Differential Calculus 7

7.1 Introduction EMCBA

Calculus is one of the central branches of mathematics and was developedfrom algebra and geometry.

Calculus is built on theconcept of limits, whichwill be discussed in this chapter. Calculus consists

of two complementary ideas: differential calculus and integral calculus.We will only be dealingwith

differential calculus in this text and will explore how it can be used to address mathematical problems

like optimisation and finding rates of change.

See introductory video:VMgxl at http://www.everythingmaths.co.za

7.2 Limits

The Tale of Achilles andthe Tortoise EMCBC

One of Zeno’s paradoxes can be summarised as:

Achilles and a tortoise agree to a race, but the tortoise is unhappy because Achilles is very

fast. So, the tortoise asks Achilles for a head-start. Achilles agrees togive the tortoise a

1000 m head start. Does Achilles overtake the tortoise?

FACT

Zeno (circa 490 BC –

circa 430 BC) was a pre-

Socratic Greek philoso-

pher of southern Italy

who is famous for his

paradoxes.

We know how to solvethis problem. We start by writing:

xA = vAt (7.1)

xt = 1000 m + vtt (7.2)

where

xA distance covered by Achilles

vA Achilles’ speed

t time taken by Achilles to overtake tortoise

xt distance covered by thetortoise

vt the tortoise’s speed

If we assume that Achilles runs at 2 m· s−^1 and the tortoise runs at 0 ,25 m· s−^1 then Achilles will

overtake the tortoise when both of them have covered the same distance. This means that Achilles

overtakes the tortoise ata time calculated as: