Question 1

Given the weekly cost function for producing an article is $$CC(x) = 50 + 2.5x$$ and the weekly income function is $$I(x) = 3x + 2 \ln(x)$$, 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, you set the derivative of the profit function to zero. What equation must you solve to find the optimal $$x$$?

Accepted Answer

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

Question 3

What is the economic interpretation of the equation $$3 + \frac{2}{x} - 2.5 = 0 in $$this context?

Accepted Answer

It is the condition where marginal income equals marginal cost.

Question 4

If you use the secant method to solve $$3 + \frac{2}{x} - 2.5 = 0$$, what is the formula for the next approximation $$x_{k+1}$$?

Accepted Answer

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

Question 5

Suppose your initial guesses for the secant method are $$x_0 = 50$ and x_1$ = 81.81. If$$ $$f(x_0$) = -17.17$$ and $$f(x_1$) = 0.28$$, what is the next approximation $$x_2$$?

Accepted Answer

$$x_2 = 81.81 - 0.28$$ \frac{81.81 - 50}{0.28 - (-17.17)}$$

Question 6

What is the main advantage of using the secant method over the Newton-Raphson method in this scenario?

Accepted Answer

The secant method does not require the explicit calculation of the derivative.

Question 7

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

Accepted Answer

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

Question 8

If the weekly fixed cost increases from $$50 to$$ $$100$$, what is the effect on the optimal value of $$x$$ that maximizes profit?

Accepted Answer

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

Question 9

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

Accepted Answer

$$3 + \frac{2}{x+1} - 2.5 = 0$$

Question 10

In a real-world scenario, why might the function $$I(x) = 3x + 2 \ln(x) be a $$more realistic model for income than a simple linear function?

Accepted Answer

It accounts for diminishing returns as $$x$$ increases.

Question 11

If the solution to $$3 + \frac{2}{x} - 2.5 = 0 is$$ $$x^*$$, what is the maximum weekly profit?

Accepted Answer

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

Question 12

If the optimal $$x$$ found is not an integer, what is the best practical recommendation for the number of kilograms to produce?

Accepted Answer

Produce the nearest integer value to the optimal $$x.$$

Optimization Using Calculus: Secant Method for Maximizing Profit

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