Q: If $$x = 82.35$$ maximizes profit, what is the maximum weekly profit (rounded to the nearest dollar)?

$$P(82.35) \approx 3 \times 82.35 + 2 \ln(82.35) - (50 + 2.5 \times 82.35) \approx 247.05 + 8.82 - 255.88 \approx 0$$

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 weekly profit $$P(x)$$?

Accepted Answer

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

Question 2

To maximize the weekly profit, which equation must be solved for $$x$$?

Accepted Answer

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

Question 3

If the Newton-Raphson method is used to solve $$f(x) = 3 + \frac{2}{x} - 2.5 = 0$$, what is the iterative formula for $$x_{k+1}$$?

Accepted Answer

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

Question 4

What is the economic interpretation of the solution to $$3 + \frac{2}{x} - 2.5 = 0 in $$this context?

Accepted Answer

It is the number of kilograms that maximizes weekly profit.

Question 5

If the initial guess for $$x is$$ $$50$$, what is the value of $$f(50)$$ for $$f(x) = 3 + \frac{2}{x} - 2.5$$?

Accepted Answer

$$0.54$$

Question 6

Suppose after two iterations of the Newton-Raphson method, the value of $$x$$ converges to approximately $$82.35. $$What does this value represent?

Accepted Answer

The optimal number of kilograms to sell weekly for maximum profit.

Question 7

If the cost per kilogram increases from $$2.5 to$$ $$3$$, how does the equation for maximizing profit change?

Accepted Answer

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

Question 8

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 9

If the fixed weekly cost increases from $$50 to$$ $$100$$, what is the new profit function?

Accepted Answer

$$P(x) = 3x + 2 \ln(x) - (100 + 2.5x)$$

Question 10

Which of the following is a correct application of the secant method for finding the root of $$f(x) = 3 + \frac{2}{x} - 2.5$$?

Accepted Answer

$$x_{k+1} = x_k - f(x_k) \frac{x_k - x_{k-1}}{f(x_k) - f(x_{k-1})}$$

Question 11

If $$x = 82.35$$ maximizes profit, what is the maximum weekly profit (rounded to the nearest dollar)?

Accepted Answer

$$P(82.35) \approx 3 	imes 82.35 + 2 \ln(82.35) - (50 + 2.5 	imes 82.35) \approx 247.05 + 8.82 - 255.88 \approx 0$$

Método de la Secante: Optimización de Funciones en Estadística

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