Problem 16 - Calculus: Early Transcendentals

Question

Average velocity Consider the position function \(s(t)=\sin \pi t\) representing the position of an object moving along a line on the end of a spring. Sketch a graph of \(s\) together with the secant line passing through \((0, s(0))\) and \((0.5, s(0.5)) .\) Determine the slope of the secant line and explain its relationship to the moving object.

Step-by-Step Solution

Answer

Answer: The slope of the secant line represents the average velocity of the object between the two time points t=0 and t=0.5, with a value of 2 units per time in the positive direction.

Step 1: Sketch the graph of the position function, \(s(t)\)

To start, we'll sketch the graph of the given position function, \(s(t)=\sin{\pi t}\). This is a sine function with an amplitude of 1, a period of 2, and no phase shift or vertical shift. The graph will oscillate between 1 and -1, making a complete cycle in a 2-unit interval along the \(t\)-axis. The graph starts and ends at 0 for every full cycle.

Step 2: Sketch the secant line passing through the points \((0, s(0))\) and \((0.5, s(0.5))\)

Now, let's find the position of the object at the given points: \(s(0) = \sin{\pi \cdot 0} = 0\) \(s(0.5) = \sin{\pi \cdot 0.5} = 1\) So, we have the points \((0,0)\) and \((0.5,1)\). We can sketch the secant line passing through these points on the same graph as the position function.

Step 3: Determine the slope of secant line

The slope of the secant line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be found using the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) Here, we have \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (0.5, 1)\). Substituting these values into the formula, we get: \(m = \frac{1 - 0}{0.5 - 0} = \frac{1}{0.5} = 2\) So, the slope of the secant line is 2.

Step 4: Explain the relationship between the slope of the secant line and the moving object

The slope of the secant line represents the average velocity of the object between the two time points \(t=0\) and \(t=0.5\). In this context, a positive slope implies that the object is moving in the positive direction along the line during this interval, and its average velocity is 2 units per time.

Key Concepts

Slope of Secant Line

Understanding the slope of a secant line is crucial in calculus as it serves as a foundation for understanding instantaneous rates of change. In simple terms, the secant line cuts through two distinct points on a curve. To find this slope, we use the formula
\(m = \frac{y_2 - y_1}{x_2 - x_1}\).
In the exercise given, the secant line intersects the graph of the position function
\(s(t) = \sin(\pi t)\)
at two points: \((0, s(0))\) and \((0.5, s(0.5))\). By applying the two points to the formula, we deduce that the slope is
\(2\).
This slope represents the average rate of change of the position of an object moving along a spring—essentially, this is its average velocity over the time interval given.

• A steeper slope indicates a higher average velocity.
• A slope of zero would imply no movement on average.
• Negative values for the slope would mean movement in the opposite direction.

The concept of secant lines also primes us for the concept of the derivative, which tells us the instantaneous rate of change at any given point.

Position Function

A position function, often denoted as \(s(t)\), maps time \(t\) to the position of an object along a line, curve, or space. It's a fundamental concept in the study of motion in physics and calculus.
For the exercise at hand, the position function is given by \(s(t) = \sin(\pi t)\), which describes the harmonic motion of an object attached to a spring. Here are some characteristics of a sine function that are reflected in a position function such as this:

• It oscillates in a regular pattern, signifying repetitive motion back and forth.
• It has a period and amplitude that affect how the object moves over time.
• Phase shifts or vertical shifts can indicate more complex motion, though none are present in this particular function.

As we saw in the graph, this function starts at the origin, \((0,0)\), then rises to its maximum value before returning to zero. Understanding the sine curve is essential to predicting and analyzing the object's motion.

Sine Function Graph

The graph of a sine function, like the one in our exercise \(s(t) = \sin(\pi t)\), is a smooth wave that ascends and descends in a regular pattern, known as sinusoidal. The graph is critical to understand because it captures the behavior of many physical phenomena, especially those involving periodic motion like sound waves, tides, or in this case, the motion of an object on a spring.
Here are the key features of a sine graph mentioned in the exercise:

• Amplitude: The height from the center line to the peak (or trough), which in this exercise is 1.
• Period: The length of one complete cycle, which is determined by the coefficient of \(t\) in the sine function. For \(\sin(\pi t)\), the period is 2.
• No Phase Shift: The graph starts at the origin, moving upwards first, which means there is no horizontal shift altering the start of the cycle.
• No Vertical Shift: The graph oscillates around the horizontal axis without moving up or down that axis, staying within the bounds defined by its amplitude.

The sine graph's smooth peaks and valleys reflect the natural undulation of the spring's motion, showing points where the spring is at rest (\(s(t)=0\)) and points where it's fully stretched or compressed (\(s(t)=\pm 1\)).