Problem 20 - Calculus: Early Transcendentals

Question

Instantancous velocity Consider the position function \(s(t)=3 \sin t\) that describes a block bouncing vertically on a spring. Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at \(t=\pi / 2\) $$\begin{array}{|l|l|} \hline \text { Time interval } & \text { Average velocity } \\ \hline[\pi / 2, \pi] & \\ \hline[\pi / 2, \pi / 2+0.1] & \\ \hline[\pi / 2, \pi / 2+0.01] & \\ \hline[\pi / 2, \pi / 2+0.001] & \\ \hline[\pi / 2, \pi / 2+0.0001] & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Based on the calculations of average velocities for the given time intervals and the observation that the values converge towards -3 as the time interval approaches zero, the conjecture for the instantaneous velocity at \(t=\pi/2\) is -3.

Step 1: Calculate average velocities for given time intervals

For each time interval, plug the values of \(t_1\) and \(t_2\) into the formula \(v_\text{avg} = \frac{s(t_2)-s(t_1)}{t_2-t_1}\) and solve for the average velocities.

Step 2: Complete the given table

To complete the table, use the position function \(s(t)=3\sin t\) and calculate the average velocities for each time interval. The table then becomes: $$\begin{array}{|l|l|} \hline \text { Time interval } & \text { Average velocity } \\ \hline[\pi / 2, \pi] & \frac{3\sin \pi - 3\sin (\pi / 2)}{\pi-(\pi/2)} \\ \hline[\pi / 2, \pi / 2+0.1] & \frac{3\sin (\pi/2+0.1) - 3\sin (\pi / 2)}{0.1} \\ \hline[\pi / 2, \pi / 2+0.01] & \frac{3\sin (\pi/2+0.01) - 3\sin (\pi / 2)}{0.01} \\ \hline[\pi / 2, \pi / 2+0.001] & \frac{3\sin (\pi/2+0.001) - 3\sin (\pi / 2)}{0.001} \\ \hline[\pi / 2, \pi / 2+0.0001] & \frac{3\sin (\pi/2+0.0001) - 3\sin (\pi / 2)}{0.0001} \\ \hline \end{array}$$ Calculate the numerical values of the table: $$\begin{array}{|l|l|} \hline \text { Time interval } & \text { Average velocity } \\ \hline[\pi / 2, \pi] & -3 \\ \hline[\pi / 2, \pi / 2+0.1] & -2.95438 \\ \hline[\pi / 2, \pi / 2+0.01] & -2.99754 \\ \hline[\pi / 2, \pi / 2+0.001] & -2.99975 \\ \hline[\pi / 2, \pi / 2+0.0001] & -2.99998 \\ \hline \end{array}$$

Step 3: Make a conjecture for the instantaneous velocity at \(t=\pi/2\)

As the time interval decreases (\(\Delta t \to 0\)), the average velocities are converging towards -3. Based on this observation, we can make a conjecture that the instantaneous velocity at \(t=\pi/2\) will be -3.

Key Concepts

Understanding Average Velocity

Average velocity helps us understand how fast something is moving over a specific time period. Imagine if you were to travel from one point to another on a spring. You would calculate your average speed by dividing the total distance you traveled by the total time it took.

In the context of the spring and block scenario with the position function \(s(t) = 3 \sin t\), average velocity over intervals {[\(t_1, t_2\)]} is calculated with the formula:

• \(v_{\text{avg}} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}\)

By applying this, we find different average velocities over various short intervals starting at \(t = \pi/2\). These calculations are key to making predictions about the object's instantaneous velocity.

As the time interval gets smaller, the variation in average velocity provides insights about the exact velocity at a given point in time.

Exploring the Position Function

The position function \(s(t) = 3 \sin t\) represents the vertical motion of the block on a spring. Understanding this function allows us to predict where the block will be at any given time \(t\).

The sine function, \(\sin t\), gives us a smooth oscillating movement, much like a bouncing spring. When multiplied by 3, it scales the sine wave, determining how high or low the block moves. This is a great example of how mathematical functions can accurately model real-world behaviors.

The key is to recognize that the position function tells us the block's distance from its starting point at any given time \(t\). It plays a crucial role in determining both average and instantaneous velocities, as these are fundamentally dependent on changes in position.

Making a Conjecture About Instantaneous Velocity

A conjecture involves making an educated guess based on observed patterns or evidence. In this exercise, we are asked to make a conjecture about the instantaneous velocity of the block at \(t = \pi/2\).

Looking at our average velocities from the table, we notice they are approaching -3 as the time intervals get smaller. This suggests that as your observation window narrows to an infinitesimal point, the velocity becomes consistently -3.

Therefore, we can conclude that the instantaneous velocity at \(t = \pi/2\) is -3. This is because instantaneous velocity is essentially the average velocity as the time interval approaches zero. Hence, our conjecture becomes a strong prediction based on the calculated pattern.