Question 1

Given the weekly cost function $$CC(x) = 50 + 2.5x$$ and the weekly income function $$I(x) = 3x + 2 \ln(x)$$, where $$x is $$the quantity of kilograms sold, what is the profit function $$P(x)$$?

Accepted Answer

$$P(x) = 3x + 2 \ln(x) - (50 + 2.5x)$$

Question 2

To maximize profit, which equation should you solve based on the given functions?

Accepted Answer

$$\frac{d}{dx} [3x + 2 \ln(x) - (50 + 2.5x)] = 0$$

Question 3

What is the derivative of the profit function $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x$$ with respect to $$x$$?

Accepted Answer

$$0.5 + \frac{2}{x}$$

Question 4

If you use the Newton-Raphson method to solve $$0.5 + \frac{2}{x} = 0$$, what is the iterative formula for $$x_{k+1}$$?

Accepted Answer

$$x_{k+1} = x_k - \frac{0.5 + \frac{2}{x_k}}{-\frac{2}{x_k^2$}}$$

Question 5

What is the economic interpretation of the solution to $$0.5 + \frac{2}{x} = 0 in $$this context?

Accepted Answer

It is the quantity of kilograms that maximizes weekly profit.

Question 6

If the initial guess for Newton-Raphson is $$x_0 = 50$, and after one iteration you get x_1$ = 81.81$$, what does this suggest about the function's behavior?

Accepted Answer

The function increases rapidly and the root is closer to $$x_1$$ than $$x_0.$$

Question 7

Which of the following best describes the role of the logarithmic term $$2 \ln(x) in $$the income function?

Accepted Answer

It models diminishing returns as more kilograms are sold.

Question 8

Suppose the cost per kilogram increases from $$2.5 to$$ $$3. $$How does this affect the profit-maximizing quantity?

Accepted Answer

The profit-maximizing quantity decreases.

Question 9

If the fixed cost increases from $$50 to$$ $$100$$, what is the effect on the profit-maximizing quantity?

Accepted Answer

There is no effect on the profit-maximizing quantity.

Question 10

Which of the following is a correct application of the Newton-Raphson method for finding the root of $$f(x) = 0.5 + \frac{2}{x}$$?

Accepted Answer

$$x_{k+1} = x_k - \frac{0.5 + \frac{2}{x_k}}{-\frac{2}{x_k^2$}}$$

Question 11

If after two iterations of Newton-Raphson you obtain $$x_2 = 82.35$, what should you do next to ensure convergence?

Accepted Answer

Continue iterating until $$|x_{k+1} - x_k| is $$sufficiently small.

Question 12

What is the main advantage of using the Newton-Raphson method in this scenario?

Accepted Answer

It provides a fast and efficient way to find the profit-maximizing quantity.

Método de la Secante: Optimización de Costos y Beneficios

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