Examples of Curve Sketching

Key Questions

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

    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):

    enter image source here

    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):

    enter image source here

    Wataru · 1 · Sep 25 2014

Questions

Graphing with the Second Derivative