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Main content
Current time:0:00Total duration:5:00
AP.CALC:
CHA‑3 (EU)
,
CHA‑3.C (LO)
,
CHA‑3.C.1 (EK)

Video transcript

I studied for an English test today and learned 80 vocabulary words in ten days I will have forgotten every word the number of words that I remember t days after studying is modeled by so W of T so this is the number of words I have in my head as a function of time is going to be equal to 80 times the 1 minus 0.1 T squared for T is between 0 and 10 including the two boundaries that's why we have back brackets right over here what is the rate of change of the number of words note of the number of known words per day two days after studying for the test and I encourage you to pause this video and try it on your own so the key here is we come up with this equation for modeling how many words have retained in my brain everyday after I first memorize them after I got the 80 of them into my head and that's this expression here and they want to know the rate of change two days after studying well the rate of change I can take the derivative of this with respect to time so let's do that so let's take the derivative the derivative of the number of words I know with respect to time is going to be equal to well we have this 80 out front that's just a constant and now I can apply the chain rule right over here so the derivative of 1 minus 0.1 T whole thing squared with respect to 1 minus 0.1 T is going to be so I'm essentially taking the derivative of this whole pink thing with this this whole expression squared with respect to the expression so that's going to be 2 times 1 minus 0.1 T and now I can find the derivative of this inner expression with respect to T so derivative of this inner expression with respect to T is just going to be 0 minus 0.1 so it's just going to be negative 0.1 so it's going to be negative zero point one and of course we can simplify this a little bit this is going to be equal to if we take 80 times 2 is 160 times negative zero point one that's going to be negative 16 160 times 0.1 is 16 so negative 16 times times 1 minus 0.1 T and if we want we could distribute the 16 or we could just leave it like this but we're ready now to answer our questions we could write it we could write this is the rate of change of the number of words we know with respect to time or we could use the alternate notation we could say this is w prime of T either way it's going to be equal to this thing it's negative let me do that same color it's equal to negative 16 negative 16 times 1 minus 0.1 T so what's this going to be what is the rate of change of the of the number of words known per day two days after studying for the test well we just have to evaluate this one T is equal to 2 so W prime of 2 do that in magenta W prime of 2 is going to be equal to whoops well I'll just go with the magenta is equal to negative 16 negative 16 times 1 minus 0.1 times 2 times 2 close parenthesis and that's going to be equal to well let's see what is this this is 1 minus essentially 0.2 this is going to be 0.8 so this is going to be equal to negative 16 times is that right 1 minus 0.2 is yep it's going to be times 0.8 and what is that going to be if I were to multiply 16 times 8 it would be 128 it's 2 times 8 times 8 so 2 times 64 128 so this is going to be negative negative 12.8 and just to really hit the point home of what we're doing this is we're going to its negative the number of words wean the rate of change is negative twelve point eight words words per day so if you believe this model for how many words we know on a given day this is saying on day two hour or right right at the day two point right after two exactly two days after studying for the test I'm going to be right at that moment I am I am essentially losing twelve point eight words per day the number of words I know is decreasing by twelve point eight words per day