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

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

Accepted Answer

It is the number of kilograms that maximizes weekly profit.

Question 4

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

Accepted Answer

$$0.54$$

Question 5

Which iterative method is used in the notes to solve for the optimal $$x$$?

Accepted Answer

Secant method

Question 6

After two iterations using the secant method, what is the approximate value of $$x$$ that maximizes profit?

Accepted Answer

$$82.35$$

Question 7

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

Accepted Answer

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

Question 8

What is the main advantage of using the secant method over the bisection method in this context?

Accepted Answer

The secant method converges faster without needing the derivative.

Question 9

If the weekly fixed cost increases from $$50 to$$ $$100$$, what happens to the optimal value of $$x$$?

Accepted Answer

It remains the same.

Question 10

Suppose the income function changes to $$I(x) = 4x + 2 \ln(x). $$What is the new equation to maximize profit?

Accepted Answer

$$0 = 4 + \frac{2}{x} - 2.5$$

Question 11

What is the derivative of the profit function $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x$$ with respect to $$x$$?

Accepted Answer

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

Question 12

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

Accepted Answer

$$156$$

Question 13

Which of the following best describes the role of the logarithmic term in the income function?

Accepted Answer

It models diminishing returns as more kilograms are sold.

Question 14

If the cost function becomes $$CC(x) = 50 + 3x$$, what is the new profit-maximizing equation?

Accepted Answer

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

Question 15

Why is it important to use an iterative method like the secant method for this problem?

Accepted Answer

Because the equation involves a logarithmic and rational term, making it difficult to solve algebraically.

Método de la Secante: Optimización de Funciones en Estadística

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