Question

What do you know about the slope when = 0 ?

A. The slope is undefined.
B. The slope is zero.
C. The slope is positive.
D. The slope is negative.

Answer 1

So, there is a sharp (discontinuous change in the slope) turn at x = 0?

Then, it's undefined. Why? Intuitively, there is no unique tangent line at that point.

Formally, the instantaneous slope is defined as the limit h->0 {[f(x+h) - f(x)]/h}. However, for a sharp turn, the limit is different when h approaches 0 from the left than when it approaches 0 from the right.

Ex: f(x) = |x| (absolute value of x)

Thus, the slope at 0 is: lim h->0 |h|/h. When approaching 0 from the left, the numerator, |h|, is positive, but the denominator, h, is negative, so |h|/h = -1. On the other hand, when approaching 0 from the right, |h|/h is 1.