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4.13. The Second Derivative
The second derivative f
of a function f is defined as (f
)
. Using
Leibniz notation, we write the second derivative of y = f(x) as
d
dx
dy
dx
=
d
2
y
dx
2
.
In general, we can interpret a second derivative as a rate of change
of a rate of change. The most famous example of this is acceleration,
which we define as follows.
If s = s(t) is the position function of an object that moves in a
straight line, we know that its first derivative is the velocity v(t) of
the object
v(t)=s
(t)=
ds
dt
.
The instantaneous rate of change of velocity with respect to time
is called the acceleration a(t) of the object. Thus, the acceleration
function is the derivative of the velocity function and is therefore the
second derivative of the position function:
a(t)=v
(t)=s
(t)
or, in Leibniz notation, a =
dv
dt
=
d
2
s
dt
2
.
5. More Applications of the Derivative
5.1. Certain Theorems About Differentiable Functions
Theorem 9 (Rolle’s Theorem)
If f(x) is continuous on [a, b], and it is differentiable on (a, b),
and f(a)=f(b)=0, then there is at least one number c inside (a, b)
with the property f
(c)=0.
Remark 9
It is enough to assume f(a)=f(b).
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