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CCSS.Math: , ,

so I've taken some screenshots of the Khan Academy exercise interpreting rate of change for exponential models in terms of change maybe they're going to change the title that seems a little bit too long but anyway let's let's actually just tackle these together so the first day of spring an entire field of flowering trees blossoms the population of locusts consuming these flowers rapidly increases as the trees blossom the relationship between the elapsed time T in days since the beginning of spring and the total number of locusts L of T so the number of locust is going to be a function of the number of days that have elapsed since the beginning of spring is modeled by the following function so locusts as a function of time it's going to be 750 times 1.85 to the teeth power complete the following sentence about the daily rate of change of the locust population every day the locust population well every day think about what's going to happen I'll draw a little table just to make it hopefully a little bit clearer so let me draw a little bit of a table so we'll put T and LF t so when T is 0 so when 0 days have elapsed well there's going to be 1 point 8 5 to the 0th power this is going to be 1 so you're going to have 750 locusts right from the get-go then when T equals 1 what's going to happen well then this is going to be 750 times 1 point 8 5 to the first power so it's going to be times 1 point 8 5 when T is equal to 2 what cell of T it's going to be 750 times 1 point 8 5 squared well that's the same thing as 1 point 8 5 times 1 point 8 5 so notice and this is just comes out of this being an exponential function every day you have one point eight five times as many as you had the day before one point eight five we essentially take what we had the day before we multiply by one point eight five and since one point eight five is larger than one this going to grow the number of locusts we have so this is going to grow I'm actually not using I'm not on the website right now so that's why I normally there would be a drop down here so I'm going to grow by a factor of well I'm going to grow by a factor of one point eight five every day let's do another one of these all right so this one tells us that Veera is an ecologist who studies the rate of change in the bear population of Siberia over time the relationship between the elapsed time T in years since viewer began studying the population and the total number of bears and of T is modeled by the following function alright fair enough it's a little got a little exponential thing going on complete the following sentence about the yearly rate of change of the bear population well just think about every year that passes T is in years now every year that passed is going to be 2/3 times the year before I can do that same table that I just did just to make that clear so let me do that whoops me let me make this clear so table so this is T and this is n of T when T is 0 n of T you're going to have 2187 bears so that's the first year that you began studying that population 0 years since this V R began setting the population the first year is going to be 2187 times 2/3 to the first power so times 2/3 the second year is going to be 2187 times 2/3 to the second power so there's just 2/3 times 2/3 so each successive year you're going to have 2/3 is the bear population of the year before you're multiplying the year before by 2/3 so every year the bear population shrinks shrinks by a factor of by a factor of 2/3 I let's do one more of these so they tell us that Akiba started studying how the number of branches on his tree change over time all right the relationship between the elapsed time T in years since the kiba study started studying history and the total number of its branches n ft is modeled by the following function complete the following sentence about the yearly percent change in the number of branches every year blank % of branches are added or subtracted from the total number of branches well I'll draw them another table although you might get used to just being able to look at this and say well look each year you're going to have 1.75 times the number of branches you had the year before and so we have 1.75 times the number of branches the year before you have grown by 75% and I'll make a little bit clearer so 75% of Brett every year 75% of branches are added to the total number of branches and I'll just draw that table again like I've done in the last two examples to make that hopefully clear ok ok so this is T and this is n of T so T equals 0 you have 42 branches T equals 1 it's going to be 42 times 1.75 times one point seven five when T equals two is going to be 42 times one point seven five squared 42 times one point seven five times one point seven five so every year you are multiplying times one point seven five so times one point seven five something funky is happening with my pen right over there but if you're multiplying by one point 75 if you're growing by a factor of one point seventy five this is the same thing as adding 75 percent it's again you are adding 75 percent think about it this way if I if you just grew by a factor of one then you're not adding anything you're staying constant if you grow by 10 percent then you're going to be 1.1 times as large if you grow by 200 percent then you're going to be 2 times as large so this right over here this right over here is is is going or if you let me be clear careful when I just said I think I just mistake that if you grow by 200% you're going to be three times as large as you were before one is constant and then another 200% would be another twofold so that make you three times as large don't want to confuse the eye my brain recognize that I said something weird right at that end all right hopefully you enjoyed that