Question 1

Given the weekly cost function for producing a chemical:

$$CC(x) = 50 + 2.5x$$

and the weekly income function:

$$R(x) = 3x + 2 \ln(x)$$

where $$x is $$the number of kilograms sold per week, what is the profit function $$P(x)$$?

Accepted Answer

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

Question 2

To maximize profit, what equation must be solved for $$x$$ using the profit function $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x$$?

Accepted Answer

$$\frac{dP}{dx} = 0$$

Question 3

Calculate $$\frac{dP}{dx}$$ for the profit function $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x.$$

Accepted Answer

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

Question 4

Set $$\frac{dP}{dx} = 0$$ and solve for $$x in $$the equation $$3 + \frac{2}{x} - 2.5 = 0. $$What is the value of $$x$$?

Accepted Answer

$$x = 4$$

Question 5

Which of the following best describes the Newton-Raphson method as used in the notebook to solve for the optimal $$x$$?

Accepted Answer

It is an iterative method to find roots of nonlinear equations using tangent lines.

Question 6

If the initial guess for $$x is$$ $$x_0 = 50$, and the Newton-Raphson update formula is x$$_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}$$, what is f(x$$)$$ in this context?

Accepted Answer

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

Question 7

What is the derivative $$f'(x)$$ needed for the Newton-Raphson method if $$f(x) = 3 + \frac{2}{x} - 2.5$$?

Accepted Answer

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

Question 8

Suppose after two iterations of the Newton-Raphson method, the values of $$x$$ are $$x_1 = 81.81$ and x_2$ = 82.35. $$What does this indicate about the convergence of the method?

Accepted Answer

The method is converging to a root near $$x = 82$$

Question 9

If the cost per kilogram increases from $$2.5 to$$ $$3$$, how does this affect the profit-maximizing quantity $$x$$?

Accepted Answer

The profit-maximizing $$x$$ decreases

Question 10

In the context of this problem, what does the solution $$x^*$$ represent?

Accepted Answer

The number of kilograms to sell weekly to maximize profit

Question 11

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

Accepted Answer

There is no effect on the profit-maximizing $$x$$

Question 12

Which of the following is a real-world application of maximizing profit using calculus as shown in the notebook?

Accepted Answer

Determining the optimal production level for a factory to maximize earnings

Question 13

If the income function changes to $$R(x) = 4x + 2 \ln(x)$$, how does this affect the profit-maximizing $$x$$?

Accepted Answer

The profit-maximizing $$x$$ increases

Question 14

Which step is necessary to confirm that the critical point found is a maximum for profit?

Accepted Answer

Check that $$\frac{d^2$P}{dx^2$} \u003c 0 at $$the critical point

Método de la Secante: Optimización de Ingresos y Costos Semanales

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