Lesson Objectives. Fast Five. (A) Derivatives of Rational Functions The Quotient Rule 5/8/2011. x 2 x 6 0
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1 5/8/0 Lesson Objectives 0. Develop the quotient rule us 0. Use the quotient rule to evaluate derivatives 0. Apply the quotient rule to an analysis o unctions 0. Apply the quotient rule to real world problems 5/8/0 IB Math SL - Santowski 0. I = /, solve or 0. Evaluate and interpret 0. Solve or i 0. Solve or i 0 5. Simpliy 6 5 Fast Five lim /8/0 IB Math SL - Santowski A Derivatives o Rational The Quotient Rule 0 Since the derivative o a product does not equal the product o the derivatives, what about a quotient? 0 Would the derivative o a quotient equal the quotient o the derivatives? 0 Since quotients are in one sense nothin more than products o a unction and a reciprocal we would uess that the derivative o a quotient is not equal to the quotient o the derivatives 5/8/0 IB Math SL - Santowski
2 5/8/0 A Derivatives o Rational The Quotient Rule 0 I a rational unction can be written in the orm,rewrite as?????? And determine the epression or d/d o R 5/8/0 IB Math SL - Santowski 5 R B Quotient Rule - Derivation 0 First, set up a division and then rearrane the division to produce a multiplication so that we can apply the product rule developed earlier ] [ 5/8/0 IB Math SL - Santowski 6 C Eamples Usin the Quotient Law 0 Dierentiate each o the ollowin rational unctions. Simpliy the derivative. 0 e. 0 e. 0 e. 5 h 5/8/0 IB Math SL - Santowski 7 C Eamples Usin the Quotient Law 0 Determine the equation o the line normal to at = 0 5/8/0 IB Math SL - Santowski 8
3 5/8/0 D Function Analysis o Rational D Function Analysis o Rational - INTS 0 Make a column worksheet. In the irst column, briely state what needs to be done and why and in the second column, do it! 0 For what intervals is the unction concave down? 0 Determine the domain o 0 Determine the vertical asymptotes and the behaviour alon the VA usin limits 0 Determine the horizontal asymptotes usin limits 0 Determine the - and y-intercepts 5/8/0 IB Math SL - Santowski 9 5/8/0 IB Math SL - Santowski 0 D Function Analysis o Rational 0 Aain, make a column worksheet. In the irst column, briely state what needs to be done and why and in the second column, simply do what you said needs to be done! 0 Find the intervals o increase/decrease and ma/min or the iven unction. Then sketch the unction based on your intervals. y 7 5 5/8/0 IB Math SL - Santowski D Function Analysis o Rational - INTS 0 Determine the domain o 0 Determine the vertical asymptotes and the behaviour alon the VA usin limits 0 Determine the horizontal asymptotes usin limits 0 Determine the - and y-intercepts 5/8/0 IB Math SL - Santowski
4 5/8/0 D Function Analysis o Rational 0 Aain, make a column worksheet. In the irst column, briely state what needs to be done and why and in the second column, simply do what you said needs to be done! 0 Find the intervals o increase/decrease and etrema or the iven unction. Sketch as well. 8 5/8/0 IB Math SL - Santowski E Applications - Economics 0 Suppose that the total cost in hundreds o dollars o producin barrels o oil is iven by the unction C = Determine the ollowin. 0 a the cost o producin 5000 barrels o oil 0 b the cost o producin 500 barrels o oil 0 c the cost o producin the 500st barrel o oil 0 d C `5000 = the marinal cost at a production level o 5000 barrels o oil. Interpret. 0 e The production level that minimizes the averae cost where AC = C/ 5/8/0 IB Math SL - Santowski E Applications G Internet Links 0 The number o used cars that Qassem sells per week, t weeks ater droppin out o Calculus class, is Nt 00t t 0 a At what rate is the number o sales chanin ater the st week? Ater the 5 th week? 0 b Does the number o sales per week decrease at any time durin the irst ten week period? 0 Calculus I Math - Derivatives - Product and Quotient Rule 0 Visual Calculus - Calculus@UTK. 0 solvin derivatives step-by-step rom Calc0 5/8/0 IB Math SL - Santowski 5 5/8/0 IB Math SL - Santowski 6
5 5/8/0 F omework 0 Tet, S., p 0 Alebra: Q-8 odds 0 Word Problems: Q6-7 0 Word problems: Q50,5,5,5,55 rom p 5 0 A Level, Worksheet p7, Q,,5,7,8 & p8, Q,, 5/8/0 IB Math SL - Santowski 7 5
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