Optimization Using Calculus: Secant Method for Maximizing Profit

Optimization and the Secant Method in Calculus

Problem Context: Maximizing Weekly Profit

This section explores how calculus can be used to optimize profit in a business scenario. The weekly cost and income functions are given, and the goal is to determine the optimal quantity of product to maximize profit using numerical methods.

• Cost Function: The weekly cost for producing and selling an article is given by $CC(x) = 50 + 2.5x$, where $x$ is the number of kilograms produced.

• Income Function: The weekly income from sales is $I(x) = 3x + 2 \ln(x)$.

• Profit Function: The profit is the difference between income and cost: $P(x) = I(x) - CC(x) = 3x + 2 \ln(x) - (50 + 2.5x)$.

• Optimization Goal: Find the value of $x$ that maximizes the profit $P(x)$.

Formulating the Optimization Problem

To maximize profit, set the derivative of the profit function to zero and solve for $x$:

• Profit Function: $P(x) = 3x + 2 \ln(x) - 50 - 2.5x$

• Derivative: $P'(x) = 3 + \frac{2}{x} - 2.5$

• Critical Point Equation: $P'(x) = 0 \Rightarrow 3 + \frac{2}{x} - 2.5 = 0$

Numerical Solution: Secant Method

The secant method is a numerical technique used to find roots of equations, especially when analytical solutions are difficult. Here, it is applied to solve $P'(x) = 0$.

• Secant Method Formula: $x_{k+1} = x_k - \frac{\delta x_k \cdot f(x_k)}{f(x_k + \delta x_k) - f(x_k)}$ where $f(x)$ is the function whose root is sought.

• Iteration Table: The notes show iterations with initial guesses and updated values for $x$ and $f(x)$.

Example: Secant Method Iterations

k	x_k	f(x_k)	x_{k+1}
0	50	-17.17	81.81
1	81.81	0.28	82.35
2	82.35	0.00023	82.35

These iterations show convergence to the optimal value of $x \approx 82.35$ kg, which maximizes weekly profit.

Applications and Importance

• Business Optimization: Calculus and numerical methods like the secant method are essential for maximizing profit and minimizing costs in real-world scenarios.

• Numerical Methods: When equations cannot be solved analytically, iterative methods provide practical solutions.

Additional info: The secant method is similar to the Newton-Raphson method but does not require the calculation of derivatives, making it useful when derivatives are difficult to compute.