a. Find the average value of ƒ shown on the interval
[1, 6] and then find the point(s) c in (1, 6) guaranteed to exist
by the Mean Value Theorem for Integrals.
Given graph,
a) To find the average of the given function without integration.
b) To find the points 'c' in (1, 6) which satisfies mean value theorem.
a) The average of the linear function occurs at the midpoint of the given interval.
The slope(m) of the given y = f(x) is,
m = (4-1) /(6-1) = 3/5
Equation => as line passes through (1, 1)
=> y - 1 = (3/5) (x - 1)
=> y = (3x/5) +(2/5)
The average of 'y' in [1, 6] occurs at x = (1+6) /2
=> at x = 3.5
And average value y(3.5) = 3.5(3/5) + (2/5)
= 2.1 + 0.4
=> Average value of f is 2.5 and occurs at x = 3.5
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