Q: If $$x = 82.35 is $$the solution found, what is the maximum weekly profit?

$$P(82.35) = 3 \times 82.35 + 2 \ln(82.35) - (50 + 2.5 \times 82.35)$$

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 mathematical method is being used to solve for the optimal value of $$x in $$the notebook?

Accepted Answer

Newton-Raphson method

Question 4

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

Accepted Answer

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

Question 5

What is the derivative $$OC'(x) of $$the function $$OC(x) = 3 + \frac{2}{x} - 2.5$$?

Accepted Answer

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

Question 6

After the first iteration of the Newton-Raphson method with $$x_0 = 50$, OC$$($$x_0) = 0.54$, and OC$$'($$x_0) = -0.0008$, what is the next approximation x_1$?

Accepted Answer

Question 7

What is the economic interpretation of the value of $$x$$ that solves $$3 + \frac{2}{x} - 2.5 = 0$$?

Accepted Answer

It is the number of kilograms that maximizes weekly profit.

Question 8

If the cost per kilogram increases from $$2.5 to$$ $$3$$, how does the equation for optimal $$x$$ change?

Accepted Answer

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

Question 9

Suppose the weekly fixed cost increases from $$50 to$$ $$100. $$How does this affect the optimal value of $$x$$?

Accepted Answer

The optimal value of $$x$$ does not change.

Question 10

If the weekly 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

Which of the following best describes the role of the Newton-Raphson method in this context?

Accepted Answer

It is used to numerically solve for the value of $$x$$ that maximizes profit.

Question 12

If $$x = 82.35 is $$the solution found, what is the maximum weekly profit?

Accepted Answer

$$P(82.35) = 3 	imes 82.35 + 2 \ln(82.35) - (50 + 2.5 	imes 82.35)$$

Método de la Secante: Optimización de Funciones en Estadística

스터디 가이드 - 연습 문제

학습 가이드 - 플래시카드