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Unformatted text preview: M10360 Exam 1: Review Guide Chapter 5 5.1: The Natural Logarithm; Differentiation Know the definition: ln( x ) = R x 1 1 t dt ; know what y = ln( x ) looks like and its properties. Know the log rules and how to use them ln( ab ) = ln( a ) + ln( b ) ln( a b ) = ln( a ) ln( b ) ln( a n ) = n ln( a ) Know the derivative d dx [ln( u )] = 1 u u where u is a function of x . Logarithmic Differentiation (take ln of both sides and differentiate implicitly after using log rules to make one side easier). Remember to plug y back in when finished. 5.2: Natural Logarithm: Integration Know how to integrate: R 1 u du = ln(  u  ) + C . Be comfortable with usubstitution to get it into this form. Remember, naming u is the same as naming x if you solve for x . Use long division if the numerator and denominator have same power. Also, you can split up the numerator to make things easier by creating two separate integrals. Know how to integrate the trigonometric functions (see page 337). 5.3: Inverse Functions If f ( a ) = b and f has an inverse, then we know that f 1 ( b ) = a . Inverse exists if function is 11 (for one output, there is only one input). Note x 2 does not have an inverse unless we restrict the domain of x 2 to the the numbers x 0. We can create inverses by restricting domains. If a function is strictly monotonic on its entire domain, it will have an inverse. We can check if a function is monotonic by using the first derivative. If f ( x ) > 0 for all x then f is always increasing (likewise for f ( x ) < 0 and f decreasing). The formula for finding derivatives of inverses is: ( f 1 ) ( a ) = 1 f ( f 1 ( a )) so long as the denominator is not 0....
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 Spring '08
 Edgar
 Calculus

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