Business Calculus Study Guide: Applications of Differentiation and Integration

Applications of Differentiation

Equilibrium Price and Quantity

In business calculus, equilibrium refers to the point where the quantity demanded equals the quantity supplied. This is crucial for determining market price and quantity.

• Demand Function (D(x)): Represents the price consumers are willing to pay for x units.

• Supply Function (S(x)): Represents the price producers are willing to accept for x units.

• Equilibrium Point: Found by solving D(x) = S(x).

• Consumer Surplus: The difference between what consumers are willing to pay and what they actually pay at equilibrium.

• Producer Surplus: The difference between the equilibrium price and the minimum price producers are willing to accept.

Example: Given D(x) = 9 - x^2 and S(x) = x^2 + 2x + 21, set D(x) = S(x) to find equilibrium.

$9 - x^2 = x^2 + 2x + 21$

Solve for x to find the equilibrium quantity, then substitute back to find the equilibrium price.

Inventory Cost Optimization

Businesses often seek to minimize inventory costs by determining optimal order sizes and timing.

• Fixed Costs: Costs that do not change with the number of units ordered (e.g., storage costs).

• Variable Costs: Costs that depend on the number of units ordered (e.g., shipping costs per unit).

• Optimization: Use calculus to minimize total cost by finding the derivative of the cost function and setting it to zero.

Example: If storage costs are $20 per table per year and shipping is $16 per table, with a fixed order cost of $40, set up a cost function and minimize it with respect to the number of shipments per year.

Differentiation and Integration

Definite and Indefinite Integrals

Integration is used to find areas under curves, total accumulated quantities, and average values.

• Definite Integral: Computes the net area under a curve between two points.

• Indefinite Integral: Represents a family of functions (antiderivatives).

• Substitution Method: Used to simplify integrals by changing variables.

Examples:

• Find $\int_{4}^{8} \sqrt{\frac{8}{x}} dx$

• Evaluate $\int_{0}^{2} \frac{2x}{x^2 - 6x} dx$

• Use substitution for $\int \frac{5}{x} dx$

Average Value of a Function

The average value of a function f(x) over [a, b] is given by:

$\text{Average value} = \frac{1}{b-a} \int_{a}^{b} f(x) dx$

Example: Find the average value of f(x) = x^3 + 2x on [2, 5].

Area Between Curves

To find the area between two curves, subtract the lower function from the upper function and integrate over the interval.

$\text{Area} = \int_{a}^{b} [f(x) - g(x)] dx$

Example: Find the area bounded by $y = x^2 - 5x + 9$ and $y = x^2 - 7x + 11$.

Applications of Integration

Business Applications: Sales and Growth

Integration can be used to calculate total sales over time when given a rate of change function.

• Sales Rate Function S'(t): Represents the rate at which sales are increasing.

• Total Sales: Integrate S'(t) over the desired interval.

Example: If $S'(t) = 14e^t$, find total sales from day 3 to day 6:

$\int_{3}^{6} 14e^t dt$

Employment Growth and Accumulation

Graphs of functions and their derivatives can be used to analyze changes in employment over time.

• Function E(t): Number of employees at time t.

• Derivative E'(t): Rate of change of employees per year.

• Integral of E'(t): Gives the net change in employees over a time interval.

Example: If E'(t) is given graphically, the area under the curve between two points gives the change in employees.

Table: Employment Data from Graph

Year since 1981	Number of Employees
0	206
?	449
?	862

Additional info: The table summarizes key points from the graph, showing employee numbers at specific years.

Key Formulas and Techniques

• Equilibrium: $D(x) = S(x)$

• Consumer Surplus: $\int_{0}^{x^*} [D(x) - p^*] dx$

• Producer Surplus: $\int_{0}^{x^*} [p^* - S(x)] dx$

• Definite Integral: $\int_{a}^{b} f(x) dx$

• Average Value: $\frac{1}{b-a} \int_{a}^{b} f(x) dx$

• Area Between Curves: $\int_{a}^{b} [f(x) - g(x)] dx$