MATHEMATICS  THEORY



In this section, the Test for Concavity is introduced along with an application. 





Concavity

Increasing Function I
Increasing Function II 

The concept of increasing and decreasing function has
been introduced in the
Monotonic Function section. However, an increasing function can have two
different bend directions as shown in Increasing Function I and Increasing
Function II graphic respectively. In order to distinguish the
two kinds of bend direction, the concept of concave upward and cave
downward is introduced.
An arc is concave upward if the arc lays above the tangent line for
every point in the function's domain. For example, the curve in Increasing
Function I is concave upward.
Similarly, an arc is concave downward if the arc lays below the tangent
line for every point in the function's domain. For example, the curve
in Increasing Function II is concave downward. 



Function s = 20 sin5t 

In engineering, data curves and relationships
are complex and in general composed of both concave upward and concave
downward. In the
Higher Order Derivatives section, the displacement and time relationship
of a springmounted slider was introduced as s = 20 sin5t. According
to the concavity definition, the curve is concave downwards from 0 to π/5,
but concave upward form π/5 to 2π/5.
To help find the concavity of a curve, the Test for Concavity can be
used. 





The Test for Concavity



The test for concavity states:
Suppose a function, f(x), is twice differentiable on an interval R.

If the second derivative of the function,
df^{2}(x)/dx^{2 }, is larger than 0 for all x in R, then
the graph of the function is concave upward on R.

If the second derivative of the function,
df^{2}(x)/dx^{2 }, is less than 0 for all x in R, then
the graph of the function is concave downward on R




Function d^{2}s/dt^{2} = 500 sin5t 

The test for concavity can be better understood
by examining the displacement and time relationship of a springmounted
slider. First, assume the displacement over time is given by s = 20 sin5t.
The first derivative of the displacement function with respect to time
is
ds/dt^{} = 100 cos5t
The second derivative of the displacement function with respect to time
is
d^{2}s/dt^{2} = 500 sin5t
This expression is negative when t is between 0 and π/5.
According to the test of concavity it is concave downward in this range.
On the other hand, This expression is positive when t is between π/5
and 2π/5, and thus it is concave upward in
this interval. This result is confirmed by Function s^{ } =
20 sin5t graphic. 





Inflection Points



The curve of springmounted slider function changes from
concave downward to concave upward when t = π/5.
In engineering this point is known as an inflection point. The inflection
point is the point where the curve changes from concave upward to concave
downward or from concave downward to concave upward. 


