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$$f(t) = \frac{100e^t}{10+e^t} \text{for all } t \in \mathbb{R}$$

(1) show that $$\frac{d}{dt}f(t)=f(t) - \frac{1}{100} (f(t)^2)$$ (2) waht is the second derivative $f^{(2)}(t)$ in term of $f(t)$?

(3) find the second order Taylor polynomial $T_2(t)$ for $f(t)$ about the point $t = \ln(10)$.

I really donot know how to solve this question.

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Let's start at the beginning. Can you find $\frac{d}{dt} f(t)$? –  Hurkyl Feb 23 '13 at 6:57
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1 Answer 1

A nifty trick you might try is to observe that

$$\frac{f'(t)}{f(t)} = \frac{d}{dt} \log{f(t)}$$

You get a form like the left hand side by dividing both sides of the equation by $f(t)$. Your equation to verify is then

$$\frac{d}{dt} \log{f(t)} = 1-\frac{1}{100} f(t)$$

Now,

$$\log{f(t)} = \log{100} - \log{(1+10 e^{-t})}$$

$$\frac{d}{dt} \log{f(t)} = \frac{10 e^{-t}}{1+10 e^{-t}}$$

Can you do the rest?

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