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Increasing and Concave Up
Decreasing and Concave Up
3. Suppose f(x) is increasing but f ' (x) is decreasing.
Whenever the rate at which a function changes decreases we say f(x) is concave down .
Since f ' (x) is a decreasing function
this means f" (x) < 0
If f(x) is incr and concave down then f ' (x)
> 0 and f " (x) < 0
4. Suppose f(x) and f ' (x) are both decreasing f ' (x)
< 0.
So here f " is still negative . Again we say f(x) is concave down
The point being
the following 3 statements are equivalent:
f concave down = f ' decreasing f
" < 0
Increasing and Concave Down
Decreasing and Concave Down
Example
Now lets consider an example of all 4 cases.
Note A pt where concavity changes is called an inflection
pt. The inflection pts are the zeroes of f " (x) or equivalently the local extrema of f ' (x)
Click below to
view
Concavity A General Example
The following diagram is a summary of the relationship between a function, it's derivative,
and its second derivative.
Summary
An Application of f "(t) to 1-D motion
As we have seen given a position function s(t) the velocity v(t) = ds/dt
The rate at which velocity
changes is called the acceleration a(t) = dv/dt = d2s/dt2.
The following animation shows
the relationship between position and acceleration for a mass on a spring.
So there are 2 important things to look
for in the following animation:
1. From a math standpoint notice the relationship between the concavity of s(t)
and the sign of d2s/dt2 .
2. In Physics the most fundamental formula is Newton's 2d Law
of Motion which relates the net force on a particle to it's acceleration in 1-D motion:
F = ma in other words
the force and acceleration are in the same direction.
Notice the spring is initially compressed so the restoring
force is down and a(t) <0
As it passes through the origin the spring is now stretched and the restoring is up
and
a(t) >0. So you'll see a(t) >0 when the spring is stretched and a(t) < 0 whenever the spring is compressed
Position and Acceleration
Be Careful-- a common misconception about motion
is that a particle is speeding up when the acceleration is positive and is slowing down when the acceleration is negative. This final animation demonstrates this is not the case. A particle is speeding up when the acceleration and velocity have the same sign i.e
are in the same direction. The animation again shows the motion of
a mass on a spring. The spring is initially compressed. When the mass is released it speeds up -both the velocity and acceleration
are negative. When the mass passes through the equillibrium its velocity is still negative but the acceleration turns positive
as the restoring force is now up. The particle slows down and comes to a rest. As it starts moving up the acceleration and
velocity are both positive until the mass reaches the equiilibrium position- the mass is speeding up. Then the acceleration
turns negative while the velocity remains positive and the mass slows down coming to rest at the top of its trajectory. This is then repeated over every 2π interval.
Position - Velocity - Acceleration
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