AP Calculus AB: Section 4.2
Intermediate Value Theorem for Derivatives: A function y = f ′(x) that is continuous on a closed interval [a, b] MUST
take on every value between f ′(a) and f ′(b).
Mean Value Theorem: If f(x) is continuous at every point of the close interval [a,b] and differentiable at every point off
its interior (a,b), then there MUST be at least one point c in (a,b) at which
f (c) 
f (b)  f (a)
ba
Examples:
1. Show f(x) = x2 satisfies the hypotheses of the MVT on the interval [0,3] and find each value c that satisfies the MVT.
2. Show f ( x) 
MVT.
x satisfies the hypotheses of the MVT on the interval [0,9] and find each value c that satisfies the
For problems 1-8: first show that the function satisfies the hypotheses of the MVT on the given interval [a, b], then find
each value of c in (a, b) that satisfies the equation f (c) 
1. f ( x)  x 2  2 x  1,
3. f ( x)  ln( x  1),
5. f ( x)  x3 ,
7. f ( x) 
x
,
x2
[0, 1]
[2, 4]
[4, 5]
[1, 4]
f (b)  f (a)
.
ba
2
2. f ( x)  x 3 ,
4. f ( x)  x 1 ,
[0, 1]
[2,8]
6. f ( x)   x 1 x  3 ,
  
8. f ( x)  sin x, - , 
 2 2
[1,3]