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
Standards:
- Identify portions of a curve that are differentiable
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 2.2.1. 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?
Instructions.
Move point A and track the slope m. Check that the secant slopes on the left approach the same value as the secant slopes on the right.
Activity 2.2.2. 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?
Instructions.
Move point A and track the slope m. Compare the secant slopes on the left to the secant slopes on the right.
