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Math1014 Calculus II
Average Value of A Continuous Function and Mean Value Theorem for Integrals

• Let f be a continuous function on the interval [a, b]. Divide the interval into subintervals of equal length by the
subdivision points
a = x0 < x1 < · · · < xn = b,
b−a
with ∆x = xi − xi−1 = n for i = 1, 2, . . . , n. Let ci be a sample point in the subinterval [xi−1 , xi ]. Then the average
value of the sample function values f (c1 ), f (c2 ), . . . , f (cn ) is given by
n
f (c1 ) + f (c2 ) + · · · + f (cn ) 1
n
=
b−a ∑ f (ci )∆x
i=1
! "# $
a Riemann sum

Thus the average value fave of the function f over [a, b] is defined as

1 n
% b
1
fave = lim
n→∞ b − a
∑ f (ci )∆x = b − a a
f (x)dx .
i=1

• M EAN VALUE T HEOREM FOR I NTEGRALS: If f is continuous on [a, b], then there exists a number c in [a, b] such
that % b
y

f (x)dx = f (c)(b − a) . y = f(x)

a

Geometrically, this means that the area under the graph of f over the
interval [a, b] is the same as the area of certain rectangle with based [a, b]
and height f (c) for some number c in [a, b].

a c b x

• In fact, by the Extreme Value Theorem, the continuous function f must reaches its absolute minimum at some point c1
in [a, b], and its absolute maximum at some point c2 in [a, b], thus

f (c1 ) ≤ f (x) ≤ f (c2 ) for all a ≤ x ≤ b

% b b% % b
1 1 1
f (c1 ) = f (c1 )dx ≤ f (x)dx ≤ f (c2 )dx = f (c2 )
b−a a b−a a b−a a
! "# $
a number between two function values

By the Intermediate Value Theorem, there must be a number c between c1 and c2 such that
% b
1
f (c) = f (x)dx
b−a a

Example-Exercise Find the average value fave of the function f (x) = 1 + x2 on the interval [−1, 2]. For which number c in
[−1, 2] do we have f (c) = fave ? y
6
% 2
1
fave = (1 + x2)dx 5
2 − (−1) −1
4
1& x3 '2
= x+ =2 3
3 3 −1
2
Now, f (c) = 2 if and only if 1 + c2 = 2, thus c = ±1. 1
x
−2 −1 1 2 3 4 5
 2

Example-Exercise Let s(t) be the position function of a particle moving along the s-axis. Then its velocity function is given
by v(t) = s′ (t). Find the average value of the velocity function of the particle over the time interval t1 ≤ t ≤ t2 .
By the Fundamental Theorem of Calculus,
% t2
1 1 & 't2 s(t2 ) − s(t1 )
vave = v(t)dt = s(t) =
t2 − t1 t1 t2 − t1 t1 t2 − t1

which agrees with the usual definition of average velocity in Physics.

Example-Exercise The temperature of a 10 m metal bar is 10 at one end and 30◦ C at the other end. Suppose the temperature
increase linearly from the cooler end to the hotter end. Find the average temperature of the bar. Find also the point on the
bar where the temperature is equal to the average temperature. ◦ C
T

Let T (x) be the liner temperature function of the metal bar. 30

Then the average temperature is

%
1 10 1 20
Tave = T (x)dx = (area of the trapezium under the linear graph)
10 0 10
10
1 1
= · · (10 + 30)10 = 20 ◦ C
10 2
x
10

( a+b )
Exercise Show that the average value of a linear function f on [a, b] is f 2 by working with (i) integration; (ii) graph of
f.

Exercise Suppose the population of certain species after t years from now is given by P(t) = 500000e0.05t . Find the average
value of the population in the next 20 years.