How to estimate the value using secant between the points?
The question is: a heart monitor is used to measure the heartbeat of a patient after surgery. It compiles the number of heartbeats after t minutes. When the data in the table are represented, the slope of the tangent represents the heart rate in beats per minute. T (min): 36, 38, 40, 42, 44 heartbeats: 2530, 2661, 2806, 2948, 3080 The following estimates this value by calculating the slope of a secant. Using these data to estimate the patient's heart rhythm after 42 minutes using the secant between points with given values of t. (a) t = 36 and T = 42 (b) t = 38 and T = 42 (c) t = 40 and t = 42 (d) t = 42 and T = 44 What are your conclusions? I do not understand how to do it at all. I tried to learn but I do not really understand. Help?
A secant line intersects a curve at two points or more. Link: http://en.wikipedia.org/wiki/Secant_line A line is tangent to a curve if they pass both by one point. The line does not intersect the curve, it is to "touch" it. The tangent is the best linear approximation of the curve at this point. Link: http://en.wikipedia.org/wiki/Tangent_line # geometry for your problem: Pt-T (min) – heartbeat 1 to 36 — 2530 2-38 — 2661 3-40 — 2806 from 4 to 42 — 2948 5 to 44 — – 3080 Slope = Y / X Let x = time, y = slope heartbeat (1 ~ 4) = (2948-2530) / (42-36) slope (1 ~ 4) = 418 / 6 = 69,666 Slope (2 ~ 4) = (2948-2661) / (42 -38) Slope (2 ~ 4) = 287 / 4 Slope = 71.75 (3 ~ 4) = (2948-2806) / (42-40) Slope (3 ~ 4) = 142 / 2 = 71 Slope (4 ~ 5) = (3080 to 2948) / (44-42) Slope (4 ~ 5) = 132 / 2 = 66 The best approximation of heart rate at 42 minutes was determined using the secant through the points closest to each side 42 minutes, ie 44 and 40 minutes. This line should be parallel to the tangent to 42 minutes. Slope (4 ~ 5) = (3080 to 2806) / (44-40) Slope (4 ~ 5) = 274 / 4 = 68.5 Note the slope of the secant is the same as the average of secants on each side of t = 42 minutes: (71 + 66) / 2 = 68.5. As for conclusions, the only thing apparent to me is that the slope of the lines before 42 minutes are about 70-71 beats per minute and the slope after 42 minutes was 66 beats per minute. This seems to be a fairly significant difference, but nothing in the data suggest a cause. Mathematically, there is no reason why the slopes should be different – the data seem to be continuous and the time interval remains constant. I'm not sure This point is made by this exercise. You may want to examine these links to see if there is something your teacher asks you to do this exercise. Http: / / en.wikipedia.org / wiki # Http://en.wikipedia.org/wiki/Tangent_line Calculus / Derivative
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