Math 160 003

Welcome to Dr. Shannon’s class

http://faculty.salisbury.edu/~kmshannon/math160/welcome.htm

" Calculus is an incredibly powerful tool which has facilitatedtechnological triumphs ... and a beautiful monument to two and a halfmillennia of human collaboration, invention and imagination.

" No one today who wishes to be a well rounded, liberally educatedcitizen should remain ignorant of its beauty and utility.

" In fact, by the 17th century, . . . accusations of plagiarism wererampant. . The most famous dispute was between Newton andLeibnitz

" Newton was perfectly correct in claiming that he stood “on theshoulders of giants.”

" Newton and Leibnitz gathered a myriad of different methods andapproaches and molded them into a single logical framework whichNewton called Fluxions and we call the Calculus.

Introduction

" Newton & Leibnitz were hampered by : the lack of a clear, rigorous formulation of the real number system, infinity and infinitesimals the difficulties inherent in blending geometry, algebra and arithmetic

" The proofs, etc. for the Calculus are mostly algebraic and arithmetic but theinspiration and intuition were primarily geometric. It is not surprising that ittook an entire century after Leibnitz and Newton to completely formulate thelogical, rigorous foundations.

" In the 17th century Viete “realized the facility to be gained in the handling ofgeometric problems by their reduction to the solution of algebraic equations ..“

" Descartes is generally credited with the development of Analytic Geometrypublished in his famous Geometrie of 1637 and the primary coordinatesystem used throughout mathematics is called Cartesian.

" Fermat, in his solution of tangent, area, and maximization problems verynearly approached the Calculus but he did not discover the FundamentalTheorem of Calculus.

Introduction Continued

Excerpt from a tour of the calculus by David Berlinski

As its campfires glow against the dark, every culture tells stories to itself about how the gods lit up the morning sky and set the wheel of being into motion. The great scientific culture of the west - our culture - is no exception. The Calculus is the story this world first told itself as it became the modern world.

The sense of intellectual discomfort by which the calculus was provoked into consciousness in the seventeenth century lies deep within memory. It arises from an unsettling contrast, a division of experience. Words and numbers are, like the human beings that employ them, isolated and discrete; but the slow and measured movement of the stars across the night sky, the rising and setting of the sun, the great ball bursting and unaccountably subsiding, the thoughts and emotions that arise at the far end of consciousness, linger for moments or for months, and then, like barges moving on some sullen river, silently disappear – these are, all of them, continuous and smoothly flowing processes. Their parts are inseparable. How can language account for what is not discrete and number for what is not divisible?

Space and time are the great imponderables of human experience, the continuum within which every life is lived and every river flows. In its largest, most architectural aspect, the calculus is a great,even spectacular theory of space and time, a demonstration that in the real numbers there is an instrument adequate to their representation. If science begins in awe as the eye extends itself throughout the cold of space, past the girdle of Orion and past the galaxies pinwheeling on their axes, then in the calculus mankind has created an instrument commensurate with its capacity to wonder.

1. Zeno’s Paradox(es)Discussion of the paradox of Achilles and theTortoise with reference to infinite sums

A B

C

A B

C

A B

C

2. The Area Problem" Beginning with a discussion of what area is “Think of area aslength times width”" Indicating how it is then possible to find the area of any regionwith straight line borders." Following with a discussion of the Method of Exhaustion used byArchimedes and his result that the area under a parabolic arch is4/3 the area of the inscribed triangle. " Foreshadowing the Fundamental Theorem of Calculus byreference to an easier, more generalizable way of computing areassimilar to a heuristic that Archimedes used but did not trust. [Boyerpps. 50 - 52]

Four Problems

We will leave this to the year long calculus sequence

We will tackle this one last….

RadiusTANGENT

Curve

HEIGHT

TIME

HEIGHT vs. TIME

Tangent, Slope = 320

3200

1600

10

3. The Tangent or Velocity Problem

Revenue

Quantity

We will begin with this problem

Area

Length25 50

625

4. Optimization Problems

The biggest, smallest, cheapest, most profitable, fastest, etc……

Profit

Price

Cost per

Item

Quantity Produced

It turns out that solving the tangent problem will help us solve this problem so we will attack this second.

Some of the tools developed to facilitate solution of these problems, you have already seen. In particular, let’s talk about functions and “analytic geometry” or the Cartesian Plane.

(x,y)

{ }y

x

Independent VariableOften

(but not always) called x

Dependent VariableOften

(but not always) called y

Function

y = f(x)