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.

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Engineering Mathematics SEM – 1Concept

**LOGARITHMS**

**Logarithm:** For ant two positive real numbers a, b, and a ≠ 1. If the real number x such then a^{x} = b, then x is called logarithm of b to the base a. it is denoted by

**Standard formulae of logarithms:**

**Logarithmic Function:**

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

**PARTIAL FRACTIONS**

**Fractions:**

If f(x) and g(x) are two polynomials, g(x) ≠ 0, then is called rational fraction.

Ex:

_{ }etc. are rational fractions.

**Proper Fraction:**

A rational fraction is said to be a Proper fraction if the degree of g(x) is greater than the degree of f(x).

Ex:

_{ }etc. are the proper fractions.

**Improper Fraction:**

A rational fraction is said to be an Improper fraction if the degree of g(x) is less than the degree of f(x).

Ex:

_{ }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) = 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:

**Case(ii): – **For every factor of g(x) of the form (ax^{2} + bx + c)^{ n}, there will be a sum of n partial fractions of the form:

∎ If R(x) = is improper fraction, then

**Case (i): – **If degree f(x) = degree of g(x), where k is the quotient of the highest degree term of f(x) and g(x).

**Case (ii): – **If f(x) > g(x)