Definition 2.2.1. Differentiable.
A function \(f(x)\) is differentiable at \(x=a\) if and only if the derivative of \(f(x)\) exists at \(x=a.\)
Section 2.2 Discovering Differentiability
Subsection 2.2.1 Terminology
The grammar of derivatives is not uniform. The noun is derivative, the verb is differentiate, and the adjective is differentiable.
Subsection 2.2.2 Differentiable Appearance
If a curve is differentiable on all points in an interval we say the curve is differentiable on that interval.
This definition is what we expect, but it does not tell us what a differentiable curve looks like. To determine this we will use the interpretation of the derivative as the slope of the tangent line and discover what a function looks like where it is differentiable and what it looks like where it is not differentiable.
The definition of derivative is a two sided limit. This means the curve will be not differentiable when the left and right sides of that limit do not match. There will be a jump in the slopes.
Activity 12. Differentiable Points.
The goal of this activity is to visualize a derivative existing by comparing left and right secants (line connecting two points on a curve).
(a)
Using Figure 2.2.2 move point A from \(x=-1\) toward point B. Note the change of the slope m (shown on the left). To what does it appear to be approaching?
(b)
Using Figure 2.2.2 move point A from \(x=1\) toward point B. Note the change of the slope m (shown on the left). To what does it appear to be approaching?
(c)
Do the two approximations of the tangent slope above match? If so, what does this imply about the (2-sided) limit?
Activity 13. Non-Differentiable Points.
The goal of this activity is to determine what the graph of a function looks like where it is not differentiable.
(a)
Using Figure 2.2.3 move point A from \(x=-1\) toward point B. Note the change of the slope m (shown on the left). To what does it appear to be approaching?
(b)
Using Figure 2.2.3 move point A from \(x=1\) toward point B. Note the change of the slope m (shown on the left). To what does it appear to be approaching?
(c)
Do the two approximations of the tangent slope above match? If not, what kind of jump is it?
Activity 14. Appearance of Differentiable Curves.
The goal of this activity is to determine the difference in appearance between an interval on a curve that is differentiable versus one that includes a non-differentiable point.
(a)
Using Figure 2.2.2 zoom in around point B until the graph becomes very boring. What does the curve look like when zoomed in this far?
(b)
Using Figure 2.2.3 zoom in around point B the same amount as in the previous task. How does the appearance of the curve near point B change on this curve?
(c)
To see this property we have to zoom in far enough. This means the trait is true on some interval (possibly small). Where have we used this concept before in calculus?
