Business Calculus: Extrema, Optimization, and Applications Study Guide

Extrema of Functions

Absolute and Relative Extrema

In calculus, extrema refer to the maximum and minimum values of a function. These can be classified as either absolute (global) or relative (local) extrema.

• Absolute Maximum: The highest value a function attains on its domain.

• Absolute Minimum: The lowest value a function attains on its domain.

• Relative Maximum: A point where the function value is higher than all nearby points.

• Relative Minimum: A point where the function value is lower than all nearby points.

Example: For $f(x) = -x^2 + 4$, the vertex at $x = 0$ is an absolute maximum.

Finding Extrema Analytically

To find extrema, set the derivative equal to zero and solve for critical points. Then, use the second derivative or test intervals to classify each point.

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

• Second Derivative Test: If $f''(x) > 0$, the point is a minimum; if $f''(x) < 0$, it is a maximum.

Example: For $f(x) = -3x^4 + 2x^3 - 3x^2 + 8$, find $f'(x)$, set to zero, and solve for $x$.

Optimization Problems

Business Applications of Optimization

Optimization in business calculus involves finding the maximum or minimum values of functions representing profit, cost, revenue, or other quantities.

• Profit Function: $P(x)$ models profit as a function of units sold.

• Revenue Function: $R(x)$ models revenue as a function of units sold.

• Cost Function: $C(x)$ models cost as a function of units produced.

Example: Given $P(x) = -0.2x^2 + 60x - 100$, find the value of $x$ that maximizes profit by solving $P'(x) = 0$.

Graphical Analysis of Extrema

Interpreting Graphs for Maxima and Minima

Graphs can be used to visually identify points of maximum and minimum values. These points are often marked by peaks (maximum) or valleys (minimum).

• Locate the highest and lowest points on the graph within the given interval.

• Label each as absolute or relative extrema based on their position.

Example: On a sales graph, the year with the highest sales is an absolute maximum.

Applied Business Problems

Revenue, Cost, and Profit Relationships

Business calculus often requires setting up equations relating cost, revenue, and profit, then using calculus to optimize these quantities.

• Revenue: $R(x) = p(x) imes x$, where $p(x)$ is price per unit and $x$ is quantity sold.

• Profit: $P(x) = R(x) - C(x)$

Example: If $C(x) = \frac{280x}{x^2 + 2}$ and $R(x) = 570x$, find the rate of change of profit when $x$ changes by $51$ units.

Elasticity of Demand

Definition and Calculation

Elasticity of demand measures how the quantity demanded responds to changes in price. It is calculated as:

• Elasticity: $E(p) = -\frac{p}{q} \frac{dq}{dp}$

• If $|E(p)| > 1$, demand is elastic; if $|E(p)| < 1$, demand is inelastic.

Example: For $q = 4000 - 100p$, find $E(p)$ at $p = 10$.

Implicit Differentiation

Finding Derivatives for Implicit Functions

When a function is not explicitly solved for one variable, use implicit differentiation to find derivatives.

• Differentiating both sides of the equation with respect to $x$.

• Apply the chain rule for terms involving $y$.

Example: For $3x^2y + y^3 = 8$, differentiate both sides to find $\frac{dy}{dx}$.

Tangent Lines to Curves

Equation of the Tangent Line

The tangent line to a curve at a point $(x_0, y_0)$ has the equation:

• $y - y_0 = f'(x_0)(x - x_0)$

• Find $f'(x)$, evaluate at $x_0$, and substitute into the equation.

Example: For $y = x^2 + 3x$, find the tangent at $x = 2$.

Related Rates

Solving Related Rates Problems

Related rates problems involve finding the rate at which one quantity changes with respect to another, often using implicit differentiation.

• Identify all variables and their rates of change.

• Differentiate the relationship between variables with respect to time.

• Solve for the desired rate.

Example: If the radius of a tank is increasing at $2$ inches/sec, find the rate at which the surface level is falling.

Tables: Sales Data Analysis

Purpose: Identifying Maxima and Minima in Sales Data

The table below summarizes sales data over several years, used to identify relative maxima and minima.

Year	Sales (thousands)
1990	171
1992	258
1994	211
1996	277
1998	258
2000	258

Additional info: Students may be asked to identify years with highest and lowest sales as relative maxima and minima.

Summary of Key Formulas

• Derivative: $f'(x)$

• Second Derivative: $f''(x)$

• Elasticity: $E(p) = -\frac{p}{q} \frac{dq}{dp}$

• Tangent Line: $y - y_0 = f'(x_0)(x - x_0)$

• Profit: $P(x) = R(x) - C(x)$