What is the growth after 1 day?

Question

The rate of growth of a microbe population is given by m’(x)=30xe^2x, where x is time in days. What is the growth after 1 day?

I am assuming that u=x, du=dx, dv=e^2xdx, and v=1/2e^2x

This is as far as I got. Not quite sure how to set this one up.

Answer 1

Question: The rate of growth of a microbe population is given by m’(x)=30xe^2x, where x is time in days. What is the growth after 1 day?

We use intgration by parts here.

m’(x)=30x.e^(2x)

integrating both sides,

m(x) = int 30x.e^(2x) dx = 15x.e^(2x) - 15*int e^(2x) dx = 15x.e^(2x) - 7.5e^(2x) + C1

m(x) = 15x.e^(2x) - 7.5e^(2x) + C1

m(x) = 7.5e^(2x)(2x - 1) + C1

Let m0 be the microbe population at the beginning of day1 (i.e. at x = 0). Then

m0 = m(x = 0) = -7.5 + C1

i.e. C1 = m0 + 7.5

Then m(x) = 7.5e^(2x)(2x - 1) + m0 + 7.5

At the end of the 1st day, x = 1, the population is

m(1) = 7.5e+ m0 + 7.5 = 7.5(e + 1) + m0

Growth is m(1) - m(0) = 7.5(e+1) + m0 - m0 = 7.5(e + 1)

Answer: Growth = 7.5(e + 1)