How do you find the average rate of change of #f(x) = sec(x)# from #x=0# to #x=pi/4#?

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Answer 1 · 2014-09-24T03:40:22

The average rate of change is the slope of the secant line through the points #(0,1)# and #(pi/4,sqrt(2))#.

Average rate of change of = #(f(b)-f(a))/(b-a)#

#f(x)=sec(x)#

#b=pi/4#

#a=0#

#(f(b)-f(a))/(b-a)=(f(pi/4)-f(0))/(pi/4-0)=(sec(pi/4)-sec(0))/(pi/4-0)#

Using the knowledge of the Unit Circle we know that #pi/4# represents the special triangle #45, 45, 90 -> 1, 1, sqrt(2)#

#sec(pi/4)=(hyp)/(adj)=sqrt(2)/1=sqrt(2)#

#sec(0)=1#

#=(sec(pi/4)-sec(0))/(pi/4-0)=(sqrt(2)-1)/(pi/4-0)=(sqrt(2)-1)/(pi/4)#

#=0.4142/0.7854=0.5274#