Q: If the cost per kilogram is $$2.5$$ and the fixed weekly cost is $$50$$, what is the total cost for producing $$x = 10 kg$$?

$$CC(10) = 50 + 2.5 \times 10 = 75$$

Q: The income function is $$I(x) = 3x + 2 \ln(x). $$What is the income for $$x = 5 kg$$?

$$I(5) = 3 \times 5 + 2 \ln(5) \approx 15 + 3.218 = 18.218$$

Q: If the weekly profit function is $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x$$, what is the profit for $$x = 10 kg$$?

$$P(10) = 3 \times 10 + 2 \ln(10) - 50 - 2.5 \times 10 \approx 30 + 4.605 - 50 - 25 = -40.395$$

Question 1

Given the weekly cost function $$CC(x) = 50 + 2.5x$$ and the weekly income function $$I(x) = 3x + 2 \ln(x)$$, what is the equation for the weekly profit?

Accepted Answer

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

Question 2

If the weekly profit function is $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x$$, what value of $$x$$ maximizes the profit?

Accepted Answer

$$x = 2$$

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

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

Question 4

If the cost per kilogram is $$2.5$$ and the fixed weekly cost is $$50$$, what is the total cost for producing $$x = 10 kg$$?

Accepted Answer

$$CC(10) = 50 + 2.5 	imes 10 = 75$$

Question 5

The income function is $$I(x) = 3x + 2 \ln(x). $$What is the income for $$x = 5 kg$$?

Accepted Answer

$$I(5) = 3 	imes 5 + 2 \ln(5) \approx 15 + 3.218 = 18.218$$

Question 6

To maximize profit, the derivative of the profit function must be zero. For $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x$$, what equation must be solved?

Accepted Answer

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

Question 7

If the Newton-Raphson method is used to solve $$f(x) = 0$$, what is the iterative formula?

Accepted Answer

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

Question 8

If you start with $$x_0 = 50$ and apply the Newton-Raphson method to f(x) = 3$$ + \frac{2}{x} - 2.5$$, what is f$$($$x_0$$)$$?

Accepted Answer

$$f(50) = 3 + \frac{2}{50} - 2.5 = 3 + 0.04 - 2.5 = 0.54$$

Question 9

What is the derivative of $$f(x) = 3 + \frac{2}{x} - 2.5$$ with respect to $$x$$?

Accepted Answer

$$f'(x) = -\frac{2}{x^2$}$$

Question 10

If the Newton-Raphson method is applied with $$x_0 = 50$, f$$($$x_0) = 0.54$, and f$$'($$x_0) = -0.0008$, what is x_1$?

Accepted Answer

$$x_1 = 50$$ - \frac{0.54}{-0.0008} = 50 + 675 = 725$$

Question 11

In a real-world scenario, if the cost per kilogram increases to $$3$$ and the fixed cost remains $$50$$, what is the new cost function?

Accepted Answer

$$CC(x) = 50 + 3x$$

Question 12

If the weekly income is $$I(x) = 3x + 2 \ln(x)$$ and the cost is $$CC(x) = 50 + 2.5x$$, what is the break-even point?

Accepted Answer

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

Question 13

If the weekly profit is negative for $$x \u003c 20$$, what does this imply about the production?

Accepted Answer

Producing less than 20 kg results in a loss.

Question 14

If the cost function is linear and the income function is logarithmic plus linear, what type of optimization problem is this?

Accepted Answer

Nonlinear optimization

Question 15

If the Newton-Raphson method converges to $$x = 82.35$$, what does this value represent in the context of the problem?

Accepted Answer

The optimal kilograms to produce weekly for maximum profit.

Question 16

If the weekly profit function is $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x$$, what is the profit for $$x = 10 kg$$?

Accepted Answer

$$P(10) = 3 	imes 10 + 2 \ln(10) - 50 - 2.5 	imes 10 \approx 30 + 4.605 - 50 - 25 = -40.395$$

Método de la Secante: Optimización de Ingresos y Costos

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