Calculus I Notes, Section 2-8

Notes, Lesson 2.8
The Derivative as a Function

As we have been calculating general derivatives, we have been getting algebraic results. These general derivative results form a new function which itself has all of the properties and characteristics of a function.

Here we take a look at the function

, and its derivative,

. The function is plotted in red, and the derivative is plotted in green. The derivative is itself a function. We have to remember what the derivative (and its curve) tell us. We note several things:

1) The derivative never negative. This tells us that the function is never decreasing.

2) The derivative is zero only once. This tells us that at x = 0, the slope of the function is zero. (For this cubic function this means that there is a point of inflection at x = 0.

3) The slope of the derivative curve first decreases, and then increases, but never goes below zero. This tells us that for the beginning of the curve (x < 0) the slope of the function is decreasing. For the second "half" of the curve (x > 0) the slope of the function is increasing.

If you have the Journey Through Calculus CD, load and run MResources/Module 3/Derivatives as Functions/Mars Rover. This module shows an animation of the relation between a function and its derivative..

If you have the Journey Through Calculus CD, load and run MResources/Module 3/Slope-a-Scope/Derivative of a Cubic. This module shows another animation of the relation between a function and its derivative.

Approximating a derivative from a function's table of values. In scientific research, all we often have is some data, and no equation. How can we approximate a derivative in this situation?

	The first two columns at left are scientific data regarding the number of yeast cells in a new laboratory culture after a certain number of hours. The last column is the slope or average rate of change from point 1 to point 2, etc.
	Here is a plot of the ordered pairs in the first two columns from the above chart. Even though this is discrete data, we can make a scatterplot of the data and get a feel for what this might look like in continuous form.
	If we plot both the set of ordered pairs in the first two columns (blue), and the set of ordered pairs with the first column paired with the last column (pink) we get the two plots at the left. Here we can also see what the derivative might look like if it was continuous. (pink) This pink set of discrete points (if connected) show that: 1) The derivative of this function is always positive, 2) The derivative is increasing up to about 10 hours, and then decreasing, 3) The derivative appears to be approaching zero, but never arriving at it (hours>15).
Important Note:	This technique assumes that there are not significant fluctuations between the given points of information.


Isaac Newton co-discoverer of calculus		Gottfried Wilhelm Leibniz, co-discoverer of calculus

Leibniz Notation

As we found in Lesson 2.6, the "f-prime" (f '(x)=...) notation for a derivative is only one way to signify differentiation. We were also introduced to Leibniz notation. Leibniz was one of the two co-discoverers of the calculus. A comparison of Leibniz notation and f-prime notation is shown below:

	Leibniz notation:	f-prime notation:	Alternate f-prime notation
1st derivative
2nd derivative
3rd derivative

An application of different degree derivatives that is commonly used in science is the following:

Scientific Application	Calculus Term	Notation	Example
Displacement or Distance Function	original function
Velocity	1st derivative
Acceleration	2nd derivative
Jerk	3rd derivative

Definition:

Differentiable

A function is differentiable at c if f '(c) exists. To be differentiable over an interval, it must be differentiable at every point in the interval. Differentiate is the verb form of derivative. For f '(c) to exist, f (c) must be defined and

Differentiability and Continuity

Using your Tools for Enriching Calculus CD (that came with your book), load and run Module 2.8A. This module guides you in determining properties of the derivative f ' by examining the graphs of a variety of functions f.

Using your Tools for Enriching Calculus CD (that came with your book), load and run Module 2.8. This module will let you see how changing the coefficients of a polynomial f affects the appearance of the graphs of f, f ', and f ''..

Check Concepts

#1: Leibniz notation is named for ....

#2: To be differentiable, all three of these must agree: 1) the left-hand limit, 2) the right-hand limit, and 3) the value of the function at that point.

#3: For a limit to exist, all three of these must agree: 1) the left-hand limit, 2) the right-hand limit, and 3) the value of the function at that point.

#4: True or False. A derivative can itself be regarded as a function on its own.

#5 For a function to be differentiable over an interval, it must be differentiable at __________ points in the interval.