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revcalc-review - STAT 427 Calculus Review Prof P Goel Note...

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STAT 427 Prof. P. Goel Calculus Review Note: Good understanding of calculus is essential for your success in Stat 427. Please review the following concepts by the begining of third week. Function, limit, continuity, differentiation; Infinite sequences and series, Taylor series expansion; Integral of a function on an interval (a, b) equals the area under the curve in (a, b); Techniques of integration - integration by parts, integration by substitution or change of variables; Multiple integration (functions of two variables). 1 Basic Differentiation Rules dx n dx = nx n - 1 de x dx = e x de u dx = e u du dx (Chain Rule) da u dx = a u ‘n ( a ) du dx (Chain Rule) d‘n ( x ) dx = 1 x d ( cf ( x )) dx = c df ( x ) dx d ( f ( x ) + g ( x )) dx = df ( x ) dx + dg ( x ) dx d ( f ( x ) g ( x )) dx = f ( x ) dg ( x ) dx + df ( x ) dx g ( x ) d ( f ( x ) /g ( x )) dx = df ( x ) dx g ( x ) - f ( x ) dg ( x ) dx g ( x ) 2 1
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2 Basic Integration Rules Let Z x a f ( u ) du = F ( x ) . Then F ( . ) is called the anti-derivative of f ( . ) , and of course, f ( . ) is the derivative of F ( . ) .
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