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P23SelectedAnswers - Project 23 MAC 2311 1. In Lecture 23...

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Project 23 MAC 2311 1. In Lecture 23 we saw that Mean Value Theorem can be used to prove the following: If f and g are two functions such that f 0 ( x ) = g 0 ( x ) on an interval ( a;b ) , then there is a constant C such that f ( x ) = g ( x ) + C for all x in ( a;b ) . How does this explain our formula for the general antiderivative of a function (that is, if F is an antiderivative of f , why must all others be F plus a constant)? You can use this property to prove interesting identities. For example. .. Look at the functions y = tan 2 ( x ) and y = sec 2 ( x ) . Show that they have the same derivative. Use this relationship to prove the identity: tan 2 ( x ) + 1 = sec 2 ( x ) on ( ± ± 2 ; ± 2 ) . Show that the function y = x 1 ln( x ) is constant on its domain. (Do this by calculating the derivative using logarithmic differentiation.) What is the constant? By taking the derivative of each side, prove that for positive functions f and g , we have ln[ f ( x ) g ( x )] = ln[ f ( x )] + ln[ g ( x )] .
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2. What is the derivative of
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P23SelectedAnswers - Project 23 MAC 2311 1. In Lecture 23...

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