Examples of Curve Sketching
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Key Questions


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 :
image courtesy Wikipedia.

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
Videos on topic View all (15)
Graphing with the Second Derivative

1Relationship between First and Second Derivatives of a Function

2Analyzing Concavity of a Function

3Notation for the Second Derivative

4Determining Points of Inflection for a Function

5First Derivative Test vs Second Derivative Test for Local Extrema

6The special case of x⁴

7Critical Points of Inflection

8Application of the Second Derivative (Acceleration)

9Examples of Curve Sketching