Examples of Curve Sketching
Key Questions
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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):
Questions
Graphing with the Second Derivative
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Relationship between First and Second Derivatives of a Function
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Analyzing Concavity of a Function
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Notation for the Second Derivative
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Determining Points of Inflection for a Function
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First Derivative Test vs Second Derivative Test for Local Extrema
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The special case of x⁴
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Critical Points of Inflection
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Application of the Second Derivative (Acceleration)
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Examples of Curve Sketching