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Problem 41 Hard Difficulty

Growth of a Population The rate of growth of a microbem population is given by
$$m^{\prime}(t)=27 t e^{3 t}$$
where $t$ is time in days. What is the total accumulated growthduring the first 2 days?


$15 e^{6}+3$


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Video Transcript

in this problem, we're being given the rate of growth of a micro population. And since we're being asked to find the total accumulated growth, we're taking the integral of this functions 27 t e to the three T from 0 to 2. Since we're being asked about the 1st 2 days, So we're gonna do this. Using integration by parts will let you be 2070. That means Devi will be on the other piece, which is e tow three t d t D'You would then be 27 d t and V would be won over three e to the three t and we're gonna use this information to evaluate the integral. So the integral from 0 to 2 of 27 t e to the three t is equal to UV. So that's gonna be 2070 times, one over three to the three t ahh Factor of three will cancel up. So we'll have nine t e to the three t minus the integral of VD. You integral of again 27 in the three canceled threesome, nine e to the three t DT. So the first part will stay the same. The second part is an integral that we can take. The anti derivative of that ends up being three e to the three t plus C. We're going to evaluate this from, uh, 0 to 2. So we get 18 he to the six minus three e to the six. And then when we sub zero in there, the first term will go away. The second term will turn into minus three. So overall, we have 15 e to the six plus three and this is the accumulated growth in two days.

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