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# P29 - Project 29 MAC 2311 YOU MUST SHOW YOUR WORK TO...

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Project 29 MAC 2311 YOU MUST SHOW YOUR WORK TO RECEIVE FULL CREDIT!! 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 ( x ) = g ( 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. What then must be the relationship between the two functions? Use this relationship to prove the identity tan 2 ( x ) + 1 = sec 2 ( x ) . Look at the function y = x 1 ln( x ) . Calculate the derivative using logarithmic differentiation. (Does the answer surprise you?) What other functions have this derivative? Use this relationship to evaluate x 1 ln( x ) for all x in its domain. (Hint: the value e simplifies nicely when you plug it in.) 1

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2. What is the derivative of sin(2
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P29 - Project 29 MAC 2311 YOU MUST SHOW YOUR WORK TO...

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