Business Calculus Study Guide: Functions, Limits, Derivatives, Integration, and Applications

Ch. 1 – Functions and Graphs

Introduction to Functions and Their Graphs

Understanding functions and their graphical representations is foundational in business calculus. This section covers definitions, types, transformations, and properties of functions relevant to business applications.

• Function: A rule assigning each input x to exactly one output y.

• Piecewise functions: Functions defined differently on separate intervals. Example: $f(x) = \begin{cases} x^2 & x < 0 \\ x+2 & x \geq 0 \end{cases}$

• Transformations:

  • Vertical shift: $f(x)+c$

  • Horizontal shift: $f(x-c)$

  • Reflection: $-f(x)$ or $f(-x)$

  • Stretch/Shrink: $af(x)$

• Exponent rules: $a^m a^n = a^{m+n}$, $a^m/a^n = a^{m-n}$

• Log properties: $\log(ab) = \log a + \log b$, $\log(a/b) = \log a - \log b$, $\log(a^n) = n \log a$

• Exponential/Log Equations: Rewrite bases or use logs. Example: Solve $3x = 20 \Rightarrow x = \log_{3}{20}$

Ch. 2 – Limits & Derivatives

Understanding Limits and the Concept of Derivatives

Limits and derivatives are essential for analyzing change and optimizing business functions. This section introduces their definitions, properties, and applications.

• Limit: $\lim_{x \to a} f(x) = L$ means the function approaches $L$ as $x$ approaches $a$.

• Continuity: A function is continuous at $a$ if $\lim_{x \to a} f(x) = f(a)$.

• Derivative as slope: $f'(x)$ represents the slope or tangent line at $x$.

• Derivative formula: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

• Example for $f(x) = x^2$: $f'(x) = 2x$; at $x=3$, slope = 6.

• Linearization: $L(x) = f(a) + f'(a)(x-a)$ gives a local linear approximation.

Ch. 3 – Derivative Rules

Rules for Differentiating Functions

Derivative rules simplify the process of finding rates of change for various types of functions, which is crucial in business modeling and analysis.

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

• Product Rule: $[fg]' = f'g + fg'$

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

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

• Implicit differentiation: Treat $y$ as a function of $x$ when differentiating equations involving both variables. Example: $2x^2 + y^2 = 2$ yields $4x + 2y \frac{dy}{dx} = 0$

• Related rates: Differentiate with respect to $t$ to find how rates change over time.

Ch. 4 – Graphing & Optimization

Analyzing Functions for Maximum and Minimum Values

Graphing and optimization techniques help identify critical points and optimal solutions in business scenarios.

• Critical points: Where $f'(x) = 0$ or undefined.

• First derivative test: The sign change of $f'(x)$ indicates local maxima or minima.

• Second derivative test: If $f''(c) > 0$, local minimum; if $f''(c) < 0$, local maximum.

• Concavity:

  • If $f''(x) > 0$, the graph is concave up.

  • If $f''(x) < 0$, the graph is concave down.

• Optimization problems: Express the target quantity in one variable, take the derivative, set equal to zero, and check endpoints.

Ch. 5 – Integration

Techniques and Applications of Integration

Integration is used to find areas, accumulated quantities, and solve business problems involving totals and averages.

• Antiderivative: The reverse process of differentiation.

• Basic rule: $\int x^n dx = \frac{1}{n+1} x^{n+1}$ (for $n \neq -1$)

• Substitution: Useful for integrating composite functions. $u$-substitution: $\int f(g(x))g'(x)dx = \int f(u)du$

• Riemann sums: Approximate area under a curve by summing rectangles.

• Definite integral: $\int_a^b f(x)dx$ gives net area between $x=a$ and $x=b$.

• FTC (Fundamental Theorem of Calculus): $\frac{d}{dx} \int_a^x f(t)dt = f(x)$

• Average value: $\text{Ave} = \frac{1}{b-a} \int_a^b f(x)dx$

Ch. 6 – More Integration

Advanced Integration Techniques and Applications

This section covers additional integration methods and their use in business contexts, such as finding areas and handling improper integrals.

• Area between curves: $\int_a^b [f(x) - g(x)]dx$; top minus bottom.

• Integration by parts: $\int u dv = uv - \int v du$

• Improper integrals: Used when limits are infinite or there is a discontinuity.

Ch. 7 – Trigonometric Functions and Calculus

Trigonometric Definitions, Identities, and Calculus

Trigonometric functions and their calculus are occasionally used in business calculus, especially in modeling periodic phenomena.

• Right triangle definitions: $\sin \theta = \frac{\text{opp}}{\text{hyp}}$, $\cos \theta = \frac{\text{adj}}{\text{hyp}}$, $\tan \theta = \frac{\text{opp}}{\text{adj}}$

• Unit circle: Coordinates are $(\cos \theta, \sin \theta)$

• Key identities: $\sin^2 \theta + \cos^2 \theta = 1$, $\tan \theta = \frac{\sin \theta}{\cos \theta}$

• Derivatives: $\frac{d}{dx}(\sin x) = \cos x$, $\frac{d}{dx}(\cos x) = -\sin x$, $\frac{d}{dx}(\tan x) = \sec^2 x$

• Integrals: $\int \sin x dx = -\cos x + C$, $\int \cos x dx = \sin x + C$