Pain in the Asymptotes: Q: f(x) = 1 / [ 1 + e^(3x) ]

matthewR

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[FONT=&quot]First year engineering checking in, revision for Semester One tests and cant find any notes on how to solve this,

Q: f(x)=1/[1+e^(3x)]

(i) Find Horizontal Asymptotes
(ii)Determine intervals where increasing/decreasing

I know im asking a lot, but please help out stressed out dude [/FONT]
:rolleyes:
 
You already know how to analyze the behavior of a ratio with fixed numerator, by considering what's happening in the denominator. That is, as the denominator grows without bound, while the numerator is fixed, the value of the ratio heads toward zero.

Are you familiar with the behavior of e^x, for all Real numbers?

If so, you could focus on the e^(3x) term, and work through the following limit analyses, step-by-step.

What happens to the value of e^(3x), as x increases without bound? How does this affect the entire denominator? Hence, what is the resulting behavior of the ratio f(x), as x heads toward positive infinity?

What happens to the value of e^(3x), as x becomes increasingly negative? What effect does this have on the denominator, and, therefore, on the value of f(x)?

What's the value of f(0)? What happens to function f, as x starts to move away from zero, in either direction?

Your conclusions ought to give you a good idea of function f's global behavior, and that's enough to answer both parts.

If you know how to determine the first derivative of f(x), you could also analyze that as well, to discover that f'(x) is always negative. Do you know: Functions increase, when their first derivative is positive; they decrease, when the first derivative is negative.

Questions about this approach? :cool:
 
Given the function \(\displaystyle f(x)\, =\, \dfrac{1}{1\, +\, e^{3x}},\)

(i) find the horizontal asymptotes and
(ii) determine intervals where the function is increasing or decreasing.


I know im asking a lot, but please help out stressed out dude
:rolleyes:
On which step of which part are you getting stuck?

(i) You remember horizontal asymptotes from algebra (here), and you know how \(\displaystyle y\, =\, e^{x}\) behaves. So what will happen to the graph of the given function, with the exponential in the denominator? What then must be the horizontal asymptotes in either direction?

(ii) This part is more complicated, coming from when you studied calculus. It is not reasonably feasible to attempt to teach differentiation within this environment. (It's the first half of the notes on this page.) Do you need instruction on the entire topic, or are you just getting bogged down at one particular spot in the process?

Please be specific. Thank you! ;)
 
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