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Average Value of a Function
AP Calculus AB Ā· Unit 8 Integration Applications
Learning Objectives
Calculate f_avg = (1/(bāa)) ā«āįµ f(x) dx
Interpret as height of equal-area rectangle
Find c where f(c) = f_avg (MVT for Integrals)
Apply to real-world contexts (velocity, temperature)
Formula
f
avg
= 1/(bāa) Ā· ā«āįµ f(x) dx
MVT for Integrals:
ā c ā [a,b] such that f(c) = f
avg
Rectangle area = f
avg
Ā· (bāa) = ā«āįµ f(x) dx
Tags
average value
MVT integrals
area
FUNCTION
f(x) = xĀ² [0,3]
f(x) = sin(x) [0,Ļ]
f(x) = āx [1,4]
f(x) = 4āxĀ² [0,2]
a
0
b
3
ā« f(x) dx
ā
f avg
ā
c (MVT)
ā
b ā a
ā
The
blue shaded area
under f(x) equals the
green rectangle
area. The green line shows f
avg
ā the "height" that creates an equal-area rectangle.