Rates of change in other applied contexts (non-motion problems)
Applied rate of change: forgetfulness
I studied for an English test today and learned 80 vocabulary words. In 10 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 1 minus 0.1t squared for t is between 0 and 10, including the two boundaries. That's why we have brackets right over here. What is the rate of change 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 every day after I first memorized 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 0.1t, the whole thing squared with respect to 1 minus 0.1t is going to be-- so I'm essentially taking the derivative of this whole pink thing, this whole expression squared with respect to the expression. So that's going to be 2 times 1 minus 0.1t. And now I can find the derivative of this inner expression with respect to t. So the 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. 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 0.1, that's going to be negative 16. 160 times 0.1 is 16, so negative 16 times 1 minus 0.1t. 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 question. We could write this as 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. Let me do that same color. It's equal to negative 16 times 1 minus 0.1t. So what's this going to be? What is the rate of change of the number of words known per day two days after studying for the test? Well, we just have to evaluate this when t is equal to 2. So W prime of 2 is going to be equal to negative 16 times 1 minus 0.1 times 2 close parentheses. 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 12.8. And just to really hit the point home of what we're doing, the rate of change is negative 12.8 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, right at the day two point, right after exactly two days after studying for the test, right at that moment, I am essentially losing 12.8 words per day. The number of words I know is decreasing by 12.8 words per day.
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