Calculus Animations,Graphics and Lecture Notes

Calculus 1 - The Second Derivative

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The Second Derivative f " (x)

There are 4 cases we want to consider. At the end of the discussions we show a general case illustrating a function which features all 4 cases.

 1. Suppose f(x)  and  f ' (x)  are increasing. Whenever the rate at which a function  changes increases we say f(x) is concave up.
Since f ' (x) is an increasing  function this means its derivative (f'(x))' or simply f" (x) > 0

f(x) incr and concave up   then f ' (x) >0  and f " (x) >0

2. Suppose f(x) is decreasing which means f ' (x) < 0. However f ' (x) can still be
increasing if the slope of the tangent line is becoming less negative. So here f " is still positive . Again we say f(x) is concave up

The point being the following 3 statements are equivalent:
    f concave up = f ' increasing   f " > 0 

The following 2 animations illustrate these 2 cases

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



The Smart Bunny-A very short story by Kurt Vonnegut Jr.

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