Business Calculus: First Exam Review Study Notes

First Exam Review: Key Topics in Business Calculus

Differentiation

Differentiation is a fundamental concept in calculus, used to determine the rate at which a function changes. In business applications, derivatives help analyze cost, revenue, and profit functions.

• Definition: The derivative of a function $f(x)$, denoted $f'(x)$ or $\frac{df}{dx}$, measures the instantaneous rate of change of $f$ with respect to $x$.

• Basic Rules:

  • Power Rule: $\frac{d}{dx}x^n = nx^{n-1}$

  • Product Rule: $\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$

  • Quotient Rule: $\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$

  • Chain Rule: $\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$

• Applications: Marginal cost, marginal revenue, and marginal profit are found using derivatives.

• Example: Find $\frac{d}{dx}(e^{2x})$. Solution: $\frac{d}{dx}(e^{2x}) = 2e^{2x}$

Critical Values and Optimization

Critical values are points where the derivative is zero or undefined. These are candidates for local maxima or minima, which are important in business for maximizing profit or minimizing cost.

• Finding Critical Values: Solve $f'(x) = 0$ or where $f'(x)$ is undefined.

• Second Derivative Test: If $f''(x) > 0$, the function has a local minimum at $x$. If $f''(x) < 0$, it has a local maximum.

• Example: For $f(x) = x^3 - 3x^2 + 2$, find critical points. Solution: $f'(x) = 3x^2 - 6x$, set $f'(x) = 0$ to get $x = 0, 2$.

Exponential and Logarithmic Functions

Exponential and logarithmic functions are widely used in modeling growth and decay in business contexts, such as compound interest and population growth.

• Derivative of Exponential: $\frac{d}{dx}e^{kx} = ke^{kx}$

• Derivative of Logarithm: $\frac{d}{dx}\ln(x) = \frac{1}{x}$

• Example: Differentiate $y = \ln(x^2 + 5x^3)$. Solution: $y' = \frac{2x + 15x^2}{x^2 + 5x^3}$

Integration

Integration is the reverse process of differentiation and is used to find areas under curves, total accumulated quantities, and solve problems involving marginal functions.

• Definition: The integral of $f(x)$ with respect to $x$ is $\int f(x)dx$.

• Basic Rules:

  • Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, $n \neq -1$

  • Exponential: $\int e^{kx} dx = \frac{1}{k}e^{kx} + C$

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

• Definite Integrals: $\int_a^b f(x)dx$ gives the net area under $f(x)$ from $x=a$ to $x=b$.

• Example: $\int_0^1 e^{2x} dx = \left.\frac{1}{2}e^{2x}\right|_0^1 = \frac{1}{2}(e^2 - 1)$

Applications of Differentiation

Derivatives are used to solve real-world business problems, such as maximizing profit, minimizing cost, and analyzing rates of change.

• Marginal Analysis: Marginal cost, revenue, and profit are the derivatives of total cost, revenue, and profit functions, respectively.

• Optimization: Use critical points and endpoints to find maximum or minimum values of functions.

• Example: If $C(x) = 1600 - e^{-0.2x}$, the minimum cost occurs where $C'(x) = 0$.

Applications of Integration

Integration is used to find total profit, consumer and producer surplus, and areas between curves in business contexts.

• Area Between Curves: $\int_a^b [f(x) - g(x)] dx$ gives the area between $f(x)$ and $g(x)$ from $x=a$ to $x=b$.

• Consumer Surplus: $\int_0^q D(x)dx - qp$ where $D(x)$ is the demand function and $p$ is the price.

• Producer Surplus: $qp - \int_0^q S(x)dx$ where $S(x)$ is the supply function.

• Example: Find the area bounded by $y=5-x$ and $y=x^2$ by solving $\int_{a}^{b} [(5-x) - x^2] dx$ over the intersection points.

Limits and Continuity

Limits are foundational to calculus, used to define derivatives and integrals. Understanding limits helps analyze function behavior near specific points.

• Definition: $\lim_{x \to a} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$.

• One-Sided Limits: $\lim_{x \to a^-} f(x)$ (from the left), $\lim_{x \to a^+} f(x)$ (from the right)

• Example: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = 4$

Business Applications: Cost, Revenue, and Profit Functions

Business calculus frequently involves analyzing cost, revenue, and profit functions to make informed decisions.

• Cost Function ($C(x)$): Total cost to produce $x$ units.

• Revenue Function ($R(x)$): Total revenue from selling $x$ units.

• Profit Function ($P(x)$): $P(x) = R(x) - C(x)$

• Marginal Functions: The derivative of each function (e.g., $C'(x)$) gives the marginal cost, revenue, or profit.

• Example: If $C(x) = 40x^{0.9}$, then $C'(x) = 36x^{-0.1}$ is the marginal cost.

Table: Key Business Calculus Concepts

Concept	Formula	Business Application
Marginal Cost	$C'(x)$	Cost to produce one more unit
Marginal Revenue	$R'(x)$	Revenue from selling one more unit
Marginal Profit	$P'(x)$	Profit from selling one more unit
Consumer Surplus	$\int_0^q D(x)dx - qp$	Extra value consumers receive
Producer Surplus	$qp - \int_0^q S(x)dx$	Extra value producers receive

Graphical Analysis

Graphs are used to visualize functions, identify intervals of increase/decrease, and locate maxima/minima. Interpreting graphs is essential for answering questions about function behavior and optimization.

• Intervals of Increase/Decrease: Where $f'(x) > 0$, the function increases; where $f'(x) < 0$, it decreases.

• Concavity: Determined by the sign of $f''(x)$. If $f''(x) > 0$, the graph is concave up; if $f''(x) < 0$, concave down.

• Inflection Points: Points where $f''(x) = 0$ and concavity changes.

Solving Business Word Problems

Many calculus problems involve translating real-world business scenarios into mathematical models, then using calculus to find optimal solutions.

• Steps:

  1. Define variables and write functions for cost, revenue, or profit.

  2. Find derivatives to determine marginal values or optimize functions.

  3. Solve equations to find critical points and interpret results in context.

• Example: A company wants to maximize profit by choosing the optimal number of units to produce. Set $P'(x) = 0$ and solve for $x$.

Additional info:

• Some questions involve using the method of Lagrange Multipliers for constrained optimization, which is an advanced topic sometimes included in business calculus.

• Questions also cover interpreting and constructing profit, cost, and revenue functions from word problems, and using calculus to analyze them.