Functions and Mathematical Models in Business Calculus

Business Decisions and Mathematical Modeling

• Role of Mathematics: Businesses use mathematical functions to model relationships between key variables, predict outcomes, and optimize operations.

• Core Relationships: Includes demand vs. price, cost vs. production volume, and revenue vs. sales.

• Market Equilibrium and Consumer Surplus: Equilibrium occurs where demand equals supply ($p_e, x_e$). Consumer surplus is the net benefit to consumers willing to pay more than the equilibrium price.

Demand and Revenue Optimization

• Linear Demand Function: Derived from empirical price-quantity data ($p, x$), e.g., $D(p) = 115 - 2.5p$

• Revenue Function: Calculated as $R(p) = p \times D(p) = 115p - 2.5p^2$

• Revenue Optimization: Maximized at the parabolic vertex $(23, 1322.5)$. Charging a price of $p = 23$ yields maximum revenue of $1322.50$

Functions and Graphical Analysis

• Definition: The graph of a function $f$ consists of ordered pairs $(x, f(x))$. A point $(a, b)$ lies on the graph if and only if $b = f(a)$

• Evaluating Graphs: Function values are found by identifying $(x, y)$ coordinates directly on parabolic, cubic, or other polynomial curves.

Linear Functions and Line Equations

• Properties: Represented by straight lines and completely defined by two distinct points.

• Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Forms of Line Equations:

  • Point-Slope Form: $y - y_1 = m(x - x_1)$

  • Slope-Intercept Form: $y = mx + b$

Tangent Lines and Rates of Change

• Concept: A tangent line touches a function curve at a specific point and shares the curve's instantaneous rate of change (slope $m$) at that point.

• Procedure: Find point $(x_1, f(x_1))$, substitute point and slope $m$ into point-slope form, and simplify to slope-intercept form.

Fundamental Business Functions

• Cost Function ($C(x)$): $C(x) = FC + VC(x)$, where $FC$ is fixed costs and $VC(x)$ is variable costs scaling with quantity $x$

• Revenue Function ($R(x)$): $R(x) = p \times x$ or $R(x) = p(x) \times x$

• Profit Function ($P(x)$): $P(x) = R(x) - C(x)$

• Case Study (Duralex Companies):

  • Annual fixed costs: $FC = (300 + 30) \times 12 = 3960$

  • Variable costs: $VC(x) = (5 + 2)x = 7x$

  • Total cost function: $C(x) = 3960 + 7x$

  • Manufacturing $1200$ toys: $C(1200) = 3960 + 7(1200) = 12360$

  • Manufacturing $1500$ toys: $C(1500) = 3960 + 7(1500) = 14460$