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The average rate of change helps to determine how one quantity changes in relation to another over a specific interval. In calculus, it mimics how fast something changes over a particular range. For the demand problem, you're finding how the unit price of tents changes as you increase the number of tents sold.

To calculate this, use the formula:

• Average rate of change = \(\frac{f(x_2) - f(x_1)}{x_2 - x_1}\).

Here, \(x_1\) and \(x_2\) represent the starting and ending quantities demanded (in thousands). Plug these into the demand function to find \(f(x_1)\) and \(f(x_2)\). This gives you the respective prices at these points.

By finding the difference in prices and dividing it by the difference in quantities, you get how much the price increased or decreased per unit sold between those quantities.

Derivatives represent the rate at which a function is changing at any given point. In the context of your demand function, the derivative tells you how the price changes as the quantity of tents sold changes. It's the slope of the tangent line at any point on the demand curve and provides immediate information about the rate of change.

To find the derivative, differentiate the function \(f(x) = -0.1 x^{2} - x + 40\) with respect to \(x\). Apply differentiation rules to gain:\(f'(x) = -0.2x - 1\). This new function \(f'(x)\) gives the rate of change of price concerning the quantity at any point \(x\).

Evaluating this derivative at a specific point, like \(x = 5000\), answers questions on how the price is precisely changing with the sale of 5000 tents.

A demand function relates the quantity of an item demanded to its price. Understanding it can help businesses predict how many units they'll sell at different price levels. In this exercise, the demand function is \(p = f(x) = -0.1 x^{2} - x + 40\). This indicator allows seeing how the price (\(p\)) adjusts as the demand in units (\(x\)) changes.

The negative coefficient of the square term \(-0.1 x^{2}\) suggests a nonlinear relationship where increasing demand decreases price, indicating inverse demand behavior typical in market settings. This function is quadratic, depicting more complex price changes than simple linear functions.

Mathematically understanding this helps in decision-making processes like setting optimal price points to maximize revenue or market penetration.

The power rule is a fundamental technique in calculus, especially useful when dealing with polynomials. It simplifies the process of differentiation, making it easy to find derivatives of functions. The rule states that if you have a term \(x^n\), its derivative is \(nx^{n-1}\).

For example, in the given demand function, applying the power rule to \(-0.1 x^2\):

• Derivative of \(-0.1 x^2\) is \(-0.2 x^1\) or \(-0.2x\).

Similarly, the derivative of \(-x\) (\(x^1\)) becomes \(-1\), and the constant \(40\) has no \(x\) to differentiate, so it drops out to 0.

Using the power rule quickly handles polynomial differentiation needed to find how prices in the tent demand function change with varying sales.