Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

An object travels along a line so that its position \(s\) is \(s=t^{2}+1\) meters after \(t\) seconds. (a) What is its average velocity on the interval \(2 \leq t \leq 3 ?\) (b) What is its average velocity on the interval \(2 \leq t \leq 2.003\) ? (c) What is its average velocity on the interval \(2 \leq t \leq 2+h ?\) (d) Find its instantaneous velocity at \(t=2 .\)

Step-by-Step Solution

Answer

(a) 5 m/s; (b) 4.003 m/s; (c) 4+h m/s.

Step 1: Understand Average Velocity Formula

The average velocity of an object over an interval \([t_1, t_2]\) is given by the formula \(\text{Average Velocity} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}\). This formula takes the change in position between two times and divides it by the time interval.

Step 2: Calculate for Interval (a)

Calculate the average velocity for \(t\) from 2 to 3. Plug \(t_1 = 2\) and \(t_2 = 3\) into the position formula: \[s(2) = 2^2 + 1 = 5\] \[s(3) = 3^2 + 1 = 10\]Then, \[\text{Average Velocity} = \frac{10 - 5}{3 - 2} = \frac{5}{1} = 5 \text{ m/s}\]

Step 3: Calculate for Interval (b)

Calculate the average velocity for \(t\) from 2 to 2.003.Plug \(t_1 = 2\) and \(t_2 = 2.003\) into the position function:\[s(2.003) = (2.003)^2 + 1 \approx 5.012009\]\[\text{Average Velocity} = \frac{5.012009 - 5}{2.003 - 2} \approx \frac{0.012009}{0.003} \approx 4.003 \text{ m/s}\]

Step 4: Calculate for Interval (c)

Calculate the average velocity with variable interval \([2, 2+h]\).Position at \(t = 2 + h\) is \[s(2 + h) = (2 + h)^2 + 1 = 4 + 4h + h^2 + 1 = 5 + 4h + h^2\]Average velocity is \[\frac{(5 + 4h + h^2) - 5}{h} = \frac{4h + h^2}{h} = 4 + h\] Thus, \text{Average Velocity} = 4 + h \text{ m/s.}

Key Concepts

Average Velocity

Average velocity is a fundamental concept in calculus, particularly when analyzing motion along a straight line. It's the measure of how fast an object moves, averaged over a specific time interval. To determine the average velocity of an object, you use the formula:

• \( \text{Average Velocity} = \frac{s(t_2) - s(t_1)}{t_2 - t_1} \)

This formula essentially calculates the change in position \( s(t) \) between two times \( t_1 \) and \( t_2 \), divided by the change in time, \( t_2 - t_1 \). By using this simple ratio, you can easily calculate how fast an object moves averagely between any given times.

For instance, when calculating the average velocity between time intervals \( t = 2 \) and \( t = 3 \), you substitute into the position function, yielding positions \( s(2) = 5 \) and \( s(3) = 10 \). This results in an average velocity of 5 meters per second, as the position changed by 5 meters over 1 second.

Similarly, when dealing with significantly smaller intervals, such as from \( t = 2 \) to \( t = 2.003 \), the formula proves helpful in illustrating small changes effectively and accurately, showcasing the precision calculus can offer.

Instantaneous Velocity

Instantaneous velocity goes a step beyond average velocity, offering insights into how fast an object is moving at a specific moment rather than over a period. In calculus, the instantaneous velocity is often found by taking the derivative of the position function, capturing how the position changes at every infinitely small interval.

To get the instantaneous velocity at a particular time \( t \), say \( t = 2 \), you first differentiate the position function \( s(t) = t^2 + 1 \). Doing so gives \( \frac{ds}{dt} = 2t \), where \( \frac{ds}{dt} \) represents the instantaneous velocity function. At \( t = 2 \), the instantaneous velocity calculates to \( 2(2) = 4 \) meters per second.

• Provides more precision than average velocity
• Requires differentiation of the position function

This measure of velocity is crucial for understanding motion in real-time, helping describe how the object's speed varies precisely at any given moment.

Position Function

Exploring the position function is essential for understanding the movement of objects over time. The position function \( s(t) \) describes where an object is at any given time \( t \). In our example, the position function \( s(t) = t^2 + 1 \) dictates that at time \( t = 0 \), the object is at position \( s(0) = 1 \).

This simple quadratic function provides a clear visualization of the object's trajectory along a line. By plugging in various time values into the function, you can determine where the object will be at those moments. For example:

• When \( t = 1 \), \( s(1) = 1^2 + 1 = 2 \) meters
• When \( t = 2 \), \( s(2) = 2^2 + 1 = 5 \) meters

Understanding the position function also underpins finding velocities, as it portrays the background against which you compute how fast or slow the object moves when other calculus tools, like derivatives, are applied.