Business Calculus Study Guide: Limits, Continuity, Rates of Change, and Derivatives

Limits and Continuity

Introduction to Limits

Limits are a foundational concept in calculus, describing the behavior of a function as its input approaches a particular value. Understanding limits is essential for analyzing functions, determining continuity, and calculating derivatives.

• Limit of a Function: The value that a function approaches as the input approaches a specific point.

• One-Sided Limits: Limits taken from the left ($x \to a^-$) or right ($x \to a^+$) of a point.

• Two-Sided Limit: The limit as $x$ approaches a point from both sides.

Example: For $g(x) = \begin{cases} x+2 & \text{for } x<2 \\ \frac{1}{2}x+1 & \text{for } x \geq 2 \end{cases}$, find:

• $\lim_{x \to 2^-} g(x)$: Approach $x=2$ from the left, use $x+2$.

• $\lim_{x \to 2^+} g(x)$: Approach $x=2$ from the right, use $\frac{1}{2}x+1$.

• $\lim_{x \to 2} g(x)$: Exists only if both one-sided limits are equal.

Graphical and Numerical Evaluation of Limits

Limits can be evaluated using graphs and tables. Look for the value the function approaches, not necessarily the value at the point.

• Graphical Approach: Observe the behavior of the function near the point of interest.

• Numerical Approach: Use values of the function close to the point.

Continuity

A function is continuous at a point $x=a$ if:

• $f(a)$ is defined

• $\lim_{x \to a} f(x)$ exists

• $\lim_{x \to a} f(x) = f(a)$

Discontinuities can be identified graphically by jumps, holes, or vertical asymptotes.

Rates of Change

Average Rate of Change

The average rate of change of a function $f(x)$ over $[x_1, x_2]$ is the change in output divided by the change in input.

• Formula:

$\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

• Secant Slope: The slope of the line connecting two points on the graph.

Example: If $f(x)$ represents position, the average rate of change is average velocity.

Instantaneous Rate of Change

The instantaneous rate of change at $x=a$ is the derivative $f'(a)$, representing the slope of the tangent line at that point.

• Formula:

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

• Tangent Slope: The slope of the line that just touches the curve at one point.

Derivatives

Definition and Notation

The derivative of a function measures its instantaneous rate of change. Common notations include $f'(x)$ and $\frac{dy}{dx}$.

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

• Constant Multiple Rule: $\frac{d}{dx} [c f(x)] = c f'(x)$

Product, Quotient, and Chain Rules

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

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

• Chain Rule: For $y = f(g(x))$, $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$

Example: Differentiate $y = (5x+8)^3$ using the chain rule.

Applications of Derivatives

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

• Velocity and Acceleration: If $s(t)$ is position, $s'(t)$ is velocity, $s''(t)$ is acceleration.

• Business Applications: Derivatives are used to find marginal cost, marginal revenue, and optimize profit.

Worked Examples and Practice Problems

Limits

• Find $\lim_{x \to 4} (6x+1)$

• Find $\lim_{x \to 5} \frac{x^2-9}{5-x}$

• Find $\lim_{x \to 10} \frac{x^2-100}{x-10}$

• Find $\lim_{x \to 5} \frac{x^2+3x-40}{x^2-25}$

Continuity

• Determine if a function is continuous at a point by checking the three conditions above.

• Use graphs to identify discontinuities.

Rates of Change

• Find the average rate of change for $f(x)$ over an interval.

• Find the instantaneous rate of change using the definition of the derivative.

Derivatives

• Apply the power, product, quotient, and chain rules to differentiate functions.

• Find the equation of tangent lines to curves at given points.

Summary Table: Rules for Differentiation

Rule	Formula	Example
Power Rule	$\frac{d}{dx} x^n = n x^{n-1}$	$\frac{d}{dx} x^3 = 3x^2$
Product Rule	$\frac{d}{dx} [u v] = u' v + u v'$	$\frac{d}{dx} (x^2 \cdot \sin x)$
Quotient Rule	$\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u' v - u v'}{v^2}$	$\frac{d}{dx} \left( \frac{x^2}{x+1} \right)$
Chain Rule	$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$	$\frac{d}{dx} (\sin(x^2))$

Business Applications of Calculus

Marginal Analysis

In business, derivatives are used to compute marginal cost, marginal revenue, and marginal profit, which represent the instantaneous rate of change of these quantities with respect to production level.

• Marginal Cost: $C'(x)$, the derivative of the cost function.

• Marginal Revenue: $R'(x)$, the derivative of the revenue function.

• Marginal Profit: $P'(x)$, the derivative of the profit function.

Example: If $C(x) = 100 + 5x$, then $C'(x) = 5$.

Key Formulas

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

• Average Rate of Change: $\frac{f(x_2) - f(x_1)}{x_2 - x_1}$

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

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

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

Practice Problems

• Evaluate limits graphically and algebraically.

• Determine continuity at a point using the three-step test.

• Calculate average and instantaneous rates of change for given functions.

• Differentiate functions using the power, product, quotient, and chain rules.

• Find equations of tangent lines to curves at specified points.

Additional info: The study materials include graphical, numerical, and algebraic approaches to limits, continuity, rates of change, and derivatives, with applications to business contexts such as population growth and cost analysis.