b - Ch.2 PHYS2041 Problems in Quantitative Methods for...

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Ch.2 PHYS2041 Problems in Quantitative Methods for Basic Physics (1 st Term 10/11) 1 Chapter 2 Differential and Integration 2.1 Concepts of Limit and Continuity 2.1.1 Definition of Limit: A function F , which is defined on the domain 1 x xc  and 2 cxx , is said to approach the limit L as x approaches c , and we write lim ( ) Fx L , ( 2 . 1 . 1 ) if, given any positive number , there is a positive number such that the values of F are within of L whenever x is within of c , x c :  Fx L  , when 0 . (2.1.2) It is noted that the function F may or may not be defined at x c . More precisely, we may have to consider the limits of   for x c and x c respectively: Left-hand Limi t of F x : lim ( ) F xL , for 1 x ; (2.1.2a) Right-hand Limit of : lim ( ) L , for 2 . (2.1.2b) lim ( ) L exists if and only if both left- and right-hand limits exist and equal, i.e.: LL L   . ( 2 . 1 . 3 ) 2.1.2 Definition of Continuity: A function F which is defined in some neighborhood of c is said to be continuous at c provided
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Ch.2 PHYS2041 Problems in Quantitative Methods for Basic Physics (1 st Term 10/11) 2 (i) the function has a definite finite   F c at c , and (ii) as x approaches c ,  Fx approaches   Fc as limit: lim ( ) xc F xF c . ( 2 . 1 . 4 ) If a function is continuous at all points of an interval axb (or  , etc.), then it is said to be continuous on, or in, that interval. 2.2 Derivatives 2.2.1 Derivative of a Function For yfx , the derivative of y with respect to x is given by: 0 lim x dy y dx x  , ( 2 . 2 . 1 ) and equivalently, the function   f x is differentiable at x :     0 lim x df x f x x f x fx dx x    . (2.2.2) Geometrically, dy dx is the slope of the curve   . In particular, we denote the derivative or slope at 0 x x by 0 x x dy dx ,   0 f x . Some derivative identities: (i)       df x dh x d hx dx dx dx     ,
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Ch.2 PHYS2041 Problems in Quantitative Methods for Basic Physics (1 st Term 10/11) 3 (ii)  df x d cf x c dx dx   , where c is a constant, (iii) Product Rule:        dg x df x d f xgx f x gx dx dx dx  , (iv) Quotient Rule:     2 df x dg x fx d dx dx dx g x  , (v) Power Rule: 1 nn d x nx dx , (vi)
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This note was uploaded on 10/14/2010 for the course PHYS 2051 taught by Professor Drkwong during the Spring '10 term at CUHK.

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b - Ch.2 PHYS2041 Problems in Quantitative Methods for...

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