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 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 $$x$$?

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

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

Question 4

Using the Newton-Raphson method, which formula is used to find the root of $$f(x)$$?

Accepted Answer

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

Question 5

If $$f(x) = 3 + \frac{2}{x} - 2.5$$, what is $$f'(x)$$?

Accepted Answer

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

Question 6

Given an initial guess $$x_0 = 50$, and using Newton-Raphson for f(x) = 3$$ + \frac{2}{x} - 2.5$$, what is the next iteration x_1$?

Accepted Answer

$$x_1 = 50$$ - \frac{3 + \frac{2}{50} - 2.5}{-\frac{2}{$$50^2$$}}$$

Question 7

What is the economic interpretation of the value of $$x$$ that maximizes profit in this context?

Accepted Answer

It is the optimal number of kilograms to sell weekly to maximize profit.

Question 8

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

Accepted Answer

The profit-maximizing quantity decreases.

Question 9

Suppose the weekly income function changes to $$I(x) = 3x + 2 \ln(x+1). $$What is the new profit function?

Accepted Answer

$$P(x) = 3x + 2 \ln(x+1) - 50 - 2.5x$$

Question 10

If the fixed cost increases from $$50 to$$ $$100$$, what happens to the profit-maximizing quantity?

Accepted Answer

It remains the same.

Question 11

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

Accepted Answer

It represents diminishing returns to scale as $$x$$ increases.

Question 12

If $$x = 82.35 is $$found as the optimal solution, what does this value represent?

Accepted Answer

The number of kilograms to sell weekly to maximize profit.

Question 13

Which of the following is a necessary condition for a maximum in the profit function?

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

Which of the following is NOT a valid step in applying the Newton-Raphson method?

Accepted Answer

Dividing by zero in the denominator.

Optimization Methods: Solving for Maximum Profit Using Calculus and Iterative Methods

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