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Mean Value Theorem for Derivatives4.2
Teddy Roosevelt National Park, North Dakota
Photo by Vickie Kelly, 2002Created by Greg Kelly, Hanford High School, Richland, WashingtonRevised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts
Mean Value Theorem for DerivativesIf f (x) is a continuous function over [a, b] and
differentiable over (a, b),
then at some point c between a and b:
f b f af
bc
a
Mean Value Theorem for Derivatives
The Mean Value Theorem only applies toa continuous function
over a closed interval…
If f (x) is a continuous function over [a, b] and
differentiable over (a, b),
then at some point c between a and b:
f b f af
bc
a
Mean Value Theorem for Derivatives
The Mean Value Theorem only applies toa well-behaved function that is also
differentiable in the interior of the interval.
If f (x) is a continuous function over [a, b] and
differentiable over (a, b),
then at some point c between a and b:
f b f af
bc
a
Mean Value Theorem for Derivatives
The Mean Value Theorem says that the average slope across an interval equals
the instantaneous slope at a point somewhere on the closed interval (where x=c).
y
x0
( )f a
( )f b
a b
Average slope of chord
from a to b:
f b f a
b a
Slope of tangent AT c: f c
y f x
Tangent parallel to chord.
c
If the derivative of a function is always positive over an interval, then the function is increasing there.
If the derivative of a function is always negative over an interval, then the function is decreasing there.
A couple of definitions:
Note the IF and the THEN…. they are NOT interchangeable!!!
Trying to swap these is a typical misunderstanding!!!
y
x0
y f x
y g x
These two functions have the same slope at each value of x, but they are vertical translations of each other.
Functions with the same derivative differ from each other by a constant.
vertical
translation
Example 6:
Find the function whose derivative is sin(x),and whose graph passes through .
f x 0,2
cos sind
x xdx
cos( )d
xdx
So f x could be cos x , or could vary by a constant .C
cosf x x C
cos 02 C 2 1 C
Recognize that we need to have an initial value to determine the precise value of C!
cos 3f x x
3 C
( sin( ))
sin( )
x
x
The process of finding the original function by working backward from the derivative is so pivotal that it has a name:
Antiderivative
A function is an antiderivative of a function
if for all x in the domain of f. The process
of finding an antiderivative is called antidifferentiation.
F x f x
F x f x
You will hear much more about antiderivatives in the future;
this section is just an introduction!
Since acceleration is the derivative of velocity, velocity must be the antiderivative of acceleration.
Example 7b: Find the velocity and position equations for a downward acceleration of 9.8 m/sec2 and an initial velocity of 1 m/sec downward; s(0) = -3.
9.8a t
9.8 1v t t
1 9.8 0 C 1 C
19.8
1
tv t C
(Let “down” mean a negative value.)
The power rule is now running in reverse: increase the exponent by one, then divide by the new exponent. 9.8v t t C
Since velocity is the derivative of position, position must be the antiderivative of velocity.
Example 7b: Find the velocity and position equations for a downward acceleration of 9.8 m/sec2 and an initial velocity of 1 m/sec downward; s(0) = -3.
9.8a t
9.8 1v t t
2 19.8 1
2 1
t ts t C
The power rule running in reverse: increase the exponent by one, then divide by the new exponent.
Example 7b: Find the velocity and position equations for a downward acceleration of 9.8 m/sec2 and an initial velocity of 1 m/sec downward; s(0) = -3.
9.8a t
9.8 1v t t
1 9.8 0 C 1 C
9.8v t t C 29.8
12
s t t t C
24.9 1s t t t C The initial position is -3 at time zero. 2
3 4.9 0 1(0) C
3 C
24.9 1 3s t t t