Question

In each exercise, a function and an interval of its independent variable are given. The endpoints of the interval are associated with the points \(P\) and \(Q\) on the graph of the function. a. Sketch a graph of the function and the secant line through P and \(Q\). b. Find the slope of the secant line in part (a), and interpret your answer in terms of an average rate of change over the interval. Include units in your answer. The volume \(V\) of an ideal gas in cubic centimeters is given by \(V=2 / p,\) where \(p\) is the pressure in atmospheres and \(0.5 \leq p \leq 2\).

Short answer

Answer: The average rate of change of the function is -2 within the interval. This means that for each increase of one atmosphere in pressure, the volume of the ideal gas decreases by 2 cubic centimeters on average. The negative sign of the slope indicates an inverse relationship between pressure and volume.

Step-by-step solution

Find the coordinates of points P and Q

We are given the function \(V= \dfrac{2}{p}\) and the interval \(0.5 \leq p \leq 2\). To find the coordinates of points P and Q, we need to plug in the values of \(p\) at the interval endpoints. For point P, \(p = 0.5\): $$V(0.5) = \frac{2}{0.5} = 4$$ So, P has coordinates \((0.5, 4)\). For point Q, \(p = 2\): $$V(2) = \frac{2}{2} = 1$$ So, Q has coordinates \((2, 1)\).

Sketch the graph of the function and the secant line

With the coordinates of points P and Q \((0.5, 4)\) and \((2, 1)\), we can sketch the graph of the function \(V(p) = \dfrac{2}{p}\) and the secant line passing through these points.

Calculate the slope of the secant line

Now, we will calculate the average rate of change by finding the slope of the secant line between points P and Q. We can find the slope with the formula: $$\text{slope} = \frac{\text{change in } V}{\text{change in } p} = \frac{V(Q) - V(P)}{p(Q) - p(P)}$$ Using the coordinates of P and Q, we get: $$\text{slope} = \frac{1 - 4}{2 - 0.5} = \frac{-3}{1.5} = -2$$

Interpret the slope of the secant line

The slope of the secant line is -2. This means that for each increase of one atmosphere in pressure, the volume of the ideal gas decreases by 2 cubic centimeters on average within the interval \(0.5 \leq p \leq 2\). The negative sign of the slope indicates that the relationship between pressure and volume is inverse. This information is useful for understanding how the gas behaves when its pressure changes.

Key concepts

Secant Line

A secant line is a straight line that connects two points on the graph of a function. In this exercise, the secant line passes through points \( P \) and \( Q \) on the volume function \( V(p) = \frac{2}{p} \), where \( p \) is the pressure in atmospheres. The coordinates of these points are essential to sketch the line and understand the function's behavior between these points.

• For point \( P \), with \( p = 0.5 \), the volume \( V \) is 4 cubic centimeters.
• Point \( Q \), at \( p = 2 \), has a volume of 1 cubic centimeter.

This line effectively becomes a visual tool for examining the average rate of change over the interval. By drawing the secant line through these points, you can see how sharply the function rises or falls between \( P \) and \( Q \), offering a snapshot of the function's behavior over that interval.

Slope Calculation

Calculating the slope of the secant line provides insight into the average rate of change of the function over a given interval. The slope formula \( \frac{\Delta V}{\Delta p} = \frac{V(Q) - V(P)}{p(Q) - p(P)} \) helps quantify this change as it shows the change in volume per change in pressure. To find the slope using the points \( P \) and \( Q \):

• Point \( P \) is at \( (0.5, 4) \).
• Point \( Q \) is at \( (2, 1) \).

Substitute these coordinates into the slope formula:\[slope = \frac{1 - 4}{2 - 0.5} = \frac{-3}{1.5} = -2\]This result shows that as pressure increases by one atmosphere, the volume of the gas decreases by 2 cubic centimeters. This gives an average rate of change and indicates how the volume reacts to changes in pressure.

Inverse Relationship

In this problem, the concept of an inverse relationship is demonstrated by the negative slope of the secant line. An inverse relationship implies that as one variable increases, the other decreases. Specifically, for the given function \( V(p) = \frac{2}{p} \):

• If the pressure \( p \) increases, the volume \( V \) decreases.
• A negative slope conveys that the variables are inversely related.

In practical terms, for an ideal gas in a fixed setting, as you increase the pressure, the volume decreases. This exercise illustrated this by calculating the slope of the secant line, which turned out to be \(-2\). Therefore, understanding the inverse nature helps in predicting how the volume will behave with changes in pressure. This type of understanding is crucial in fields like chemistry and physics, where such relationships frequently occur.