Problem 10 - Active Calculus

Question

In Chapter \(1,\) we showed that for an object moving along a straight line with position function \(s(t),\) the object's "average velocity on the interval \([a, b]^{\prime \prime}\) is given by $$ A V_{[a, b]}=\frac{s(b)-s(a)}{b-a} $$ More recently in Chapter \(4,\) we found that for an object moving along a straight line with velocity function \(v(t),\) the object's "average value of its velocity function on \([a, b]^{\prime \prime}\) is $$ v_{\mathrm{AVG}[a, b]}=\frac{1}{b-a} \int_{a}^{b} v(t) d t $$ Are the "average velocity on the interval \([a, b]^{\prime \prime}\) and the "average value of the velocity function on \([a, b]^{\prime \prime}\) the same thing? Why or why not? Explain.

Step-by-Step Solution

Answer

No, they are not the same. The average velocity considers net change in position, while the average value of the velocity function considers all instantaneous velocities in the interval.

Step 1 - Understanding Average Velocity

The average velocity on the interval \([a, b]\) for an object moving along a straight line is defined by the expression \(\frac{s(b) - s(a)}{b - a}\). This formula calculates the net change in position (\(s(b) - s(a)\)) over the time interval (\(b - a)\).

Step 2 - Understanding the Average Value of Velocity Function

The average value of the velocity function on the interval \([a, b]\) for an object moving along a straight line is given by \(\frac{1}{b - a} \int_{a}^{b} v(t) dt\). This formula finds the integral of the velocity function \(v(t)\) over \([a, b]\) and normalizes it by the length of the interval (\(b - a)\).

Step 3 - Comparing Both Concepts

To understand whether these two definitions are the same, observe that average velocity \( \frac{s(b) - s(a)}{b - a} \) depends on the net change in position over time. Whereas the average value of the velocity function \( \frac{1}{b - a} \int_{a}^{b} v(t) dt \) calculates the mean value of the velocity function over the interval.

Step 4 - Analyzing Differences

In general, these two quantities can be different unless \(v(t) \) is a constant function. This is because \( \frac{1}{b-a} \int_{a}^{b} v(t) dt \) takes into account all the instantaneous velocities within the interval, while \( \frac{s(b) - s(a)}{b - a} \) only depends on the starting and ending positions.

Step 5 - Conclusion

In conclusion, the average velocity on the interval and the average value of the velocity function on the interval are generally not the same. The average velocity considers only initial and final positions, whereas the average value of the velocity function considers all velocities within the interval.

Key Concepts

Position Function

The position function, denoted as \(s(t)\), describes the location of an object at any given time \(t\). For example, if an object is moving in a straight line, \(s(t)\) can tell us where the object is at time \(t\). Understanding the position function helps in calculating other aspects of motion such as velocity and acceleration.

• The key idea is that the position function gives us a snapshot of the object's location over time.
• This function is crucial because it allows us to find the average velocity over a specific time interval \([a, b]\).

Consider an object moving along a straight path, where its position at time \(t\) is given by \(s(t)\). To find out how far the object has traveled over a period—from time \(a\) to time \(b\)—we use the change in the position: \(s(b) - s(a)\).
This difference highlights the net change in position of the object over the specific interval. If you divide this change by the length of the interval \(b - a\), you get the average velocity. That brings us to our next concept.

Velocity Function

The velocity function is denoted as \(v(t)\) and indicates the speed and direction of an object at time \(t\). It's essentially the rate of change of the position function \(s(t)\). For instance, if \(s(t) = t^2\), then the velocity function \(v(t) = ds/dt = 2t\).

• The velocity function tells us how quickly or slowly the position is changing at any point in time.
• This is useful for understanding how the motion of an object varies over time, rather than just its starting and ending points.

To find out how the object's velocity changes over a time interval, we compute the average value of the velocity function. This is done by integrating the velocity function over the interval \([a, b]\) and then dividing by the length of that interval:
\ \ \ v_{\text{AVG}[a, b]} = \frac{1}{b - a} \int_{a}^{b} v(t) dt \ \ By integrating, we sum up all the instantaneous velocities across the interval and normalize by its length. This concept leads us to the integral.

Integral

Integration is a mathematical technique used to find the total or accumulated value, such as area under a curve or the total velocity over time. When we integrate the velocity function \(v(t)\) with respect to time \(t)\), we find the change in position.

• The integral of \(v(t)\) over \([a, b]\) calculates the total distance traveled between these two times.
• This accumulated change when normalized by \(b - a)\) provides the averaged effect over that interval.

Mathematically, the integral is represented as:
\ \ \ \ \ \int_{a}^{b} v(t) dt = s(b) - s(a) \ \ This tells us that integrating the velocity function over an interval gives the net change in the position function. When we divide the integral by \(b - a\), we get the average value of the velocity function. This normalized sum helps us understand the behavior of the object over time.
This differentiation between average velocity and average value is crucial. Average velocity considers only initial and final positions, whereas the average value of the velocity function takes into account the entire range of velocities over the interval.