Be sure you also submit these problems using the form. I went ahead and posted this one, received through email, for you.
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#1
Determine the average rate at which the quadratic function y
= .02 t2 + -2.4 t + 77 changes between clock times t = 32.2
and t = 32.2 + .1 sec; between clock times t = 32.2 and t =
32.2 + .01 sec; and between clock times t = 32.2 and t = 32.2
+ .001 sec. At what rate do you conjecture this function will
be changing at the instant t = 32.2 sec?
.02t2 + -2.4t + 77
Solve for t = 32.2
.02(32.2)2 + -2.4(32.2) +77
.02(1036.84) + -77.28 +77
y = 20.4568
Solve for t=32.2+.1= 32.3
.02t2 + -2.4t +77
.02(32.3)2 + -2.4(32.3) + 77
.02(1043.29) + -77.52 +77
20.8658 + -77.52 +77
y= 20.3458
Solve for t=32.2 + .01 = 32.21
.02(32.21)2 + -2.4(32.21) + 77
.02(1037.48) + -77.304 + 77
20.7496 + -.304
y= 20.4456
Solve for t = 32.2 + .001 = 32.201
.02(32.201)2 + -2.4(32.201) + 77
.02(1036.9044) + -77.2824 + 77
20.7380 + -.2824
y = 20.4556
Computing for the average rate between the clock times you
get the following:
t = 32.2 → y = 20.4568 (20.3458-20.4568)/(32.3-32.2)
t = 32.3 → y = 20.3458 = -.111/.1 = -1.11 = Averge rate
between 32.2 & 32.3
t = 32.2 → y = 20.4568 (20.4456-20.4568)/(32.21-32.2)
t = 32.21 → y = 20.4456 = -0.112/.01 = -1.12 = Averge rate
between 32.2 & 32.21
That 0.112 is really 0.0112.
You are losing significant figures here. Your difference in depth has only three significant figures; the difference in your rates will appear in any beyond the fourth significant figure. An average rate for this time interval, based on 6-significant-figure calculation of depths, is only accurate to 2 significant figures.
t = 32.2 → y = 20.4568 (20.4556-20.4568)/(32.201-32.2)
t = 32.201 → y = 20.4556 = -.0012/.001 = -1.2 = Averge
rate between 32.2 & 32.201
Due to significant figure considerations this result is only good to one significant figure. Your depths would have to be calculated accurate to many more significant figures in order to show the difference in rate between the preceding and the present result.
At what rate do you conjecture this function will be changing
at the instant t=32.2 sec? Using the Universal Formula for
rate of change you get the following:
2at + b
2(.02)(32.2) + -2.4 = -1.112
"
end of document
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This looks good. Let me know if you have questions.