Project in CalculusSTART

Members:

Angelo Gabriel FavisCarl John MellaJoselito Monzon

Patrick Lotan Katriel PazJazon RoqueGuile Sabino

John Paulo TolentinoJaine Cruz

Danica Eve EsquilonaCheska Topacio

Contents:Contents

Review*Functions*Zeno*Mathematicians*Real Number System*Inequality*Interval Notation

LimitsDerivativesProduct and Quotient RuleTypes of Infinity

Review

In This section we’re going to make sure that you’re familiar with functions and function notation. Both will appear in almost every section in a Calculus class and so you will need to be able to deal with them.

Functions

FunctionsWhat exactly is a function? An equation will be a function if for any x in the domain of the equation (the domain is all the x ’ s that can be plugged into the equation) the equation will yield exactly one value of y.

This Topic needs an Example to understand more. here is the example:

Example 1 Determine if each of the following are functions.(a)  y = x2 + 1(b)  y2 = x + 1Solution(a) This first one is a function. Given an x, there is only one

way to square it and then add 1 to the result. So, no matter what value of x you put into the equation, there is only one possible value of y.

y2=3+1=4 (b) The only difference between this equation and the first

is that we moved the exponent off the x and onto the y. This small change is all that is required, in this case, to change the equation from a function to something that isn’t a function.

To see that this isn’t a function is fairly simple. Choose a value of x, say x=3 and plug this into the equation. Now, there are two possible values of y that we could use here. We could use y = 2 or y = -2 . Since there are two possible values of y that we get from a single x this equation isn’t a function.

Paradox- logical steps which gives an illogical conclusion / contrary to

intuitionMethod of Exhaustion - approximation technique

area of 0 ≈ area of n-gon

Zeno

Infinite number of points

Finite length

∞

A B

Deals with infinity- Jean Le Rond d’Alembert

Dichotomy

Infinity

Limits

-English Mathematician-Born in 1642-Differential approach -Realized that in using infinite series in approximation gave the exact value

Isaac Newton

Approximation ∞ exact-Discovered the inverse relationship of the slope of the tangent to a curve and the area under the curve -Wrote in 1687. Philosophiae Naturalis Principia Mathematica Principia Calculating m of tangent line to a curve, v c+1-Find the velocity and calculation ac+1 function from position function s c+1-Calculation arc lengths and volume of solids -Calculation relative and absolute extrema -Calculating m of tangent line to a curve, v c+1-Find the velocity and calculation ac+1 function from position function s c+1-Calculation arc lengths and volume of solids -Calculation relative and absolute extrema

-German polymath -Born in1646-Integral approach -Published own version of calculus before Newton-Superior notation

dx dy

Gottfried Wilheim Leibniz

Real Number System

Counting Numbers, IN = {1,2,3…}Whole Numbers, W= {0,1,2,3,…}Integers, Z={…..-2,-1,0,1,2,…}Rational Numbers, R= {1/2,-3/4,0.45,-03.23,0.4444,……,-2.75}-rational can be expressed as a simple functionTerminating DecimalsNon-terminating, separating decimalsIrrational Numbers, Q1 = { √2,∏,e}Non-terminating, non-separating decimals

Set- wall destined-defined collection of objects

Elements-objects that make up set

{} A={1,2,3,4} roster methodA = {x/x is a counting number less than 5}Seven that set- builder notational A is a subset of B it all elements at A one element of BA& B

Def: If a,b є IR1.) a<b if b-a is positive2.) a>b if a-b is positive3. ) a>b if a=b or a>b4.) a<bif a=b or a<b

Inequalities

Theorem of Inequalities1. If a > 0 and b> 0, then a + b > 02. If a > 0 and b> 0, then ab > 03. Transitive Property of Inequality• If a > b and b < c, then a < c4. If a < b, then a + c < b + c5. If a < b and c > 0, then ac < bc

If a < b and c < 0, then ac > bc

Interval notation1. Open interval

( a, b) = {x/a < x < b }2. Closed Interval

( a, b) = { x/a ≤ x ≤ b}3. Half-Open Interval

[ a , b) = { x/a ≤ x < b}( a, b] = { x/a} < x ≤ b}

Limits

In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows one to, in complete space, define a new point from a Cauchy sequence of previously defined points. Limits are essential to calculus and are used to define continuity , derivatives and integrals.

Definitions

The concept of the limit of a function is further generalized to the concept of topological net, while the limit of a sequence is closely related to limit and direct limit in category theory.

In formulas, limit is usually abbreviated as lim as in lim(an) = a or represented by the right arrow (→) as in an → a.

The derivative of f(x) with respect to x is the function f’(x) and is defined as,

f’(x)=lim f(x+h)-f(x)

h 0 h

Limit of a Sequence

Limit of a SequenceConsider the following sequence: 1.79, 1.799, 1.7999,... We could observe that the numbers are "approaching" 1.8, the limit of the sequence.Formally, suppose x1, x2, ... is a sequence of real numbers. We say that the real number L is the limit of this sequence and we write to mean

For every real number ε > 0, there exists a natural number n0 such that for all n > n0, |xn − L| < ε.

Lim xn = Ln ∞

Intuitively, this means that eventually all elements of the sequence get as close as we want to the limit, since the absolute value |xn − L| is the distance between xn and L. Not every sequence has a limit; if it does, we call it convergent, otherwise divergent. One can show that a convergent sequence has only one limit.The limit of a sequence and the limit of a function are closely related. On one hand, the limit of a sequence is simply the limit at infinity of a function defined on natural numbers. On the other hand, a limit of a function f at x, if it exists, is the same as the limit of the sequence xn = f(x + 1/n).

One-Sided Limit

One-Sided Limit

In calculus, a one-sided limit is either of the two limits of a function f(x) of a real variable x as x approaches a specified point either from below or from above. One should write either: for the limit as x decreases in value approaching a (x approaches a "from the right" or "from above"), and similarly for the limit as x increases in value approaching a (x approaches a "from the left" or "from below").

The two one-sided limits exist and are equal if and only if the limit of f(x) as x approaches a exists. In some cases in which the limitdoes not exist, the two one-sided limits nonetheless exist. Consequently the limit as x approaches a is sometimes called a "two-sided limit". In some cases one of the two one-sided limits exists and the other does not, and in some cases neither exists.

Lim f(x) or lim f(x)x a + x a

Lim f(x) or lim f(x)x a x a

Lim f(x)x a

CONVERGENCE AND FIXED POINT

A formal definition of convergence can be stated as follows. Suppose pn as n goes from 0 to  ∞   is a sequence that converges to a fixed point p, with  λ  for all n. If positive constants λ and α exist with

lim = λn ∞

then pn as n goes from 0 to   converges to p of order α, with asymptotic error constant λGiven a function f(x) = x with a fixed point p, there is a nice checklist for checking the convergence of p.1) First check that p is indeed a fixed point:f(p) = p2) Check for linear convergence. Start by finding  .f’(p) If....                                                    3) If we find that we have something better than linear we should

check for quadratic convergence. Start by finding  f’”(p) If....                                                          

Pn+1-PPn - P a

Derivatives

the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity

Definition:

Derivative Rules 

d dx

ddx

ddx

c = 0 , where c = constant

(Ax) = A, where a = constant

(uv) = uv’ + vu’

ddx

(xa ) = axa-

1

ddx

ux

= xu’ – ux’x2

Derivative Rules 

d (sin(x))=cos(x)

dx

d (sec(x))=sec(x)tan(x)

dx

d (tan(x))=sec

2(x)

dx

d (cos(x))=sin(x)

dx

d (csc(x))=csc(x)cot(x)

dx

d (cot(x))=cos

2(x)

dx

Product Rule

Product Rule

-A formula used to determine the derivatives of products of functions.

The definition of a derivative

(fx(x)g(x))’ =lim f(x+h)g(x+h) – f(x)g(x)H 0 h

Factoring a f(x+h) on the first limit and g(x) from the second limit we get:

(fx(x)g(x))’ =lim f(x+h)[g(x+h) – g(x)]

+ lim g(x)[f(x+h)-f(x)

h 0 h h 0 h

The key is to subtract and add a term:f(x+h)g(x)

By doing this, you can get the following:

(fx(x)g(x))’ =lim f(x+h)g(x+h) – f(x+h)g(x)+f(x+h)g(x)-f(x)g(x)

h 0 h

Based on the property of limits, we can the limit can be broken into two because the limit of a sum is the sum of the limits.

(fx(x)g(x))’ =lim f(x+h)g(x+h) – f(x+h)g(x)

+ lim f(x+h)g(x)-f(x)g(x)

h 0 h h 0 h

(f(x)g(x))’ =lim f(x+h)lim g(x+h)-g(x)

+ lim g(x)lim f(x+h)-f(x)

h 0 h h 0 h 0 h

Another property of limits says that the limit of a product is a product of the limits. Using this fact, the limit can be written like this:

By definition

lim f(x+h)-(x+h) – f(x) =f’(x)h 0 h

And

lim g(x+h)-g(x) =g’(x)h 0 h

The Quotient Rule

If the functions f and g are differentiable at x, with g(x) 0, then the quotient f/g is differentiable at x, and

The Quotient Rule

d f f’ (x)g(x) – f(x)g’(x)dx g (x) = [g(x)]2

lim f(x+h) = f (x)h 0

lim g(x) = g (x)h 0

(f(x) g (x))’ = f (x) g’(x) + g (x) f’(x)

Also,

And

Since they do not defend on h

And the last one, what is supposed to be shown:

Proof By the definition of the derivative

d f f’ (x+h) f(x)dx g (x) =

lim g (x+h)

- g (x)

h

= lim f(x+h)g(x) – f(x)g(x+h) h 0 g(x+h)g(x)h

= lim f(x+h)g(x) – f(x)g(x) + f(x)g(x) – f(x)g(x+h) h 0 g(x+h)g(x)h

= lim f(x+h) – f(x)g(x) - f(x)[(g(x+h) – g(x)] h 0 g(x+h)g(x)h

= lim f(x+h) -f(x) g(x) - f(x) g(x+h) – g(x)

h 0 h

g(x+h)g(x) h

h 0

If we recognize the difference quotients for f and g in this last expression, we see that taking the limit as h0 replaces them by the dreivatives f'(x) and g'(x). Further, since g is differentiable, it is also continuous, and so g(x+h)g(x) as h0. Putting this all together gives

And that is the quotient rule.

d f f’ (x)g(x) – f(x)g’(x)dx g (x) = [g(x)]2

Types of infinity

means that f(t) does not bound a finite area from a to b

f (t) dt = ∞∫b

a

means that the area under f(t) is infinite.

f (t) dt = ∞∫∞

-∞

means that the area under f(t) equals 1

f (t) dt = 1∫∞

-∞

Thank You!HOPE YOU LEARN MORE