Math Problem Statement
Homework:Section 2.1: The Idea of Limits
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Part 2 of 3
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Part 1
Consider the position function
s left parenthesis t right parenthesis equals sine left parenthesis pi t right parenthesiss(t)=sin(πt)
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
left parenthesis 0 comma s left parenthesis 0 right parenthesis right parenthesis(0,s(0))
and
left parenthesis 0.5 comma s left parenthesis 0.5 right parenthesis right parenthesis(0.5,s(0.5)).
Determine the slope of the secant line and explain its relationship to the moving object.
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Part 1
Sketch the graph of s together with the secant line. Choose the correct graph below.
A.
01.501.5ts
A coordinate system has a horizontal t-axis labeled from 0 to 1.5 in increments of 0.1 and a vertical s-axis labeled from 0 to 1.5 in increments of 0.1. From left to right, a smooth curve rises at a decreasing rate from (0, 0) to (0.5, 1), then falls at an increasing rate to (1, 0). Two points are plotted on the curve, at (0, 0) and at (0.5, 1). A line, rising from left to right, passes through the two plotted points.
Your answer is correct.
B.
01.501.5ts
A coordinate system has a horizontal t-axis labeled from 0 to 1.5 in increments of 0.1 and a vertical s-axis labeled from 0 to 1.5 in increments of 0.1. From left to right, a smooth curve rises at a decreasing rate from (0, 0) to (0.5, 1), then falls at an increasing rate to (1, 0). Two points are plotted on the curve, at (0.5, 1) and at (1, 0). A line, falling from left to right, passes through the two plotted points.
C.
01.501.5ts
A coordinate system has a horizontal t-axis labeled from 0 to 1.5 in increments of 0.1 and a vertical s-axis labeled from 0 to 1.5 in increments of 0.1. From left to right, a smooth curve rises at a decreasing rate from (0, 0) to (0.5, 1), then falls at an increasing rate to (1, 0). Two points are plotted on the curve, at (0, 0) and at approximately (0.8, 0.58). A line, rising from left to right, passes through the two plotted points.
D.
01.501.5ts
A coordinate system has a horizontal t-axis labeled from 0 to 1.5 in increments of 0.1 and a vertical s-axis labeled from 0 to 1.5 in increments of 0.1. From left to right, a smooth curve rises at a decreasing rate from (0, 0) to (0.5, 1), then falls at an increasing rate to (1, 0). Two points are plotted on the curve, at (0, 0) and at (0.5, 1). A horizontal line passes through the plotted point at (0.5, 1).
Part 2
Determine the slope of the secant line.
m Subscript secant Baseline equals nothingmsec=enter your response here
(Round to the nearest thousandth as needed.)
Choose the correct graph below.input field 1What is the relationship of the slope to the velocity of the moving object?Choose the correct graph below.input field 1
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Solution
Part 1: Graph Interpretation
You are asked to sketch the position function and a secant line passing through two points on the graph: and .
The position function describes a sinusoidal motion, which is typical for an object moving along a spring or oscillating. The graph of starts at , rises to , and will return to zero at .
Given this, the correct graph choice that corresponds to the description of the curve, along with the secant line passing through and , is:
- Option A: The curve rises from to , then decreases after that. The secant line passes through the points and , which corresponds to the right answer.
Part 2: Slope of the Secant Line
To find the slope of the secant line, we need to calculate the change in position (change in ) divided by the change in time (change in ) between the points and .
The formula for the slope of the secant line is:
Given , we can substitute the values for and :
Now, calculating the slope:
So, the slope of the secant line is .
Relationship of the Slope to the Velocity of the Moving Object
The slope of the secant line represents the average rate of change of the position over the interval from to . In the context of motion, this slope corresponds to the average velocity of the object during this time interval.
Thus, the average velocity of the object between and is 2 units per time.
Additional Questions to Explore:
- What is the instantaneous velocity of the object at ?
- How would the graph of change if the function were ?
- What would the slope of the secant line be if we chose points at and ?
- How does the velocity of the object relate to the slope of the tangent line at any point on the graph?
- How do the concepts of limits and derivatives help explain the instantaneous velocity of the object?
Tip:
When working with sinusoidal functions, be sure to recall key values for sine and cosine at standard angles like , , , etc., to make calculations faster!
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Calculus
Motion
Secant Line
Average Velocity
Formulas
m_{secant} = (s(t2) - s(t1)) / (t2 - t1)
s(t) = sin(πt)
Theorems
Slope of secant line
Sinusoidal functions
Average rate of change
Suitable Grade Level
Grades 11-12
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