04Final Exam - MATH150 Final Exam GIVE DETAILED AND...

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MATH150 Final Exam 12/9/04 GIVE DETAILED AND COMPLETE SOLUTIONS. FULLY SIMPLIFY YOUR ANSWERS. NO CALCULATORS PERMITTED! 1. (12 marks) Evaluate the following limits or show that they don’t exist: (a) lim x 0 1 - sec x tan 2 x (b) lim x →∞ (ln x ) 1 /x (c) lim ( x,y ) (2 , 1) x 2 - 4 y 2 x - 2 y 2. (10 marks) (a) Apply the mean value theorem to the function f ( x ) = x on the interval [49 , 50] to show that 50 = 7 + 1 2 c for some number c in (49 , 50). (b) Use (a) to show that 7 < 50 < 99 14 . 3. (12 marks) Let f ( x ) = x 2 / 3 (5 - x ), defined for all real numbers x . (a) Find and classify the critical and singular points (if any) and find the intervals of increase and decrease of f . (b) Find all inflection points (if any) and the intervals of concavity of f . (c) Find the global maximum and minimum values of f on the interval [ - 1 , 4]. 4. (10 marks) Consider the implicitly defined curve C : 2 x 3 - x 2 y + 2 y 2 = 3. Compute an equation of the tangent line to C at P = (1 , 1) by (a) using implicit differentiation.
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