How do you find the average value of #f(x)=2^x# as x varies between #[1,4]#?

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Answer 1 · 2017-03-30T16:07:23

#6.7#

The formula for average value of a continuous function #F(x)# on #[a, b]# is #1/(b - a) int_a^b F(x)dx#

Our formula here will be

#A = 1/(4 - 1) int_1^4 2^x dx#

#A = 1/3 int_1^4 2^x dx#

#A = 1/3[2^x/ln2]_1^4#

#A = 1/3[2^4/ln2 - 2^1/ln2]#

#A = 1/3[14/ln2]#

#A = 14/(3ln2)#

This can be approximated as #6.7#. The average value of #y = 2^x# on #[1, 4]# is #6.7#.

Hopefully this helps!