1999final

# 1999final - niversity of oronto t rorough hysil ienes...

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University of Toronto at Scarborough Physical Sciences Division, Mathematics MATA26Y April 18, 2000 3 hours FINAL EXAMINATION [10] 1. Sketch the graph of the function f ( x ) = x ( x +2) 2 / 3 showing extrema, points of inﬂection, intervals of increase and decrease and intervals of concavity. Show your work. 2. Calculate the derivatives of [4] (a) g ( x ), at x = 2, where g ( x ) is the inverse function of f ( x ) = e x - e - x . [ Note: f (0) = 2 ] [2] (b) f ( x ) = sin ( cos(tan x 2 ) ) x ± - π 2 , π 2 ² [4] (c) f ( x ) = Z x 2 tan x 1 2 + t 4 dt . 3. Find the following limits: [2] (a) lim x π - | π - x | x - π [4] (b) lim x →∞ x 3 / 2 + 2 x 2 - 4 x ln x [4] (c) lim x 0 tan x - sin x x 3 tan x [6] 4. Find exactly how many roots f ( x ) = 3 x 5 - 5 x 3 + 1 has. [Do not attempt to calculate the roots.] [4] 5. (a) Give the n th degree Taylor polynomial of the function f ( x ) at a and the n th degree remainder term R n ( x ) associated with it. [3]

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1999final - niversity of oronto t rorough hysil ienes...

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