## Monthly Archives: August 2016

### When Does a Function NOT have a Derivative at a Point?

A function has a derivative at a point $x_{0}$ if the slopes of the secant lines through $P(x_{0}, f(x_{0}))$ and a nearby point Q on the graph approach a limit as Q approaches P. Whenever the secant fail to take a limiting position or become vertical as Q approaches P, the derivative does not exist. A function, whose graph is other wise “smooth” will fail to have a derivative at a point where the graph has:

1. a Corner, where the one-sided derivatives differ
2. A Cusp where the slope of PQ approaches $\infty$ from one side and $-\infty$ from the other.
3. A Vertical Tangent, when the slope of PQ approaches $\infty$ from both sides or approaches $-\infty$ from both sides.
4. A Discontinuity.

More later,

Nalin Pithwa

### Two versions of Zeno’s paradox

1. The story of the hare and the tortoise is well-known. The hare lost because he took a long nap on the way. But, here is an argument that shows that even if the hare were running fast, he could not overtake the tortoise, provided that the tortoise was ahead to begin with. Suppose the hare was at a point A and the tortoise was at point B ahead of A on the race track. By the time, the hare reached B, the tortoise would have moved to C ahead of B. Again when the hare reaches C; the tortoise would have moved to D ahead of C. And, so on, ad fnfinitum. So, the tortoise will always be ahead of the hare! What is the fallacy in this argument? This is one of the paradoxes of the Greek philosopher Zeno relating to infinity.
2. Here is another of Zeno’s paradoxes. A runner wishes to cover a distance of 1 km. Zeno uses the following argument to show that he/she will never succeed if, at each stage, he/she aims at covering half the remaining distance. For to do so, first he/she will have to cover half the remaining distance. That leaves $\frac{1}{4}$ km, of which he/she next covers half the distance. if he/she adopts the strategy of covering half the remaining distance each time. He/she has an infinity of operations lying ahead of him/her, with the destination always lying just that little bit ahead. What is wrong with this reasoning?

Ref: Fun and Fundamentals of Mathematics, Jayant V Narlikar and Mangala Narlikar, Universities Press.

More fun later,

Nalin Pithwa