Mathematics - All Important Notes, Formula and Concept

Mathematical Reasoning - Notes, Concept and All Important Formula

MATHEMATICAL REASONING 1. STATEMENT : A declarative sentence which is either true or false but not both, is called a statement. A sentence which is an exclamatory or a wish or an imperative or an interrogative can not be a statement. If a statement is true then its truth value is \(T\) and if it is false then its truth value is \(F\) . All Chapter Notes, Concept and Important Formula 2. SIMPLE STATEMENT: Any statement whose truth value does not depend on other statement are called simple statement. 3. COMPOUND STATEMENT : A statement which is a combination of two or more simple statements are called compound statement. Here the simple statements which form a compound statement are known as its sub statements. 4. LOGICAL CONNECTIVES : The words or phrases which combines simple statements to form a compound statement are called logical connectives. \(\scriptsize{ \begin{array}{|l|l|l|l|l|} \hline \text { S.N. } & \text { Connectives } & \text { Symbol } & \text { Use } ...

Function (Mathematics)- Notes, Concept and All Important Formula

FUNCTION 1. DEFINITION : If to every value (considered as real unless other-wise stated) of a variable \(\mathrm{x}\) , which belongs to a set \(\mathrm{A}\) , there corresponds one and only one finite value of the quantity \(y\) which belong to set \(B\) , then \(y\) is said to be a function of \(x\) and written as \(f: A \rightarrow B, y=f(x), x\) is called argument or independent variable and \(y\) is called dependent variable. Pictorially: \( \underset{\text { input }}{\stackrel{\mathrm{x}}{\longrightarrow}}\boxed{\mathrm{f}}\,\, \underset{\text { output }}{\stackrel{\mathrm{f(x)=y}}{\longrightarrow}}\) \(y\) is called the image of \(x\) and \(x\) is the pre-image of \(y\) , under mapping \(\mathrm{f}\) . Every function \(\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}\) satisfies the following conditions. (i) \(f \subset A \times B\) (ii) \(\forall \,\, a \in A \,\, \exists\,\, b \in B\) such that \((a, b) \in f\) and (iii) If \((a, b) \in f \) &...

Definite Integration - Notes, Concept and All Important Formula

DEFINITE INTEGRATION Table Of Contents 1. (a) The Fundamental Theorem of Calculus, Part 1: If \(\mathrm{f}\) is continuous on \([\mathrm{a}, \mathrm{b}]\) , then the function \(\mathrm{g}\) defined by \(g(x)=\displaystyle \int_{a}^{x} f(t) d t,\) \( a \leq x \leq b\) is continuous on \([\mathrm{a}, \mathrm{b}]\) and differentiable on \((\mathrm{a}, \mathrm{b})\) , and \(g^{\prime}(\mathrm{x})=\mathrm{f}(\mathrm{x})\) . (b) The Fundamental Theorem of Calculus, Part 2: If f is continuous on \([a, b]\) , then \(\displaystyle \int_{a}^{b} f(x) d x=F(b)-F(a)\) where \(F\) is any antiderivative of \(\mathrm{f}\) , that is, a function such that \(\mathrm{F}^{\prime}=\mathrm{f}.\) Note : If \(\displaystyle \int_{a}^{b} f(x) d x=0 \Rightarrow\) then the equation \(f(x)=0\) has atleast one root lying in \((a, b)\) provided \(f\) is a continuous function in \((a, b)\) . All Chapter Notes, Concept and Important Formula 2. Representation of Definite Int...

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