Mean Value Theorem and Rolles Theorem – An Introduction
AP Calculus
Name:
1) Draw the graph of any continuous and differentiable function you want in the interval [ -5 , 5]
2) Draw the line that connects the end-points of the function
a) What is the name of that line?
b) What does the slope of that line represent?
3) Is there at least one x-value in the interval [-5,5] whose tangent line is parallel to that secant line?
MEAN VALUE THEOREM
The instantaneous rate of change of a function, F’(x), will equal that function’s average rate of change over
some interval [a,b] ,
F (b)  F (a )
, . . . . .at least once at x = c as long as the function is continuous and
ba
differentiable in that interval [a,b] and c will be in the interval [a,b]…symbolically it can be written as
F’(c) =
F (b)  F (a )
ba
1) Find the c value(s) guaranteed by the mean value theorem given each
a) f(x) = 2 – 3x2 [-1,2]
b) f(x) =
2  x [-7,2]
c) f(x) = ln(2x+1) [0,
e 1
]
2
d) a. Why wouldn’t the mean value theorem apply to f ( x) 
x 1
on [-1,2]?
x 1
3) A special case of the mean value theorem is called Rolle’s theorem and it is basically the mathematics behind
the phrase “what goes up must come down” Rolle’s theorem states: Given a function f(x), if it is continuous
and differentiable over the closed interval [a,b] and f(a) = f(b) then there is at least one point x = c where
f (c)  0 .
a) Can you see how Rolle’s theorem is a special case of the MEAN VALUE THEOREM? Discuss or
show how…
b) Determine if Rolle’s theorem applies and if it does, use it to determine the c-value guaranteed given
2
3
f(x) = x  1 [-8,8]