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 Position
Row A
Row B
Row C
X
X
X X
X X
X X X
Second Position
X X X
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:
n = .9
n = 3.9
Limit:
9
9
9
n=
+
+
+ ...
10 100 1000
28
n
“ Behaves like” . . . . . “Very near” . . . .
What is Calculus?
•Calculus is a Language of Change.
•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
A P Calculus
• Calculus was discovered independently by
NEWTON (ENGLAND) and LEIBNIZ (GERMANY)
L’Hopital (1696)
The Analysis of the Infinitely Small
• First Text Book
• “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.
y2  y1
m
x2  x1
Slope of a Tangent Line
y2  y1
m
x2  x1
Tangent provides only one point !
Limit of the slope
( x2  x1 )
y2  y1
lim
x2  x1 x  x
2
1
Area
All our formulas require sides
that are line segments!
A=bh
A curve has none!
Area under a Curve
Total Area = Sum of
A1+A2+A3+ . . . .
Total Area = Limit as the base  x 
goes to 0 (number of rectangles
goes to infinity)
Limit
(x  0)
Last Update:
• 9/05/07