Business Calculus: Limits, Derivatives, and Integrals – Course Outline and Key Concepts

Limits

Evaluation of Limits

Limits are foundational in calculus, describing the behavior of functions as inputs approach a specific value. Evaluating limits often involves recognizing indeterminate forms and applying algebraic simplification or L'Hôpital's Rule.

• Indeterminate Forms: Expressions like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ that require special techniques to evaluate.

• L'Hôpital's Rule: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ yields an indeterminate form, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ (if the latter limit exists).

Asymptotes

• Vertical Asymptotes: Occur where a function grows without bound as $x$ approaches a certain value.

• Horizontal Asymptotes: Describe the end behavior of a function as $x$ approaches $\infty$ or $-\infty$.

Continuity and Instantaneous Rates

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

• Instantaneous Rate of Change: Defined as the derivative at a point, representing the slope of the tangent line.

Derivatives

Basic Rules and Techniques

• Power Rule: For $f(x) = x^n$, $f'(x) = n x^{n-1}$.

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

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

Derivatives of Exponential and Logarithmic Functions

• $\frac{d}{dx} e^x = e^x$

• $\frac{d}{dx} a^x = a^x \ln a$

• $\frac{d}{dx} \ln x = \frac{1}{x}$

Chain Rule

• For composite functions: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

Implicit Differentiation

• Used when functions are not solved explicitly for $y$ in terms of $x$.

Tangent Lines

• The derivative at a point gives the slope of the tangent line to the curve at that point.

Marginal Analysis

• Marginal Cost, Profit, Revenue: The derivative of cost, profit, or revenue functions with respect to quantity, representing the rate of change per unit.

Elasticity of Demand

• Measures the responsiveness of quantity demanded to changes in price.

• Elasticity formula: $E(p) = \left| \frac{p}{q} \cdot \frac{dq}{dp} \right|$

Graphing and Optimization

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

• Intervals of Increase/Decrease: Determined by the sign of the first derivative.

• Concavity and Inflection Points: Determined by the second derivative; inflection points occur where concavity changes.

• Optimization: Finding maximum or minimum values of functions, often subject to constraints.

• Related Rates: Problems involving rates at which related variables change over time.

Integrals

Antiderivatives and General Antiderivatives

• Finding a function whose derivative is the given function.

• For $f(x) = x^n$ (with $n \neq -1$): $\int x^n dx = \frac{x^{n+1}}{n+1} + C$

• For $f(x) = e^x$: $\int e^x dx = e^x + C$

Method of Substitution (u-substitution)

• Used to simplify integrals by substituting part of the integrand with a new variable $u$.

Exponential Growth/Decay Differential Equations

• General solution: $y = Ce^{kt}$, where $k$ is the growth/decay rate.

Definite Integrals and Properties

• Represents the net area under a curve from $a$ to $b$.

• $\int_a^b f(x) dx$

Fundamental Theorem of Calculus

• Connects differentiation and integration: If $F(x)$ is an antiderivative of $f(x)$, then $\int_a^b f(x) dx = F(b) - F(a)$.

• Represents the net amount of change over an interval.