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Using the algorithm for curve sketching i.e first derivative and second derivatives. ...

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lbawa | Student, Undergraduate | eNoter

Posted March 29, 2012 at 12:40 AM via web

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Using the algorithm for curve sketching i.e first derivative and second derivatives.  Explain what the graph of y=x^(5/3)-2x^(2/3) would look like.

My teacher said there would be a cusp in the graph.  How would I know that? 

 

 

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rcmath | High School Teacher | (Level 1) Associate Educator

Posted March 30, 2012 at 3:31 AM (Answer #1)

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Usually you have a cusp, when 

or 

i) Let's start by finding the inflection points.

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so ``

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Point of inflection

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I will use the unsimplified y' to find y''

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since the denominator is always non negative we only need to study the numerator.

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x=4/5 falls in the firt interval, thus we know that we have a relative minimum.

ii) For the Cusp.

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we notice that x=0 will make the denominator zero, thus y' undefined. let's study the limit at this point.

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because the numerator will be negative, as well as the denominator.

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because the numerator will be negative but denominator positive

Hence we have a cusp at x=0, in other words we have a vertical tangant.

 

 

 

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