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408Npracticemid2[1] - MATH 408N PRACTICE MIDTERM 2...

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MATH 408N PRACTICE MIDTERM 2 Name: 10/21/2011 Bormashenko TA session: Show your work for all the problems. Good luck! (1) Use the limit definition of the derivative for the following questions. You will get no points for using the rules learned later! (a) [5 pts] Find f 0 (1) if f ( x ) = x . (b) [5 pts] Find f 0 ( x ) if f ( x ) = x 2 + x + 1.
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2 MATH 408N PRACTICE MIDTERM 2 (2) Differentiate the following functions, using whatever methods you think are best (you are now allowed to use all the rules): (a) [5 pts] f ( x ) = x 3 + 3 x + 5 (b) [5 pts] f ( x ) = arctan ( x 2 ) e x
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MATH 408N PRACTICE MIDTERM 2 3 (c) [5 pts] f ( x ) = 2 x sin( x ) ln( x + 1) (d) [5 pts] f ( x ) = sin( x ) cos( x )
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4 MATH 408N PRACTICE MIDTERM 2 (3) Find the equations of the tangent lines to the following graphs at the given points: (a) [7 pts] y = x 2 ln( x ) + 1 at (1 , 1). (b) [8 pts] x 2 + y 2 = xe y at (1 , 0).
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MATH 408N PRACTICE MIDTERM 2 5 (4) Let f ( x ) and g ( x ) satisfy f (1) = 3 , g (1) = 2 , f 0 (1) = 1 , g 0 (1) = - 1, and f 0 (2) = - 2. (a) [5 pts] If F ( x ) = f ( x ) g ( x ), calculate F 0 (1). (b) [5 pts] If G ( x ) = f ( g ( x )), calculate G 0 (1).
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