Question 1

Given the weekly cost function for producing an article $$CC(x) = 50 + 2.5x$$ and the weekly income function $$I(x) = 3x + 2 \ln(x)$$, 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 the weekly profit, which 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

Which equation should be solved to find the critical points for maximizing profit?

Accepted Answer

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

Question 5

Using the secant method, which formula is used to find the next approximation $$x_{k+1}$$ for the root of $$f(x)$$?

Accepted Answer

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

Question 6

If the initial guesses for the secant method are $$x_0 = 50$ and x_1$ = 81.81$$, and $$f(x) = 3 + \frac{2}{x} - 2.5$$, what is $$f(x_0$)$$?

Accepted Answer

Question 7

What is the economic interpretation of the value of $$x$$ that maximizes the profit function?

Accepted Answer

It is the number of kilograms that should be sold weekly to maximize profit.

Question 8

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

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

Accepted Answer

It remains unchanged.

Question 11

Which of the following best describes the role of the natural logarithm in the income function $$I(x) = 3x + 2 \ln(x)$$?

Accepted Answer

It models diminishing returns as more kilograms are sold.

Question 12

If $$x = 82.35 is $$found as the solution to the profit-maximizing equation, what does this value represent?

Accepted Answer

The optimal number of kilograms to sell per week.

Question 13

Which of the following is a necessary condition for a maximum in the profit function $$P(x)$$?

Accepted Answer

$$\frac{d^2$P}{dx^2$} \u003c 0$$

Question 14

If the weekly cost function is $$CC(x) = 50 + 2.5x$$, what is the marginal cost?

Accepted Answer

$$2.5$$

Question 15

If the weekly income function is $$I(x) = 3x + 2 \ln(x)$$, what is the marginal income?

Accepted Answer

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

Optimization Using the Secant Method: Calculus Applications

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