Name ___________________
4.2 The Mean Value Theorem
AP Calculus AB
Chapter 4 day 2
Mean Value Theorem
(for derivatives)
If f(x) is continuous at every point of the interval [a, b] and differentiable at every
point of its interior (a, b), then there is at least one point c, in (a, b) at which:
f ' ( c) 
f (b)  f (a )
ba
Example 1: Show that the function 𝑦 = 𝑥 2 , satisfies the requirements for the Mean
Value Theorem for derivatives on the interval [0, 2] then find a solution, c, to the
equation:
f (b)  f (a )
f ' ( c) 
ba
Name ___________________
4.2 The Mean Value Theorem
AP Calculus AB
Chapter 4 day 2
Example 2: 1993 AB 3 a
Consider the curve 𝑦 2 = 4 + 𝑥 and chord AB joining points A (-4,0) and B (0,2) on the
curve.
Find the x- and y- coordinates of the point on the curve where the tangent line is parallel
to the chord AB.
Example 3: Multiple Choice question
If c is the number that satisfies the conclusion of the Mean Value Theorem for
f(x)= x3-2x2, on the interval 0 ≤ x ≤ 2, then c=
(A) 0
(B) ½
(C) 1
(D) 4/3
(E) 2
Example 4: Free response Question
Cal says that according to the Mean Value Theorem, it is not possible to find a
polynomial function such that: f(0) =-1 , f(2) =4, and f '( x) ≤ 2 for all x in the interval
[0, 2]. Explain how Cal might support his argument both numerically and graphically.
Name ___________________
4.2 The Mean Value Theorem
AP Calculus AB
Chapter 4 day 2
Increasing/Decreasing Functions:
Definition: If f is defined on the interval I and x1 and x2 are any two points in I then:
1) f increases on I if when x1  x2  f ( x1 )  f ( x2 )
2) f decreases on I if when x1  x2  f ( x1 )  f ( x2 )
f(x1)
f(x2)
f(x2)
f(x1)
X1
X2
X1
I
Theorem: If f is continuous on [a, b] and differentiable on (a,b) then:
1) If f ' > 0 at each point of (a, b) then f increases on [a,b]
2) If f ' < 0 at each point on (a,b) then f decreases on [a, b].
Example 2: Where is f(x)= x 3  4 x increasing and where is it decreasing?
Homework
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