Examples of Curve Sketching
Key Questions

Information from
#f(x)# #f(0)=1/{1+1}=1/2 Rightarrow# yintercept:#1/2# #f(x) > 0 Rightarrow# xintercept: 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 yintercept (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)^2e^xcdot2(1+e^x)e^x}/{(1+e^x)^4}# #={e^x(1+e^x)(1e^x)}/{(1+e^x)^4}={e^x(1e^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

Relationship between First and Second Derivatives of a Function

Analyzing Concavity of a Function

Notation for the Second Derivative

Determining Points of Inflection for a Function

First Derivative Test vs Second Derivative Test for Local Extrema

The special case of x⁴

Critical Points of Inflection

Application of the Second Derivative (Acceleration)

Examples of Curve Sketching