Another math problem

Briggs and Cochran page 377

A cultore of cells in a lab has a population of 100 cells when nutrients are added at time t=0. Suppose population N(t) increases at a rate given by

Find N(t) for t≥0

N'(t)=90e^-0.1t cells per hour.

N(t)=N(0) + t⌠0 N'(x)dx

100 + t⌠0 90e^-0.1x dx

100 + [(90/-0.1)e^-0.1x]t|0

Now I get confused as to how the next step is 1000. Also, I thought when you integrate an exponential function, you do x^n+1 divided by n+1. But 1 is not added to -0.1 either on the exponent of e or on the -.01 in the denominator. Here’s what the book came up with:

= 1000-900^e-0.1t

Thanks.

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About mindweapon

A mind weapon riding along with Four Horsemen of the Apocalypse.https://en.gravatar.com/profiles/edit/#
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4 Responses to Another math problem

  1. FD says:

    With respect to your question, the exponential function e^f(x) preserves its exponent through differentiation or integration, as distinct from x^f(x).

    Where is the x coming from? Time is the dependent variable here.

    Starting with N’(t)=90e^-0.1t cells per hour and N(0) = 100:
    N(t) = ⌠(90e^-0.1t )dt = 90⌠( e^-0.1t )dt = 90 * (-10) * e^-0.1t + C = -900 e^-0.1t + C
    N(0) = 100 = -900*(e^0) + C
    e^0 = 1, so
    100 = -900 + C
    C = 1000
    N(t) = -900 e^(-0.1t) + 1000

    Feel free to email me questions, I’m happy to help.

  2. Peter Blood says:

    I really like that you’re doing this. How about you explain to us what this population growth looks like?

    I see this stuff and realize how much is recessed deep in the brain. I remember it when I need to. When I can’t remember I can drag it back out, by figuring it out.

  3. FD says:

    Glad to help, mindweapon.

    Peter Blood:
    Start at (0, 100). You will see rapid growth tapering off to a horizontal asymptote at N = 1000 as t gets large. Its first derivative is positive on [0,infinity), so it is always increasing. Its second derivative is negative on [0,infinity) so it is concave downward. IOW, it’s increasing at a decreasing rate.

    The function shows the bacteria rushing toward a Malthusian state at N = 1000.

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