Q: If the weekly cost function is $$CC(x) = 50 + 2.5x$$ and you produce 100 kg, what is the total weekly cost?

$$CC(100) = 50 + 2.5 \times 100 = 300$$

Q: If the weekly income function is $$I(x) = 3x + 2 \ln(x)$$ and you sell 100 kg, what is the total weekly income?

$$I(100) = 3 \times 100 + 2 \ln(100) \approx 300 + 9.21 = 309.21$$

Question 1

Given the weekly cost function $$CC(x) = 50 + 2.5x$$ and the weekly income function $$I(x) = 3x + 2 \ln(x)$$, what is the correct equation for the weekly profit?

Accepted Answer

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

Question 2

If the weekly profit function is $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x$$, what value of $$x$$ maximizes the profit according to the method of successive approximations shown in the notes?

Accepted Answer

$$x \approx 82.35$$

Question 3

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

Accepted Answer

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

Question 4

To find the maximum profit, which equation must be solved?

Accepted Answer

$$3 + \frac{2}{x} - 2.5 = 0$$

Question 5

If $$x_0 = 50$ is the initial guess for the iterative method, what is the next value x_1$ calculated using the Newton-Raphson method for f(x) = 3$$ + \frac{2}{x} - 2.5$$?

Accepted Answer

$$x_1$$ \approx 81.81$$

Question 6

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

Accepted Answer

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

Question 7

If the cost per kilogram increases from $$2.5 to$$ $$3$$, how does this affect the optimal value of $$x$$?

Accepted Answer

The optimal value of $$x$$ decreases.

Question 8

Which of the following best describes the method used to solve for the optimal $$x in $$the notes?

Accepted Answer

Successive approximations (Newton-Raphson method)

Question 9

If the weekly income function changes to $$I(x) = 4x + 2 \ln(x)$$, what is the new profit function?

Accepted Answer

$$P(x) = 4x + 2 \ln(x) - 50 - 2.5x$$

Question 10

Suppose the fixed cost increases to $$100. $$What is the new cost function?

Accepted Answer

$$CC(x) = 100 + 2.5x$$

Question 11

If the profit function is $$P(x) = 3x + 2 \ln(x) - 50 - 2.5x$$, what is the second derivative with respect to $$x$$?

Accepted Answer

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

Question 12

What does a negative second derivative of the profit function indicate about the maximum?

Accepted Answer

The maximum is a local maximum (the function is concave at that point).

Question 13

If the weekly cost function is $$CC(x) = 50 + 2.5x$$ and you produce 100 kg, what is the total weekly cost?

Accepted Answer

$$CC(100) = 50 + 2.5 	imes 100 = 300$$

Question 14

If the weekly income function is $$I(x) = 3x + 2 \ln(x)$$ and you sell 100 kg, what is the total weekly income?

Accepted Answer

$$I(100) = 3 	imes 100 + 2 \ln(100) \approx 300 + 9.21 = 309.21$$

Question 15

If the weekly profit for selling 100 kg is $$P(100) = 309.21 - 300 = 9.21$$, what does this value represent?

Accepted Answer

The net gain after subtracting costs from income for 100 kg sold.

Question 16

If the cost function is linear and the income function is nonlinear, what type of optimization problem is this?

Accepted Answer

A nonlinear optimization problem.

Question 17

If the optimal $$x is $$found to be $$82.35$$, what is the approximate weekly profit at this value?

Accepted Answer

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

Optimization Methods: Secant Method for Maximizing Weekly Profit

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