course
Melissa Mcelheny
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What function represents the depth?
What would this function be if it was known that at clock time t = 0 the depth is 200 ?
If the function y = .028 t2 + -1.3 t + 61 represents depth y vs. clock time t,
then what is the average rate of depth change between clock time t = 6.7 and clock time t = 13.4?
Y= .028t^2 + -1.3t + 61
x = 6.7 .028(6.7)^2 -1.3(6.7) +61 Y= 53.54692
x = 13.4 .028(13.4)^2 -1.3(13.4) +61 Y= 48.60768
(6.7, 53.54692) and (13.4, 48.60768)
48.60768 - 53.54692 divided by 13.4 - 6.7 equals -0.7372 avg rate of depth change
What is the rate of depth change at the clock time halfway between t = 6.7 and t = 13.4?
Y= .028t^2 + -1.3t + 61
Y' = 2(.028)t + (-1.3)
Y' = .056t - 1.3
y' = .056 (6.7) - 1.3
y' = -.9248 -.9248 + -.5496 divided by 2 equals -.7372 rate of depth change
-.7372 times 6.7 equals -4.93924 depth change
Good. Note that the average of the beginning and ending rates is equal to the average rate you calculated earlier. THis is because the rate function is linear, as will always be the case for a quadratic depth function.
y'= .056(13.4) - 1.3
y'= -.5496
What function represents the rate r of depth change at clock time t?
R(t) = .056(t) - 1.3
What is the clock time halfway between t = 6.7 and t = 13.4, and what is the rate of depth change at this instant?
Clock time 1/2 way between 6.7 and 13.4
13.4 + 6.7 divided by 2 10.05
rate of depth change at 10.05
y' = .056(10.05) - 1.3 equals -0.7372
If the function r(t) = .289 t + -2.7 represents the rate at which depth is changing at clock time t,
then how much depth change will there be between clock times t = 6.7 and t = 13.4?
t=6.7 .289(6.7) - 2.7 equals -.7637
t=13.4 .289(13.4) - 2.7 equals 1.1726
-.7637 + 1.1726 divided by 2 equals .20445 rate of change
Good, but best to call this 'average rate of change' to distinguish between this and an instantaeous rate. That distinction becomes very important in this course.
13.4 - 6.7 = 6.7 time interval
rate of change x time interval equals amount of depth change
More accurately, average rate of change x time interval equals amount of depth change
.20445 x 6.7 equals 1.369815
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Good. Let me know if you have questions.