Problem 23 - Calculus Early Transcendentals

Question

For the following position functions, make a table of average velocities similar to those in Exercises \(19-20\) and make a conjecture about the instantaneous velocity at the indicated time. $$s(t)=40 \sin 2 t \quad \text { at } t=0$$

Step-by-Step Solution

Answer

Answer: Based on the table of average velocities and the pattern of convergence, we can estimate the instantaneous velocity at \(t=0\) to be approximately 80 m/s.

Step 1: Write down the position function and the given time

The position function is given by \(s(t) = 40 \sin 2t\). We have to find the instantaneous velocity at \(t=0\).

Step 2: Create a table of intervals around the given time

To estimate the instantaneous velocity at \(t=0\), we will take small intervals around \(t=0\), such as \([-0.1, 0]\), \([-0.01, 0]\), \([-0.001, 0]\), and their right-side counterparts \([0, 0.1]\), \([0, 0.01]\), and \([0, 0.001]\).

Step 3: Calculate the average velocities over the intervals

For each interval, calculate the average velocity using the position function and the formula for average velocity: \(\overline{v} = \frac{s(t_2)-s(t_1)}{t_2-t_1}\). Interval \([-0.1, 0]\): $$\overline{v} = \frac{40 \sin(2(0))-40 \sin(2(-0.1))}{0-(-0.1)}$$ Interval \([-0.01, 0]\): $$\overline{v} = \frac{40 \sin(2(0))-40 \sin(2(-0.01))}{0-(-0.01)}$$ Interval \([-0.001, 0]\): $$\overline{v} = \frac{40 \sin(2(0))-40 \sin(2(-0.001))}{0-(-0.001)}$$ Interval \([0, 0.1]\): $$\overline{v} = \frac{40 \sin(2(0.1))-40 \sin(2(0))}{0.1-0}$$ Interval \([0, 0.01]\): $$\overline{v} = \frac{40 \sin(2(0.01))-40 \sin(2(0))}{0.01-0}$$ Interval \([0, 0.001]\): $$\overline{v} = \frac{40 \sin(2(0.001))-40 \sin(2(0))}{0.001-0}$$

Step 4: Observe the table and make a conjecture

Now, observe the table of average velocities and notice how they converge towards a certain value as the interval becomes smaller. Using this pattern, you can make a reasonable conjecture about the instantaneous velocity at \(t=0\). Remember that you can also find the instantaneous velocity using the derivative of the position function, but for this exercise, the table of average velocities should provide a good approximation.

Key Concepts

Average Velocity

The concept of average velocity is crucial for understanding movement over a certain period. It essentially refers to the total displacement divided by the total time taken for that displacement.
This is expressed mathematically as:

• For any time interval \[t_1, t_2\], average velocity \(\overline{v}\) is calculated using \(\overline{v} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}\).

In our exercise, average velocities are computed over several decreasing intervals around \( t=0 \).
This approach helps to approximate the instantaneous velocity by observing how average velocities behave as intervals shrink.

Position Function

The position function, in this context, provides a mathematical description of an object's position relative to time.
In the exercise, we are given the position function \( s(t) = 40 \sin 2t \).
This function tells us how the position changes periodically over time.

• The function is based on a sine function, indicating a wave-like motion.
• Multiplying sine by 40 scales the amplitude of the wave.
• The argument of sine, \( 2t \), affects the frequency of its oscillations.

Understanding this function helps us in calculating displacements needed to find average velocities.

Velocity Calculation

Velocity calculation is a key concept in determining how an object's speed and direction change over time. In calculus, this is often addressed through differentiation. For our problem, we focus on velocity at very small or instantaneous intervals around \( t=0 \).

• We calculate the average velocity over tiny intervals to approximate instantaneous velocity.
• For smaller intervals like \([-0.1, 0] \), \([-0.01, 0] \), and \([-0.001, 0] \), calculation involves evaluating the sine function for points at the boundaries of these intervals.

This method helps us observe how velocities change and how they converge to provide what we call instantaneous velocity.

Conjecture in Calculus

In calculus, forming a conjecture involves predicting a limit or behavior based on observed data. In this exercise, we use a table of average velocities to make an informed guess about instantaneous velocity.

• A conjecture is based on noticing the trend as intervals become smaller.
• As intervals \([-0.1, 0] \) and others decrease towards zero, average velocities approach a specific value.

This limit helps us conjecture what the instantaneous velocity should be. In more advanced calculations, the derivative of the position function can be used as it directly gives the velocity, confirming our conjecture.