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 as a function of $$x$$?

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

Question 3

To maximize the weekly profit, you set the derivative of the profit function to zero. What is the derivative of $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x$$?

Accepted Answer

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

Question 4

If $$P'(x) = 3 + \frac{2}{x} - 2.5$$, what equation must be solved to find the optimal $$x$$?

Accepted Answer

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

Question 5

Solve $$3 + \frac{2}{x} - 2.5 = 0$$ for $$x. $$What is the value of $$x$$?

Accepted Answer

$$x = 4$$

Question 6

If the weekly cost function is $$CC(x) = 50 + 2.5x$$, what is the marginal cost?

Accepted Answer

$$2.5$$

Question 7

If the weekly income function is $$I(x) = 3x + 2 \ln(x)$$, what is the marginal income?

Accepted Answer

$$3 + \frac{2}{x}$$

Question 8

Suppose you want to use the secant method to solve $$f(x) = 0$$ for $$f(x) = 3 + \frac{2}{x} - 2.5. $$What is the formula for the next approximation $$x_{k+1}$$?

Accepted Answer

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

Question 9

If you start the secant method with $$x_0 = 50$ and x_1$ = 81.81$$, what is the value of $$f(x_0$)$$ for $$f(x) = 3 + \frac{2}{x} - 2.5$$?

Accepted Answer

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

Question 10

In the context of maximizing profit, why is it important to set the derivative of the profit function to zero?

Accepted Answer

Because it identifies the critical points where profit could be maximized or minimized.

Question 11

If the secant method produces $$x_2 = 82.35$ after two iterations, what does this value represent in the context of the problem?

Accepted Answer

The estimated optimal number of kilograms to produce weekly to maximize profit.

Question 12

If the weekly cost function is $$CC(x) = 50 + 2.5x$$, what is the fixed cost?

Accepted Answer

$$50$$

Question 13

If the weekly income function is $$I(x) = 3x + 2 \ln(x)$$, what is the income when $$x = 10$$?

Accepted Answer

$$I(10) = 3 \cdot 10 + 2 \ln(10) \approx 30 + 4.605 = 34.605$$

Question 14

If the weekly cost function is $$CC(x) = 50 + 2.5x$$, what is the cost for producing 20 kg?

Accepted Answer

$$CC(20) = 50 + 2.5 \cdot 20 = 100$$

Question 15

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

Accepted Answer

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

Question 16

If the secant method is used to solve $$f(x) = 0$$ and the values $$x_0 = 50$, x_1$ = 81.81$$, $$x_2 = 82.35$ are obtained, what does this sequence indicate?

Accepted Answer

The method is converging to the optimal solution for $$x.$$

Question 17

If the weekly profit function is $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x$$, what is the second derivative $$P''(x)$$?

Accepted Answer

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

Question 18

Why is the second derivative $$P''(x)$$ important in determining whether the critical point is a maximum or minimum?

Accepted Answer

If $$P''(x) \u003c 0$$, the critical point is a maximum.

Método de la Secante: Optimización de Ingresos y Costos Semanales

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