivative Test

1. Find critical points: $f'(x) = 0$

2. Test intervals around critical points

3. $f' > 0$: increasing, $f' < 0$: decreasing

Second Derivative Test

At critical point $x = c$:

$f''(c) < 0$ ⟹ Local maximum

$f''(c) > 0$ ⟹ Local minimum

$f''(c) = 0$ ⟹ Test inconclusive

Extreme Value Theorem

Continuous function on $[a,b]$ has absolute max/min

Check: critical points in $(a,b)$ and endpoints

Applied Optimization

1. Define variables and objective function

2. Write constraint equation

3. Express as single-variable function

4. Find critical points and evaluate

Related Rates in Business

Strategy

1. Identify related quantities and their rates

2. Write equation relating the quantities

3. Differentiate both sides with respect to time

4. Substitute known values and solve

Common Business Cases

Production: Units produced over time

Cost Growth: Rate of cost increase

Revenue Growth: Rate of revenue change

Example Setup

If $C(x) = $ cost and $x$ = units, and $\frac{dx}{dt} = $ rate:

$\frac{dC}{dt} = C'(x) \cdot \frac{dx}{dt}$

Always identify what you know and what you're solving for before differentiating.

Exponential & Logarithmic

Exponential Properties

$a^m \cdot a^n = a^{m+n}$

$\frac{a^m}{a^n} = a^{m-n}$

$(a^m)^n = a^{mn}$

$a^0 = 1$, $a^{-n} = \frac{1}{a^n}$

Logarithm Properties

$\log_a(xy) = \log_a(x) + \log_a(y)$

$\log_a(\frac{x}{y}) = \log_a(x) - \log_a(y)$

$\log_a(x^n) = n\log_a(x)$

$\log_a(a) = 1$, $\log_a(1) = 0$

Change of Base

$\log_a(x) = \frac{\ln(x)}{\ln(a)} = \frac{\log(x)}{\log(a)}$

Solving Exponential Equations

$a^x = b$ ⟹ $x = \log_a(b)$

$e^x = b$ ⟹ $x = \ln(b)$

Compound Interest

Periodic Compounding

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

$P$ = principal, $r$ = annual rate

$n$ = compounding periods per year

$t$ = time in years

Continuous Compounding

$A = Pe^{rt}$

Most common in finance models

Effective Annual Rate

$r_{eff} = \left(1 + \frac{r}{n}\right)^n - 1$

$r_{eff} = e^r - 1$ (continuous)

Doubling Time

When $A = 2P$:

$t = \frac{\ln(2)}{r}$ (continuous)

Present & Future Value

Future Value

$FV = PV \cdot e^{rt}$

Value of investment at time $t$

Present Value

$PV = FV \cdot e^{-rt}$

Today's value of future amount

Discount factor: $e^{-rt}$

Present Value of Annuity

Regular payments $PMT$ at rate $r$:

$PV = \frac{PMT}{r}\left(1 - e^{-rt}\right)$

Present Value of Income Stream

$PV = \int_0^T f(t) e^{-rt} dt$

$f(t)$ = income at time $t$

Integration Basics

Antiderivative Definition

$F$ is an antiderivative of $f$ if $F'(x) = f(x)$

Indefinite Integral

$\int f(x) dx = F(x) + C$

$C$ = arbitrary constant of integration

Power Rule for Integration

$\int x^n dx = \frac{x^{n+1}}{n+1} + C$ $(n \neq -1)$

Basic Antiderivatives

$\int e^x dx = e^x + C$

$\int \frac{1}{x} dx = \ln|x| + C$

$\int a^x dx = \frac{a^x}{\ln(a)} + C$

$\int \sin x dx = -\cos x + C$

$\int \cos x dx = \sin x + C$

Definite Integrals & Area

Fundamental Theorem of Calculus

$\int_a^b f(x) dx = F(b) - F(a)$

$F$ = antiderivative of $f$

Net Area

$\int_a^b f(x) dx$ = signed area

Positive where $f(x) > 0$

Negative where $f(x) < 0$

Total Area Between Curves

$A = \int_a^b |f(x) - g(x)| dx$

Or split at intersection points

Area Under Curve

$A = \int_a^b f(x) dx$ (if $f(x) \geq 0$)

Total Accumulation

Distance traveled: $\int v(t) dt$

Total cost: $\int MC(x) dx$

Total revenue: $\int MR(x) dx$

Surplus Analysis

Consumer Surplus

$CS = \int_0^{x^*} [D(x) - p^*] dx$

Benefit to consumers from market price

$D(x)$ = demand function

$p^*$ = equilibrium price

$x^*$ = equilibrium quantity

Producer Surplus

$PS = \int_0^{x^*} [p^* - S(x)] dx$

Benefit to producers from market price

$S(x)$ = supply function

Total Surplus

$TS = CS + PS$

Total welfare from market transaction

Equilibrium

Where $D(x) = S(x)$

Find $(x^*, p^*)$: solution to system

Average Value & Integral Apps

Average Value of Function

$\bar{f} = \frac{1}{b-a} \int_a^b f(x) dx$

Mean value over interval $[a,b]$

Mean Value Theorem for Integrals

There exists $c \in [a,b]$ such that:

$f(c) = \bar{f}$

Business Applications

Avg cost over time: $\frac{1}{t} \int_0^t C(x) dx$

Avg revenue: $\frac{1}{b-a} \int_a^b R(x) dx$

Total from Marginal

$C(x) = C(0) + \int_0^x C'(u) du$

$R(x) = \int_0^x R'(u) du$

$P(x) = P(0) + \int_0^x P'(u) du$

Multivariable Calculus

Partial Derivatives

$\frac{\partial f}{\partial x}$ : treat $y$ as constant

$\frac{\partial f}{\partial y}$ : treat $x$ as constant

Critical Points

Solve the system:

$\frac{\partial f}{\partial x} = 0$

$\frac{\partial f}{\partial y} = 0$

Second Partial Derivative Test

At critical point $(a,b)$, compute:

$D = f_{xx}f_{yy} - (f_{xy})^2$

$D > 0, f_{xx} < 0$: Local maximum

$D > 0, f_{xx} > 0$: Local minimum

$D < 0$: Saddle point

$D = 0$: Test inconclusive

Lagrange Multipliers

Problem Setup

Optimize $f(x,y)$ subject to constraint $g(x,y) = k$

Lagrange Method

Solve the system:

1. $\nabla f = \lambda \nabla g$

2. $g(x,y) = k$

Component Form

$f_x = \lambda g_x$

$f_y = \lambda g_y$

$g(x,y) = k$

Business Example

Maximize profit $P(x,y)$ given fixed budget constraint

$x$ = units of product A

$y$ = units of product B

Constraint: Cost limit

$\lambda$ = marginal utility (value of relaxing constraint)

Integration Techniques

Substitution (u-substitution)

Let $u = g(x)$, then $du = g'(x)dx$

$\int f(g(x)) g'(x) dx = \int f(u) du$

Integration by Parts

$\int u dv = uv - \int v du$

Choose $u$ = function easier to differentiate

Partial Fractions

For rational functions $\frac{P(x)}{Q(x)}$

Factor denominator

Write as sum of simpler fractions

Integrate each term

Common Substitutions

$\sqrt{a^2-x^2}$: use $x = a\sin\theta$

$\sqrt{a^2+x^2}$: use $x = a\tan\theta$

$e^{ax}$: substitute $u = ax$

Economic Integration Apps

Total Cost from Marginal Cost

$C(x) = C(0) + \int_0^x MC(q) dq$

Total Revenue from Marginal

$R(x) = \int_0^x MR(q) dq$

Total Profit

$P(x) = R(x) - C(x)$

Accumulated Savings

$S(t) = \int_0^t s(u) du$

$s(u)$ = savings rate at time $u$

Present Value of Stream

$PV = \int_0^T f(t) e^{-rt} dt$

$f(t)$ = income/cash flow rate

$r$ = discount rate

BUSINESS CALCULUS Complete Formula Reference