A tour of calculus : A tour of calculus i)Basic and prime theme:Concept of real numbers.
ii)Basic axiom: Between any two reals there exits a rational
iii)A profound concept:concept of ifinitesimals.
iv)An important property:Theory of continuity
v)Crucial definitions:Instantatenous speed and area under the curve-concept of function
vi)An aside:concept of limit
vii)Major theorem:The mean value theorem
viii)Fundamental theorem:the fundamental theorem of calculus
The real number System : The real number System Integers
Natural Numbers
Whole numbers
Fractions
Rationals
Irrationals
Real number system
Dedekind’s cut-irrationals : Dedekind’s cut-irrationals There is no rational number r such that r^2 = 2.Proof:(this is a proof bycontradiction)
We divide a line into two classes and have a cut at r^ = 2.class A ={ r:r^2<2}.Class B = {r:r^2>2}.
Assume that the theorem is false.There does exist a rational number whose square is 2. By definition of a rational number, that number can be expressed in the form c/d, where c and d are integers, and d is not zero. Moreover, those integers, c and d, have a greatest common divisor, and by dividing each by that GCD, we obtain an equivalent fraction a/b that is in lowest terms: a and b are integers, b is not zero, and a and b are relatively prime (their GCD is 1). (here we account for the fact that the definition of rational numbers do not end with the fact that they can be expressed as fractions …we further account for their reducibility)
Slide 4 : Now we have
[1] (a/b)^2 = 2
Multiplying both sides by b^2
, we have
[2] a^2 = 2b^2
The right side is even (2 times an integer), therefore a^2 is even. But in order for the square of a number to be even, the number itself must be even.(How?)
Therefore we can write
[3] a = 2f
Using this to replace a in [2], we obtain
[4] (2f)^2 = 2b^2
[5] 4f^2 = 2b^2 [6] 2f^2 = b^2 The left side is even, therefore b^2 must be even, and by the same reasoning as before, b must be even. But now we have found that both a and b are even, contradicting the assumption that a and b are relatively prime. Therefore the assumption is incorrect, and there must NOT be a rational number whose square is 2.
Functions : Functions i)constantsii)power functionsiii)root functionsiv)polynomial functionsv)algebraic functionsvi)exponential functionsvii)logarithmic functionsviii)trigonometric functions
Galileo’s Law of freely falling bodies : Galileo’s Law of freely falling bodies D proportional to the squre of time.ie.the distance covered by an object falling freely in the air is proportional to time.ie..D =ct^2.experiments show the value of c as 16.Distance is a number that’s inevitably positive.
Galileo’s law is redefined as P(t)=-16t^2 .why?
Zeno’s Dichotomy paradox-Limits : Zeno’s Dichotomy paradox-Limits Suppose u want to cross a room. Before u can get there, u must get halfway there. Before u can get halfway there, u must get a quarter of the way there. Before moving a fourth, u must travel one-eighth; before an eighth, one-sixteenth; and so on.
The resulting sequence can be represented as:
{…,1/16,1/8,1/4,1/2,1}
So eventually he maintains that u wont be able to reach the opposite wall of the room at all.
Limit of a sequence : Limit of a sequence A sequnce converges to a limit L, if for any postive real number epsilon, there exists some n positive , such that for a;; terms of the sequnce greater than n. the term |sn-L|Continuity : Continuity continuity can defined at a point a if
Lim f(x) = f(a).x->a
i)a should be in the domain of function fii)f(a) should be defined or make sense.iii)lim f(x) as x approaches a must be well definediv)and finally the limit shd merge with the function at that point.
Boundedness : Boundedness A function is bounded on an interval [a,b] if there exists some number N such that f(x)<=N for evry x in [a,b].An exiom that follows that “if f is continuous on an interval [a,b], then f is bounded there as well.
Betweenness : Betweenness If f(x) is continuous in [a,b] and f(a) and f(b) exists and are of opposite signs , there exists a point k in [a,b] such that f(k) = 0
Intermediate Value theorem. : Intermediate Value theorem.