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course Mth 173
9/30 21:00
Question Sketch and completely label a trapezoidal approximation graph for the function y = x 2/ 5 + 1, for x = 0 to 2.7 by increments of .9.
Answer
The given graph is y = x^2 / 5 + 1. This graph thus is a concave parabola whose vertex is (0, 1).the graph is ever increasing from x = 0 to x = 2.7 and with increasing slope for that x interval. Now on the x axis markings are made from x = 0 to x = 2.7 with increments of 0.9 (that is 0, 0.9, 1.8 and 2.7). From each point marked on the x axis a line parallel to y axis is drawn to the graph to meet their respective point on the graph. Now this points on the graph are joined to one another by straight lines. That is the point ( 0 , 1 ) is connected to the point ( 0.9 , 1.16 ) which is connected to ( 1.8 , 1.65 ) which is in turn connected to ( 2.7 , 2.46 ). Now we have 3 trapezoid constructed, one trapezoid with parallel lines with x = 0 and x = 0.9, second trapezoid with parallel lines with x = 0.9 and x = 1.8, and the third trapezoid with parallel lines x = 1.8 and x = 2.7.
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Good.
Rather than reporting the coordinates of the points, you would simply report the 'altitudes' of the trapezoids as
1, 1.16, 1.65, 2.46.
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Consider the 1st trapezoid
If we consider the function to represent a quantity Q(x) which depends on clock time x, then
the width `dx of the trapezoid which is 0.9 - 0 = 0.9 represents the time interval associated with trapezoid, and the rise associated with the trapezoid represents the change `dQ in the quantity Q over that time interval which here is 1.16 - 1 = 0.16 so that the slope of the trapezoid is slope = rise / run = `dQ / `dx = 0.16/0.9 = 0.177, which represents the average rate at which quantity Q is changing for that time interval.
Now if the function represent the average rate dQ / dx at which some quantity Q changes with respect to clock time,
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You're a little ambiguous here. The function in question is the quantity function Q(x), and your statement could be taken to mean that the function if dQ/dx.
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then the average altitude of the trapezoid represent the average rate of change of quantity over that time interval which in this case is 1.08. T
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You would want to say that if Q happens to be the rate of change of some other quantity, then the average rate of change on this interval is 1.08 and the change in that quantity (not in Q) would be 1.08 * 0.9.
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he width of the trapezoid represents the change of the time interval `dx = 0.9 and thus the area of the trapezoid represents the change of the quantity over that time interval which thus is area = 1.08 * 0.9 = 0.972 which is the change in the quantity.
The slope of the trapezoid is therefore slope = rise / run = `d (dQ / dt) / `dx = (1.16 - 1) / 0.9 = 0.16 / 0.9 = 0.1778 which represents the average rate at which the rate of quantity change changes.
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If Q is a rate of change, then the slope represents the average rate at which that rate changes. That rate is itself the rate of change of some other unspecified quantity.
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Consider the 2nd trapezoid
If we consider the function to represent a quantity Q(x) which depends on clock time x, then
the width `dx of the trapezoid which is 1.8 - 0.9 = 0.9 represents the time interval associated with trapezoid, and the rise associated with the trapezoid represents the change `dQ in the quantity Q over that time interval which here is 1.65 - 1.16 = 0.49 so that the slope of the trapezoid is slope = rise / run = `dQ / `dx = 0.49/0.9 = 0.544, which represents the average rate at which quantity Q is changing for that time interval.
Now if the function represent the average rate dQ / dx at which some quantity Q changes with respect to clock time, then the average altitude of the trapezoid represent the average rate of change of quantity over that time interval which in this case is 1.405. The width of the trapezoid represents the change of the time interval `dx = 0.9 and thus the area of the trapezoid represents the change of the quantity over that time interval which thus is area = 1.405 * 0.9 = 1.2645 which is the change in the quantity. The slope of the trapezoid is therefore slope = rise / run = `d (dQ / dt) / `dx = (1.65 - 1.16) / 0.9 = 0.49 / 0.9 = 0.544 which represents the average rate at which the rate of quantity change changes.
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Having given a very good explanation of the first trapezoid, you could proceed to just list the values you got for the remaining trapezoids.
Then a simple table showing the various quantities would be sufficient.
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Consider the 3rd trapezoid
If we consider the function to represent a quantity Q(x) which depends on clock time x, then
the width `dx of the trapezoid which is 2.7 - 1.8 = 0.9 represents the time interval associated with trapezoid, and the rise associated with the trapezoid represents the change `dQ in the quantity Q over that time interval which here is 2.46 - 1.65 = 0.81 so that the slope of the trapezoid is slope = rise / run = `dQ / `dx = 0.81/0.9 = 0.9, which represents the average rate at which quantity Q is changing for that time interval.
Now if the function represent the average rate dQ / dx at which some quantity Q changes with respect to clock time, then the average altitude of the trapezoid represent the average rate of change of quantity over that time interval which in this case is 2.055. The width of the trapezoid represents the change of the time interval `dx = 0.9 and thus the area of the trapezoid represents the change of the quantity over that time interval which thus is area = 2.055 * 0.9 = 1.8495 which is the change in the quantity. The slope of the trapezoid is therefore slope = rise / run = `d (dQ / dt) / `dx = (2.46 - 1.65) / 0.9 = 0.81 / 0.9 = 0.9 which represents the average rate at which the rate of quantity change changes.
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Very good, but check my notes for some terminology.
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