Problem 35 - Calculus an Applied Approach

Question

A firm's weekly profit in marketing two products is given by \(P=192 x_{1}+576 x_{2}-x_{1}^{2}-5 x_{2}^{2}-2 x_{1} x_{2}-5000\) where \(x_{1}\) and \(x_{2}\) represent the numbers of units of each product sold weekly. Estimate the average weekly profit if \(x_{1}\) varies between 40 and 50 units and \(x_{2}\) varies between 45 and 50 units.

Step-by-Step Solution

Answer

The average weekly profit is the result of the calculation from Step 2.

Step 1: Calculate Profit for every single unit

Use the given profit function to calculate the profit obtained for each possible number of units sold for the two products. This would involve substituting \(x_1 = 40, 41, ..., 50\) and \(x_2 = 45, 46, ..., 50\) into the equation \(P = 192x_{1} + 576x_{2} - x_{1}^{2} - 5x_{2}^{2} -2x_{1}x_{2} - 5000\). Then calculate the values of \(P\) for each combination of \(x_1\) and \(x_2\).

Step 2: Calculate the Average Profit

Now, an average of these profits needs to be calculated to estimate the average weekly profit. Add up all the calculated profit values and divide it by the total number of units considered (\(11 \times 6 = 66\)), since \(x_1\) varies between 40 and 50 (that's 11 values) and \(x_2\) varies between 45 and 50 (6 values).

Key Concepts

Profit Optimization

Understanding how to maximize profit is a cornerstone of any business strategy. Profit optimization involves determining the combination of product quantities that generates the highest profit. In a mathematical sense, this is achieved by finding the maximum value of a profit function, which is typically dependent on several variables representing the quantities of products sold.

In our example, a firm's profit from selling two products is described by a specific function with terms representing revenue, costs, and the interaction between the two products. The critical point for profit maximization occurs where there is neither an increase nor decrease in profit with slight variations in product quantities. These stationary points are typically found using techniques from calculus, such as setting the partial derivatives to zero. The goal is to find the 'sweet spot' where profit peaks considering all influencing factors.

Partial Derivatives

In multivariable calculus, partial derivatives play a decisive role in understanding the behavior of functions with more than one variable. They measure how a function changes as each variable is slightly varied while all other variables are held constant.

To calculate a partial derivative, you treat all other variables as constants and differentiate with respect to the variable of interest. For instance, the partial derivatives of the profit function with respect to both products, represented as \(x_1\) and \(x_2\), reveal how slight changes in the number of units sold of each product influence the profit. The calculation and interpretation of these partial derivatives are critical in determining the direction of change for optimising profit.

Multivariable Calculus

This branch of mathematics is essential for analyzing and solving problems that involve more than one variable. In the context of profit optimization, multivariable calculus enables us to understand how different levels of product sales (variables) affect the total profit of a company.

Calculus tools, such as partial derivatives, the gradient vector, and critical points analysis, are vital for identifying profit-maximizing strategies. Techniques like the second derivative test or the Hessian matrix can further help determine whether a point is a local maximum, minimum, or a saddle point, thus helping businesses make informed decisions about their product sales strategy.

Economic Functions

An economic function is a mathematical representation of the relationships between various economic variables. In our example, the profit function \(P\) represents the relationship between the quantities of two products sold and the resulting profit. These functions not only aid in profit prediction but provide insights into cost structure, revenue potential, and how different components of the business interact with each other.

By analyzing economic functions, we get a clearer view of how altering one part of the system can lead to changes in the overall profitability. This makes these functions incredibly useful for strategic planning and financial analysis within the context of economics and business.