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 specific values. Understanding limits is essential for defining derivatives and integrals.

• Indeterminate Forms: When direct substitution in a limit yields forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$, algebraic simplification or L'Hôpital's Rule is used to evaluate the limit.

• 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).

• Vertical and Horizontal Asymptotes: Vertical asymptotes occur where a function grows without bound as $x$ approaches a certain value. Horizontal asymptotes describe the behavior as $x$ approaches $\infty$ or $-\infty$.

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

Definition of Instantaneous Rate of Change

• The derivative at a point $x = a$ is defined as $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$.

Derivatives

Rules and Techniques

Derivatives measure the rate at which a function changes. Several rules simplify the process of differentiation.

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

• Product Rule: For $f(x) = u(x)v(x)$, $f'(x) = u'(x)v(x) + u(x)v'(x)$.

• Quotient Rule: For $f(x) = \frac{u(x)}{v(x)}$, $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$.

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

Differentiation of $e^x$, $a^x$

• $\frac{d}{dx} e^x = e^x$

• $\frac{d}{dx} a^x = a^x \ln a$

Implicit Differentiation

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

Tangent Line to a Curve at a Point

• The equation of the tangent line at $x = a$ is $y = f(a) + f'(a)(x - a)$.

Marginal Analysis

• Marginal cost, profit, and revenue are approximated by the derivative of the respective function.

• For example, marginal cost $C'(x)$ estimates the cost of producing one more unit.

Elasticity of Demand

• Elasticity measures the responsiveness of demand to changes in price: $E(p) = -\frac{p}{q} \frac{dq}{dp}$.

Graphing and Optimization

Critical Numbers

• Values of $x$ where $f'(x) = 0$ or $f'(x)$ does not exist.

Intervals of Increase/Decrease

• Where $f'(x) > 0$, the function is increasing; where $f'(x) < 0$, it is decreasing.

Concavity and Inflection Points

• Where $f''(x) > 0$, the function is concave up; where $f''(x) < 0$, concave down.

• 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 with respect to time.

Integrals

Antiderivatives and General Antiderivatives

• The antiderivative of $f(x)$ is a function $F(x)$ such that $F'(x) = f(x)$.

• For $n \neq -1$, $\int x^n dx = \frac{x^{n+1}}{n+1} + C$.

• $\int \frac{1}{x} dx = \ln|x| + C$

• $\int e^x dx = e^x + C$

Method of Substitution (u-substitution)

• Used to simplify integrals by substituting $u = g(x)$, then $du = g'(x)dx$.

Exponential Growth/Decay Differential Equations

• Solutions to $\frac{dy}{dt} = ky$ are $y(t) = y_0 e^{kt}$, modeling growth or decay.

Definite Integrals and Properties

• The definite integral $\int_a^b f(x) dx$ represents the net area under $f(x)$ from $x = a$ to $x = b$.

Fundamental Theorem of Calculus

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