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 number of kilograms sold, what is the equation for the profit function $$P(x)$$?

Accepted Answer

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

Question 2

To maximize profit, you set the derivative of the profit function to zero. What equation must you solve to find the optimal $$x$$?

Accepted Answer

$$0 = 3 + \frac{2}{x} - 2.5$$

Question 3

What is the economic interpretation of the value of $$x$$ that solves $$0 = 3 + \frac{2}{x} - 2.5$$?

Accepted Answer

It is the number of kilograms that maximizes weekly profit.

Question 4

If you use the Newton-Raphson method to solve $$f(x) = 3 + \frac{2}{x} - 2.5 = 0$$, what is the iterative formula?

Accepted Answer

$$x_{k+1} = x_k - \frac{3 + \frac{2}{x_k} - 2.5}{-\frac{2}{x_k^2$}}$$

Question 5

Suppose your initial guess for $$x is$$ $$50. $$After one iteration of the Newton-Raphson method, what is the next value of $$x ($$rounded to two decimals)?

Accepted Answer

$$81.81$$

Question 6

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

Accepted Answer

It quickly finds the root of a nonlinear equation for optimal production.

Question 7

If the cost per kilogram increases, how does the optimal value of $$x$$ change?

Accepted Answer

The optimal $$x$$ decreases.

Question 8

If the selling price per kilogram increases, what happens to the profit-maximizing quantity?

Accepted Answer

The profit-maximizing quantity increases.

Question 9

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 product is sold.

Question 10

If the fixed cost increases from $$50 to$$ $$100$$, what is the effect on the optimal $$x$$?

Accepted Answer

The optimal $$x$$ does not change.

Question 11

Why is it important to check the second derivative when finding the maximum profit?

Accepted Answer

To confirm that the critical point is a maximum, not a minimum.

Question 12

If $$x = 82.35 is $$found as the optimal solution, what does this value represent in the context of the problem?

Accepted Answer

The number of kilograms to sell weekly to maximize profit.

Método de la Secante: Optimización de Costos Semanales

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