P19 - is a fairly “wild” function Find the x-values of...

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Project 19 MAC 2311 YOU MUST SHOW YOUR WORK TO RECEIVE FULL CREDIT!! 1. Rewrite the function f ( x ) = ln | x | as a piecewise deﬁned function. Draw its graph on the axes below. f ( x ) = Calculate the slopes of the tangent lines to f ( x ) at x = 2 and x = - 2. Draw these two tangent lines on the axes above. Find the derivative of g ( x ) = ln( ax ) where a is any positive number using chain rule. Does the answer surprise you? To make sense of it, note that ln( ax ) = ln( a )+ln( x ). What then is the relationship between the graphs of ln( x ) and ln( ax )? Why then should they have the same derivative (think in terms of slope)? 2. Examine the function f ( x ) = sin(ln( x )). Show that for every x > 0, x 2 f 00 ( x )+ xf 0 ( x )+ f ( x ) = 0. (We would say that f ( x ) solves this diﬀerential equation on the interval x > 0, since the right-hand side and left-hand side are the same function 0n that interval.) Although this function is a solution to a very basic diﬀerential equation, it

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Unformatted text preview: is a fairly “wild” function. Find the x-values of its horizontal tangent lines and show there are inﬁnitely many of them on the interval (0 , 1). (Hint: to ﬁnd where f ( x ) = 0, use the idea of Project 1.) 3. For which of these functions would logarithmic diﬀerentiation truly be beneﬁcial/necessary? x e x 2 ln( x 2 e x 2 ) x 2 + e x 2 x 2 e x 2 4. Examine the function f ( x ) = xe 1 2 ( x-2) 2 ( x + 2) 3 . Calculate f ( x ) using chain, power, and exponential rules for diﬀerentiation but DO NOT try to simplify. This time use logarithmic diﬀerentiation to calculate the derivative of f ( x ). At what values does logarithmic diﬀerentiation fail? What is the slope of the tangent line to f ( x ) at x = 2? x = 0? Where does f ( x ) have horizontal tangent lines (add up your fractions. ..)?...
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P19 - is a fairly “wild” function Find the x-values of...

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