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Engineering Mathematics SEM – 1Concept

Polytechnic Engineering Maths Feature Image for Sem 1 Concept

Mathematics Notes for Polytechnic SEM – 1 is Designed by the ” Basics in Maths” team. Here we can learn Concepts in Basic Engineering mathematics Polytechnic Sem – I.

 This Material is very Useful for Basic Engineering Mathematics Polytechnic Sem – I Students.

By learning These Notes, Basic Engineering Mathematics Polytechnic Sem – I Students can Write their Exam successfully and fearlessly.


Engineering Mathematics SEM – 1Concept

 

LOGARITHMS

Logarithm: For ant two positive real numbers a, b, and a ≠ 1. If the real number x such then ax = b, then x is called logarithm of b to the base a. it is denoted by Polytechnic SEM - I image 1

Polytechnic SEM - I image 2Standard formulae of logarithms:

Polytechnic SEM - I image 3

Logarithmic Function:

Let a be a positive real number and a ≠ 1. The function f: (o, ∞) → R Defined by f(x) = Polytechnic SEM - I image 4

Polytechnic SEM - I image 5

Polytechnic SEM - I image 6


PARTIAL FRACTIONS

Fractions:

If f(x) and g(x) are two polynomials, g(x) ≠ 0, then Polytechnic SEM - I Partial Fractions1  is called rational fraction.

Ex:

Polytechnic SEM - I Partial Fractions 2   etc.  are rational fractions.

Proper Fraction:

A rational fractionPolytechnic SEM - I Partial Fractions1 is said to be a Proper fraction if the degree of g(x) is greater than the degree of f(x).

Ex:

Polytechnic SEM - I Partial Fractions 3  etc. are the proper fractions.

Improper Fraction:

A rational fractionPolytechnic SEM - I Partial Fractions1 is said to be an Improper fraction if the degree of g(x) is less than the degree of f(x).

Ex:

Polytechnic SEM - I Partial Fractions 4 etc. are the Improper fractions.

Partial Fractions:

Expressing rational fractions as the sum of two or more simpler fractions is called resolving a given fraction into a partial fraction.

∎ If R(x) =Polytechnic SEM - I Partial Fractions1  is proper fraction, then

Case(i): – For every factor of g(x) of the form (ax + b) n, there will be a sum of n partial fractions of the form:

Polytechnic SEM - I Partial Fractions 5

Case(ii): – For every factor of g(x) of the form (ax2 + bx + c) n, there will be a sum of n partial fractions of the form:

Polytechnic SEM - I Partial Fractions 6

∎ If R(x) =Polytechnic SEM - I Partial Fractions1 is improper fraction, then

Case (i): – If degree f(x) = degree of g(x),Polytechnic SEM - I Partial Fractions 6   where k is the quotient of the highest degree term of f(x) and g(x).

Case (ii): – If f(x) > g(x)

R(x) = Polytechnic SEM - I Partial Fractions 8

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