course mth 173
4/7 7:30
If the function y = .015 t2 + -1.7 t + 93 represents depth y vs. clock time t, then what is the average rate of depth change between clock time t = 13.9 and clock time t = 27.8? What is the rate of depth change at the clock time halfway between t = 13.9 and t = 27.8?
The avg. rate of depth change between the given points is -1.074. This was found by first evaluating the two clock times to determine depth at the given time, done by substituting a given time into the original equation. I found the rate of depth change at the halfway point to be -1.0745.
What function represents the rate r of depth change at clock time t? What is the clock time halfway between t = 13.9 and t = 27.8, and what is the rate of depth change at this instant?
r= 2at + b, or r = .03t + (-1.7). I found the clock time to be 20.85 at the halfway point, using the function I found the rate of depth change at this point to be -1.0745.
If the function r(t) = .193 t + -2.1 represents the rate at which depth is changing at clock time t, then how much depth change will there be between clock times t = 13.9 and t = 27.8?
All I could think of was evaluate to find rates at the two clock times, which were .5827 at 13.9, and 3.2654 at 27.8. I then took those two rates and added them together and divided by two to find an average rate, which was 1.924, and multiplied by the 13.9 second interval. This gave me a depth change of 26.74
What function represents the depth? The quadratic function is the depth function.
What would this function be if it was known that at clock time t = 0 the depth is 130 ?
y = (.0965)t^2 + (-2.1)t + 130
Good. If you were to evaluate this function at the two given clock times, and subtract, you would get the same result you obtained by averaging the two rates, etc.. You would get a depth change of 26.74.
In other words, the antiderivative of the rate function is the depth function. The change in the antiderivative of the rate function is equal to the average value of the rate function, multiplied by the length of the interval.
This is essentially the fundamental theorem of calculus.
You've submitted some very good work today.