Section 4.2: The Mean Value Theorem
We continue our lectures by introducing a very important theorem in calculus:
Mean Value Theorem:
Suppose that y = f (x) is continuous on the closed interval [a, b] and differentiable in its interior (a, b). Then there
is at least one point c in (a, b) at which:
f (b) − f (a)
= f 0 (c)
b−a
Basically, this tells us that the average rate of change of f (x) on an interval will be the same as the instantaneous
rate of change of f (x) at some point inside the interval, provided that the Mean Value Theorem applies.
While it is not readily obvious how this theorem is helpful, it gives us two follow-up corollaries that will aid us in
anti-derivatives, which is the next topic.
Corollary 1:
If f 0 (x) = 0 at each point x of an open interval (a, b), then f 0 (x) = C for all x in (a, b) where C is a constant.
Corollary #1 tells us what we are hoping: If a function has a derivative of zero, then the original function must have
been some constant.
Corollary 2:
If f 0 (x) = g 0 (x) at each point x in an open interval (a, b) then there exists a constant C such that f (x) = g(x) + C
for all x in (a, b).
Corollary #2 tells us that if two functions have the same derivative, then they will differ by only a constant. Thus,
there are infinitely many functions that have the same derivative, and that this infinite set of functions will only differ
by a constant. For example, there are many functions whose derivative is 2x. ( Examples of such functions are x2 ,
x2 + 8, x2 − 7, etc.)
Examples: Determine whether or not the functions listed below satisfies the hypotheses of the Mean Value Theorem
on the indicated interval.
1. f (x) = x3 ; [−1, 1].
√
2. g(x) = x − 1, [2, 5].
3. h(x) = x +
1
; [2, 3].
x
4. j(x) = x2/3 , [−1, 1].
Section 4.8: Antiderivatives
Given a function f (x) that is differentiable on an interval I, then a function g(x) is called the antiderivative of f (x)
if g 0 (x) = f (x).
As mentioned in Section 4.2, given a function f (x), there are infinitely many antiderivatives of f (x), but they all
differ from each other by a constant.
Anti-Derivative Rules:
1. Anti-Derivative Power Rules: For a real number n, we have:
xn+1
+C
n+1
• If f 0 (x) = n, then f (x) = nx + C.
• If f 0 (x) = xn , then f (x) =
(as long as n 6= −1.)
• If f 0 (x) = 0, then f (x) = C.
2. Anti-Derivative Exponential/Logarithmic Rules:
• If f 0 (x) = ex , then f (x) = ex + C
1
• If f 0 (x) = , then f (x) = ln |x| + C.
x
3. Anti-Derivative Trig Rules:
• If f 0 (x) = cos(x), then f (x) = sin(x) + C
• If f 0 (x) = sin(x), then f (x) = − cos(x) + C
• If f 0 (x) = sec2 (x), then f (x) = tan(x) + C
• If f 0 (x) = csc2 (x), then f (x) = − cot(x) + C
• If f 0 (x) = sec(x) tan(x), then f (x) = sec(x) + C
• If f 0 (x) = csc(x) cot(x), then f (x) = − csc(x) + C
Examples: Compute an anti-derivative of the following functions:
4
0
2
1. f (x) = x + 3x − 4 + 2
x
2. f 0 (x) = sin(x) + csc2 (x)
5
3. f 0 (x) = 3ex +
3x
4. f 0 (x) = (3x − 1)(2x + 3)
√
7
5. f 0 (x) = 2 x + √
x
√
13
6. f 0 (x) = x x − √
53x
4
2x − 3x3 + 5
0
7. f (x) =
7x2