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Unformatted text preview: Lecture V Calculus of One-Variable Functions Let us first review some definitions in calculus on real numbers. In order to define limit on the real numbers we will use the concept of funnel functions . Definition 1 A function ( t ) on [0 , d ] is called a funnel function if it has the following properties: 1. It is strictly increasing. 2. For every a > , there exists t , 0 < t ≤ d , such that 0 < ( t ) < a . Definition 2 Let f ( x ) be a scalar function and let l be a real number. We say that the limit of f at c is l and we denote this by lim f ( x ) = l x c → if for some d > 0 there exists a funnel function ( t ) on [0 , d ] such that for every x with 0 < x − c ≤ d it follows that f ( x ) − l ( x − c ) . | | | | ≤ | | | f ( c + h ) − f ( c ) For a real function f and c, h real numbers, the fraction h is called difference quotient and is denoted by Δ f . Δ x f ( c + h ) − f ( c ) Definition 3 For a real function f , if lim h 0 h exists and has a real → df value, we call this value the derivative of f at c and denote it by f ( c ) or dt | c ....
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