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

$$\frac{d}{dx}P(x) = 0$$

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

$$\frac{dP}{dx} = 0.5 + \frac{2}{x}$$

Question 4

If you set $$\frac{dP}{dx} = 0$$ for $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x$$, what equation do you get?

Accepted Answer

$$0.5 + \frac{2}{x} = 0$$

Question 5

Solve $$0.5 + \frac{2}{x} = 0$$ for $$x.$$

Accepted Answer

$$x = -4$$

Question 6

Why is the solution $$x = -4$$ not valid in the context of this problem?

Accepted Answer

Because the number of kilograms sold cannot be negative.

Question 7

If the Newton-Raphson method is used to solve $$0.5 + \frac{2}{x} = 0$$, what is the iterative formula?

Accepted Answer

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

Question 8

If you start with $$x_0 = 50$, what is the value of x_1$ using the Newton-Raphson method for 0.5$$ + \frac{2}{x} = 0$$?

Accepted Answer

$$x_1 = 81.81$

Question 9

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

Accepted Answer

It converges quickly to the root of a nonlinear equation.

Question 10

Suppose the cost per kilogram increases to $$3. $$How does the profit function change?

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

$$2.5$$

Question 13

Which of the following best describes the role of the logarithmic term in the income function $$I(x) = 3x + 2 \ln(x)$$?

Accepted Answer

It models diminishing returns as $$x$$ increases.

Question 14

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

Accepted Answer

The value of $$x$$ where $$3x + 2 \ln(x) = 50 + 2.5x$$

Método de los Secantes: Optimización de Costos y Beneficios

Guida di studio - Domande di pratica

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