1994test2 - -x + 2) x 3 and f 00 ( x ) = 36( x-3) x 4 [20]...

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SCARBOROUGH CAMPUS UNIVERSITY OF TORONTO MATA26Y January 30, 1995 Term Test II [25] 1. (a) State the Chain Rule. (b) Find the derivatives of the following functions. i) f ( x ) = cos( x + e x ) ii) f ( x ) = ( x 3 + 3 x )arctan x iii) f ( x ) = x ln x , x > 0 iv) f ( x ) = (2 + sin x ) 1 3 1 + x 2 [30] 2. Let f ( x ) = 18( x - 1) x 2 . Find (a) the domain of f , (b) the zeros of f , (c) the intervals on which f is positive/negative, (d) the intervals of increase/decrease, (e) the minima and maxima, (f) the intervals on which f is concave up/down, (g) the inflection points, (h) the vertical asymptotes, and (i) the horizontal asymptotes. Sketch the graph of f on the next page , and indicate all of the above information on the graph. ± Hint: f 0 ( x ) = 18(
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Unformatted text preview: -x + 2) x 3 and f 00 ( x ) = 36( x-3) x 4 [20] 3. Find the following limits. (a) lim x cos 25 ( x ) x (b) lim x 3 x 3-5 x 2 + 3 x + 9 x 2-6 x + 9 (c) lim x 1 sin 1 x-1 ln( x 2 ) (d) lim x + (sin x ln x ) 1 [8] 4. Find the equation of the tangent line to the graph of f ( x ) = x cos x at 6 , 3 12 ! . [5] 5. Solve for x : ln( x 2-2 x-3)-ln( x + 1) = 0. [12] 6. (a) Use implicit dierentiation to nd y for the equation xy 5 + x 5 y = 1. (b) Show that the graph of xy 5 + x 5 y = 1 has no horizontal tangents. 2...
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1994test2 - -x + 2) x 3 and f 00 ( x ) = 36( x-3) x 4 [20]...

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