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 with the secant line passing through (0,s(0)) and (0.5,s(0.5)). Determine the slope of the secant line and its relationship to the moving object.

Steve, I already tried 0.5 and that was incorrect.

what do you mean you "tried 0.5"? Tried for what? In my solution, the slope is 2, not 1/2.

To sketch the graph of the position function `s(t)`, we can start by plotting a few points on the graph.

Let's consider the given values of `t` to find the corresponding `s(t)` values:
1. For `t = 0`, we have `s(0) = sin((π)(0)) = sin(0) = 0`.
2. For `t = 0.5`, we have `s(0.5) = sin((π)(0.5)) = sin(π/2) = 1`.

Now we can plot these points on a graph:

```
|
| * (0.5, 1)
|
| (0,0)
---------------------------------------
```

The points `(0, 0)` and `(0.5, 1)` are plotted on the graph.

Next, we can draw a straight line passing through these two points. This line is the secant line.

```
|
| * -- (0.5, 1) /
| /
| /
---------------------------------------
```

Now, to determine the slope of the secant line, we can use the formula:

slope = (change in y) / (change in x)

The change in `y` is given by `1 - 0 = 1`, and the change in `x` is given by `0.5 - 0 = 0.5`.

So, the slope of the secant line is `1 / 0.5 = 2`.

The slope of the secant line represents the average rate of change of the position function over the interval from `t = 0` to `t = 0.5`. In this case, the slope is positive, which indicates that the object is moving upward in the positive direction along the line on the end of the spring.

Thus, the slope of the secant line represents the velocity or speed of the moving object at that particular interval.

clearly,

s(0) = 0
s(1/2) = 1

So, the line is y=2t.

since t is time and y is position, say, cm, then the slope of the line (cm/s) is the average velocity during the interval [0,0.5]

See

http://www.wolframalpha.com/input/?i=plot+y%3Dsin%28pi*x%29%2C+y%3D2x