.
This is called Leibniz notation. Under this notation, the second
derivative takes this notation
.
Even though this notation looks like a fraction, it isn't. The
thing is a single entity, it cannot be broken up.4.4 The Second Derivative
The second derivative is a function which tells you the rate of change of the derivative of a function.
The second derivative of f(x) takes the notation f''(x).
If
the function is denoted by y
then the derivative notation looks like
.
This is called Leibniz notation. Under this notation, the second
derivative takes this notation
.
Even though this notation looks like a fraction, it isn't. The
thing is a single entity, it cannot be broken up.
As discussed the second derivative tells us how the first derivative is changing with respect to x. The main application for this is acceleration, the change in velocity with respect to time.
Ex 1 Calculate the second derivative.
(a)
(b)
Monotonic
Function: a function that is always increasing or always decreasing.
In order for a function to be monotonic, its derivative cannot change
sign for any value of x.
is a monotonic function since
is always positive. **Increasing or decreasing always refers to what
the function is doing as you move from left to right.**
Analyzing
where a function is increasing and decreasing is a very important
calculus application. Let's examine
and
determine where the function is increasing and decreasing. To do
this, we find all the points where the derivative is equal to zero.
These points represent the maximums and minimums, all the other
points represent the intervals or increase or decrease.
Ex
2
Use the intervals or increase and decrease to graph
.
Stationary Points
This term refers to to global minimums/maximums and local minimums/maximums as well as horizontal inflections.
A local min/max point is the lowest or highest point in a region of the curve. Any point where the derivative equals zero could be a local min or max, provided there is a sign change in the derivative.
A global min/max point is the lowest or highest possible point on a graph. Any point that is a local min/max or graph endpoints could be a global min or max. (Infinity is not a global max)
A horizontal inflection is a point where the derivative equals zero but isn't a local max/min. What occurs here is that the curve changes concavity.
Ex
3
Classify all stationary points of
p. 620#1-2 p.622#1 p. 624#1 p. 626#1-5