Lesson 20 - Rolle's Theorem and MVT, Slides of Calculus for Engineers

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Lesson 9

Rolle’s Theorem and

Mean Value Theorem

ROLLE’S THEOREM

This theorem states the geometrically obvious

fact that if the graph of a differentiable function

intersects the x-axis at two places, a and b there

must be at least one place where the tangent line

is horizontal.

EXAMPLE

f   x x 5 x 4

2   

Find the two x-intercepts of the function and confirm

that f’(c) = 0 at some point between those Intercepts.

Solution:

 

at which f'   c 0.

isa pointontheinterval 1, 2 5 , soc 2 5 f' x 2 x 5 0 ; x 1,4 suchthat f' c 0' Thusweareguaranteed theexistenceof atleast one point c intheinterval thehypothesesof Rolle'sTheoremaresatisfiedontheinetrval 1,4. sincethe plolynomial f iscontinuous and differentiable everywhere, f x x 5 x 4 x 1 x 4 , sothex-interceptsarex 1 and x 4 2               

(^1 2 3 ) 1 2

• 1
• 2 x y 0 2 5 f '      

Note that the slope of the secant line joining A ( a,f(a) ) and B ( b,f(b) ) is     b a f b f a m    and that the slope of the tangent line at c in Figure 4. 5. 8 a is f’(c). Similarly, in Figure 4. 5. 8 b the slopes of the tangent lines at joining A ( a,f(a) ) and B ( b,f(b) ) is Since nonvertical parallel lines have the same slope, the Mean- Value Theorem can be stated precisely as follows

c 1 and c 2 are f'  c 1  and f'  c 2  , respectively.

EXAMPLE

Show that the function satisfies the hypotheses of the mean-value theorem over the inteval [ 0 , 2 ], and find all values of c in the interval ( 0 , 2 ) at which the tangent line to the graph of f is parallel to the secant line joining the points ( 0 , f ( 0 )) and ( 2 , f ( 2 )).   x 1 4 1 f x 3  

Solution:

1. 15 , only 1.15liesintheinterval  0,2 

3 2 3 3 4 Therefore c 3 c 4 2 0 3 1 4 3 c b a f b f a Thus f' c 4 3 c , and f' c 4 3 x But f a f 0 1 , f b f 2 3 f' x of theMean-ValueTheoremaresatisfiedwitha 0 andb 2. In particular f iscontinuouson 0,2and differentiableon 0,2 ,sothehypotheses f iscontinuous anddifferentiable everywherebecauseitisa polynomial. 2 2 2 2                      