How do you sketch the curve #f(x)=e^x/(1+e^x)# ?

Answer 1

Information from #f(x)#

#f(0)=1/{1+1}=1/2 Rightarrow# y-intercept: #1/2#

#f(x) > 0 Rightarrow# x-intercept: none

#lim_{x to infty}e^x/{1+e^x}=1 Rightarrow# H.A.: #y=1#

#lim_{x to -infty}e^x/{1+e^x}=0 Rightarrow# H.A.: #x=0#

So far we have the y-intercept (in blue) and H.A.'s (in green):

Information from #f'(x)#

#f'(x)={e^xcdot(1+e^x)-e^xcdot e^x}/{(1+e^x)^2}=e^x/(1+e^x)^2>0#

#Rightarrow# #f# is always increasing.

Information from #f''(x)#

#f''(x)={e^x cdot (1+e^x)^2-e^xcdot2(1+e^x)e^x}/{(1+e^x)^4}#

#={e^x(1+e^x)(1-e^x)}/{(1+e^x)^4}={e^x(1-e^x)}/{(1+e^x)^3}#

#f''(x)>0# on #(-infty,0)# and #f''(x)<0# on #(0, infty)#

#f# is concave upward on #(-infty,0)# and downward on #(0, infty)#.

Hence, we have the graph of #f# (in blue):

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Answer 2

To sketch the curve of ( f(x) = \frac{e^x}{1 + e^x} ):

  1. Find the domain of the function.
  2. Determine the behavior of the function as ( x ) approaches positive and negative infinity.
  3. Find the ( y )-intercept by setting ( x = 0 ).
  4. Find any vertical asymptotes by solving ( 1 + e^x = 0 ).
  5. Find any horizontal asymptotes by analyzing the behavior of the function as ( x ) approaches positive and negative infinity.
  6. Find the critical points by finding where the derivative of the function is zero or undefined.
  7. Determine the concavity of the function by analyzing the second derivative.
  8. Sketch the curve using the information gathered from the above steps.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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