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.1
Hmm -- your school subject appears to be math, not chabot college.
s = sin (pi t) is a sin function starting at (0,0)the period T is when pi t = 2 pi
or in other words when t = 2
when t = .5
s = sin (pi/2) = 1
at t=0 the slope = cos 0 = 1
at t = .5 the slope = cos pi/2 = 0
in other words the function climbs up from (0,0) to a peak at (.5,1) and then heads down
To sketch the graph of the position function s(t) = sin(πt), we will focus on the interval from t = 0 to t = 0.5.
Step 1: Plot the points (0, s(0)) and (0.5, s(0.5)).
- s(0) = sin(π * 0) = sin(0) = 0, so the point (0, 0) is on the graph.
- s(0.5) = sin(π * 0.5) = sin(π/2) = 1, so the point (0.5, 1) is on the graph.
Step 2: Draw the secant line passing through the two points.
- The secant line connects the points (0, 0) and (0.5, 1).
- Since the x-coordinate changes by 0.5 and the y-coordinate changes by 1, the slope of the secant line is rise/run = 1/0.5 = 2.
Step 3: Determine the relationship of the slope to the moving object.
- The slope of the secant line represents the average rate of change of the position function over the given interval.
- In this case, the slope of 2 indicates that the object is moving at an average velocity of 2 units per time unit from t = 0 to t = 0.5.
- The positive slope indicates that the object is moving in the positive direction (i.e., to the right) along the line.
Here is a rough sketch of the graph with the secant line passing through the points (0, 0) and (0.5, 1):
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0 0.5 t
Please note that this sketch may not be to scale, but it represents the general shape of the graph and the secant line.
To sketch the graph of the position function s(t) = sin(πt), we can first plot a few points and then connect them to form a smooth curve.
Let's start with finding the position of the object at two different times: t = 0 and t = 0.5.
s(0) = sin(π * 0) = sin(0) = 0
s(0.5) = sin(π * 0.5) = sin(π/2) = 1
So, we have two points: (0, 0) and (0.5, 1). Now, let's plot these points on a graph:
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(0, 0) | | (0.5, 1)
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Now, let's draw a line passing through these two points:
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._____ _______________
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(0, 0) | | (0.5, 1)
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Observe that the line passes through the points (0,0) and (0.5,1). This line is called a secant line.
Next, let's find the slope of this secant line. The slope is given by the formula:
slope = change in y / change in x
Let's calculate the change in y and change in x:
change in y = 1 - 0 = 1
change in x = 0.5 - 0 = 0.5
Now, we can calculate the slope:
slope = 1 / 0.5 = 2
So, the slope of the secant line is 2.
The slope of the secant line represents the average rate of change of the position function between the two points (0,0) and (0.5,1). In this case, since the slope is positive (2), it means that the object is moving in the positive direction along the line on the end of the spring.
It's important to note that the slope of the secant line is an average value and does not provide information about the object's instantaneous velocity or acceleration at any specific time. To determine that, we would need to find the derivative of the position function.