Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PRELIMINARY CALCULUS

In each case, as might be expected, the application of Rolle’s theorem does no more than
focus attention on particular ranges of values;it does not yield precise answers.

Direct verification of the mean value theorem is straightforward when it is

applied to simple functions. For example, iff(x)=x^2 , it states that there is a

valuebin the intervala<b<csuch that

c^2 −a^2 =f(c)−f(a)=(c−a)f′(b)=(c−a)2b.

This is clearly so, sinceb=(a+c)/2 satisfies the relevant criteria.

As a slightly more complicated example we may consider a cubic equation, say

f(x)=x^3 +2x^2 +4x−6 = 0, between two specified values ofx, say 1 and 2. In

this case we need to verify that there is a value ofxlying in the range 1<x< 2

that satisfies

18 −1=f(2)−f(1) = (2−1)f′(x)=1(3x^2 +4x+4).

This is easily done, either by evaluating 3x^2 +4x+4−17 atx= 1 and atx= 2 and

checking that the values have opposite signs or by solving 3x^2 +4x+4−17 = 0

and showing that one of the roots lies in the stated interval.

The following applications of the mean value theorem establish some general

inequalities for two common functions.

Determine inequalities satisfied bylnxandsinxfor suitable ranges of the real variablex.

Since for positive values of its argument the derivative of lnxisx−^1 , the mean value
theorem gives us

lnc−lna

c−a

=

1

b

for somebin 0<a<b<c. Further, sincea<b<cimplies thatc−^1 <b−^1 <a−^1 ,we
have

1

c

<

lnc−lna

c−a

<

1

a

,

or, multiplying through byc−aand writingc/a=xwherex>1,

1 −

1

x

<lnx<x− 1.

Applying the mean value theorem to sinxshows that

sinc−sina

c−a

=cosb

for someblying betweenaandc.Ifaandcare restricted to lie in the range 0≤a<c≤π,
in which the cosine function is monotonically decreasing (i.e. there are no turning points),
we can deduce that

cosc<

sinc−sina

c−a

<cosa.