AP CALCULUS 1002 Introduction

Purpose of Mathematics To explain To predict To control

ALL UNDERSTANDING BEGINS WITH….. QUESTIONS and DOUBT ! ZENO’S PARODOXES (450? BCE) I.The DICHOTOMY: (Regressive Version) Motion cannot exist because before that which is in motion can reach its destination, it must reach the midpoint of its course, but before it can reach the middle, it must reach the quarterpoint, but before it reaches the quarterpoint, it must reach the eigthpoint, etc. Hence, motion can never start. (Progressive Version) It is impossible for a runner to traverse a race course. The runner must first traverse half the distance and have half the distance remaining. From there he must traverse half the remaining distance and half remains, ad infinitum. Hence motion can never end (The runner will always have half the remaining distance left.).

ALL UNDERSTANDING BEGINS WITH….. QUESTIONS and DOUBT ! ZENO’S PARODOXES (450? BCE) II. The ACHILLES: The running Achilles can never catch a crawling tortoise ahead of him because he must first reach where the tortoise started. However, when he reaches there, the tortoise has moved ahead, and Achilles must now run to the new position, which by the time he reaches the tortoise has moved ahead, etc. Hence the tortoise will always be ahead.

ZENO’S PARODOXES III. The ARROW: Time is made up of instants, which are the smallest measure of time and indivisible. An arrow is either in motion or at rest. An arrow cannot move, because for motion to occur, the arrow would have to be at one position at the start of an instant and at another at the end of the instant. However, this means that the instant is divisible which is impossible because of the definition, instants are indivisible. Hence the arrow is always at rest.

ZENO’S PARODOXES IV. The Stadium: Half the time is equal to twice the time. Take the three rows below: First PositionSecond Position Row A X X X X X X Row B X X X X X X Row C X X X X X X They begin at the first position. Row A stays stationary while rows B and C move at equal speeds in opposite directions. When they have reached the second position, each B has passed twice as many C’s as A’s. Thus it takes Row B twice as long to pass Row A as it does to pass Row C. However, the time for Rows B and C to reach the position of Row A is the same. So half the time is equal to twice the time.

Now Numbers: Limit: “ Behaves like”..... “Very near”....

What is Calculus? CALCULUS is a LIMIT MACHINE. CALCULUS takes Pre-Calculus formulas, allows all the variables to vary, then applies the LIMIT to create new concepts. Slope of a Tangent Line Area under a Curve Calculus is a Language of Change.

A P Calculus Calculus was discovered independently by NEWTON (ENGLAND) and LEIBNIZ (GERMANY) First Text Book L’Hopital (1696) The Analysis of the Infinitely Small “The Calculus takes still shots of a graph through a math projector at infinite speed.” ((as a Motion Picture produces motion with still shots running at 40 shots/ min.))

Slope Slope formula requires two points.

Slope of a Tangent Line Tangent provides only one point ! Limit of the slope

Area All our formulas require sides that are line segments! A = b h A curve has none!

Area under a Curve Total Area = Sum of A 1 +A 2 +A Total Area = Limit as the base goes to 0 (number of rectangles goes to infinity) Limit

Last Update: 9/05/07