Examples of Curve Sketching

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Calculus: Graphing Using Derivatives
20:31 — by Khan Academy

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Key Questions

  • enter image source here

  • Use differentiation to identify turning points.
    Input f(0) to find y intercept. ( 29 in this case )
    Use factor theorem to help you locate zeroes ; that is test for values that result in f(a) = 0.
    Test either side of zeroes and turning points to confirm behaviour.
    A quartic function is even, and this one is positive so will open upwards.
    . The shape will generally look something like :
    enter image source here

    image courtesy Wikipedia.

  • 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

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