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Applied Engineering Mathematics

AritmetikAritmetik

In general, an arithmetic expression, containing numbers, ( ),x, , +, , must beevaluated according to the following priorities:

If an expression contains only multiplication and division we work from left toright. If it contains only addition and subtraction we again work from left to right. Ifan expression contains powers or indices then these are evaluated after anybrackets.


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(3 + 7) 5 + (7 3) (2 4)=?= 10 5 + (4) (2)= 2 8= 6

AritmetikAritmetik


Factorial and combinatorial notation permutations and combinations

The factorial notation is a shorthand for a commonly-occurring expression involvingpositive integers. It provides some nice practice in manipulation of numbers andfractions, and gently introduces algebraic ideas. Ifn is some positive integer 1 then wewrite

n! = n(n 1)(n 2) . . . 2 1 read as n factorial.

For example5! = 5 4 3 2 1 = 120

Note that 1! = 1.Also, while the above definition does not define 0!, the convention is 0! = 1

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AritmetikAritmetikThe factorial notation is useful in the binomial theorem and in statistics. It can beused to count the number permutations ofn objects, i.e. the number of ways ofarranging n objects in a given order:

First object can be chosen in n waysSecond object can be chosen in (n 1) waysThird object can be chosen in (n 2) ways...

Last object can only be chosen in 1 way.

So the total number of permutations ofn objects is

n (n 1) (n 2) . . . 2 1 = n!

Note that n! = n (n 1)!

For 3 objects A, B, C, for example, there are 3! = 6 permutations, which are:ABC, ACB, BAC, BCA, CAB, CBA

Each of these is the same combination of the objects A, B, C that is a selection ofthree objects in which order is not important.


AritmetikAritmetikNow suppose we select just r objects from the n. Each such selection is a differentcombination ofr objects from n. An obvious question is how many different

permutations ofr objects chosen from n can be formed in this way? This number isdenoted bynPr.

It may be evaluated by repeating the previous counting procedure, but only until wehave chosen r objects:

The first may be chosen in n waysThe second may be chosen in (n 1) waysThe third may be chosen in (n 2) ways

...The rth may be chosen in (n (r 1)) ways


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Number Systems

1-) Decimal (Denary) System


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2-) Binary System


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3-) Octal System (Base 8)


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=?

4-) Hexadecimal System (Base 16)


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Example: Convert 110112 to a denary number.

Example:


Conversion of denary to binary

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AlgebraAlgebra

Algebra is that part of mathematics in which the relations and properties ofnumbers are investigated by means of general symbols. For example, the area of arectangle is found by multiplying the length by the breadth; this is expressedalgebraically asA = lxb, whereA represents the area, lthe length and b the width.The basic laws introduced in arithmetic are generalized in algebra.


AlgebraAlgebraAn algebraic expression is any quantity built up from such a finite number ofsymbols using only the arithmetic operations ofaddition, subtraction, multiplicationor division. This includes integer powers which are simply successive multiplication,and roots of variables, such as x.

Example:

Be careful to distinguish between an algebraic expression, such asx2 1, and analgebraic equation such as x2 1 = 0.

An expression tells you nothing about the variables involved, it stands alone, whereasan equation can fix the values of the variables.

Aslnda algebray u ekilde de grebiliriz.

Bir say dnn Ona 15 ekleyin Bunun iki katn aln Bunu ilk dndnz sayya ekleyin Sonucu 3e bln lk dndnz sayy kartn

Cevap?


AlgebraAlgebra

Dndmz say aBuna 15 ekleyelim a+15Bunu iki katna kartalm 2x(a+15)=(2xa)+30Buna ilk dndmz sayy ekleyelim a+(2a)+30=3a+30Bunu 3 ile blelim (3a+30)/30=a+10Bundan ilk dndmz sayy kartalm a+10-a=10


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Example:


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Example:


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AlgebraAlgebra

Example:


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Laws of Indices


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Example:


Limits of Function Values

Le t (x) be defined on an open interval about except possibly at itself. If (x) getsarbitrarily close to L (as close toL aswe like) for all x sufficiently close towe saythat approachesthe limit L asx approachesand wewrite

which is read the limit of (x) as x approaches is L. Essentially, the definition says thatthe values of (x) are close to the number L whenever x is close to (on either side of ).

rnek: fonksiyonu x=1 civarnda nasl davranr?

The given formula defines for all real numbers x except (we cannot divide by zero). Forany we can simplify the formula by factoring the numerator and canceling commonfactors:

The graph of isthus the l inex+1with the point (1, 2) removed.

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We say that (x) approaches the limit 2 as xapproaches 1, and write

The function has limit x2 as even though is notdefined at x=1.

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Bir baka rnek olarak:

The function g has limit 2 as x-> 1 even though 2 g(1).

This holds, for example, whenever (x) is an algebraic combination of polynomials andtrigonometric functions for which f(x0) is defined.

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4 3 10-3=7

The Identity and Constant Functions Have Limits at Every Point

Some ways that limits can fail to exist are illustrated below.

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rnek:

So what is the answer? Is i t0 .05 or0, or someother value?

Problems such as these demonstrate the powerof mathematical reasoning, once it is developed,over the conclusions we might draw frommaking a few observations. Both approaches

have advantages and disadvantages inrevealing natures realities.

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Calculating Limits Using the Limit Laws

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rnekler

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Theorem 2: Limits of Polynomials Can Be Found by Substitution

Theorem 3: Limits of Rational Functions Can Be Found by Substitution If the Limitof the Denominator Is Not Zero

rnek:

We cannot substitute because it makes the denominator zero. We test the numerator tosee if it, too, is zero at It is, so it has a factor of in common with the denominator.Canceling the gives a simpler fraction with the same values as the original for x1

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rnek:

We cannot subs ti tu te x=0 and the numerator and denominator have no obv iouscommon factors. We can create a common factor by multiplying both numerator anddenominator.

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The Sandwich Theorem

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One-Sided Limits

To have a limit L as x approaches c, a function must be defined on both sides of c andits values (x) must approach L as x approaches c from either side. Because of this,ordinarylimits are called two-sided.If fails to have a two-sided limit at c, it may still have a one-sided limit, that is, a limit ifthe approach is only from one side. If the approach is from the right, the limit is a right-hand limit. From the left, it is a left-hand limit.

has limit 1 asx approaches 0 from the right, and limitasx approaches 0 from the left. Since these one-sided limit values are not the same, there is no singlenumber that (x) approaches as x approaches 0. So(x) does nothave a (two-sided) limit at 0.


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rnek:

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Limits Involving (sin )/

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Theorem


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Asymptotes

Horizontal Asymptote

Bir nceki rnekteki fonksiyonuna geri dnersek

has the line as a horizontal asymptote on y=5/3 both the right and the left because

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rnek: Finding an Oblique Asymptote

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Vertical Asymptote

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Continuity (Sreklilik)Continuity (Sreklilik)

Whenwe plot functionvaluesgenerated in a laboratoryor collected in the field, we oftenconnectthe plottedpoints with an unbrokencurve to showwhat the functionsvalues arelikely to have been atthe times we did notmeasure.

In doing so, we are assuming that we are working with a continuous function, so itsoutputs vary continuouslywith the inputs and do not jump f rom one value to another w ithout tak ing on the values in between. The l im it o f a continuous funct ion as xapproaches c canbe foundsimply bycalculatingthe value of the function at c.

Continuity at a PointThe function is continuous at every point in itsdomain [0, 4] except at x=0, x=2 and x=4. At thesepoints, there are breaks in the graph. Note therelationship between the limit of and the value of ateach point of the f unctions domain. Note therelationship between the limit of and the value of ateach point of the functions domain.


To define continuity at a point in a functions domain, weneed to def ine continu it y at an inter ior point (whichinvolves a two-sided limit) and continuity at an endpoint(which involvesa one-sided limit)




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Polynomial and Rational Functions Are Continuous



Tanjantlar ve Trevler (Tangents and Derivatives)Tanjantlar ve Trevler (Tangents and Derivatives)



The dynamic approach to tangency. The tangent to the curve at P is the line through Pwhose slope is the limit of the secant slopes as Q: P from either side.

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DifferentiationDifferentiation




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rnek: Aadaki ekildeki gibi bir y=f(x) fonksiyonunun trevini iziniz.

What can we learn from the graph of Ata glancewe can see

1. where the rate of change of is

positive, negative, or zero;2.the rough size of the growth rateat any x and its size in relation tothesizeof (x);

3.where the rate ofchangeitself isincreasing or decreasing.




rnek:

There can be no derivative at the origin because the one-sided derivatives differ there:



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Applied Engineering Mathematics Applied Engineering Mathematics

Applied Engineering Mathematics Applied Engineering Mathematics


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