2000test2

# 2000test2 - 3 and has values f(1 0 = 0 1860 f(1 5 = 0 9411...

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UNIVERSITY OF TORONTO AT SCARBOROUGH PHYSICAL SCIENCES DIVISION, Mathematics MATA26Y February 7, 2001 Term Test II [14] 1. Sketch the graph of the function f ( x ) = x 4   2 x 2 , showing extrema, points of inﬂection, intervals of increase and decrease and intervals of concavity. 2. Calculate the following. [4] (a) lim x →∞ x ± 2 1 x   1 ² [4] (b) lim x 1 p x 2 + x + 1   3 x [6] (c) lim x 0 x cos x   sin x x sin 2 x [6] (d) d dx ± (ln x ) arctan x ² [12] 3. Find the length of the shortest ladder that can extend from a vertical wall, over a verti- cal fence p 3 m high located 9 m away from the wall, to a point on the ground outside the fence. b b b b b b b b b b b b b b ± ± ² ² ² 3 9 m wall ladder fence y x [4] 4. (a) State the Fundamental Theorem of Calculus. [8] (b) Find the area bounded by f ( x ) = 4   x 2 and the x –axis on [   4 , 4]. [4] (c) Find f 0 ( x ) where f ( x ) = Z 1+ x 2 sin x e t 2 dt . [8] 5. A function f satisﬁes j f (4) ( x ) j 7 on the interval [1

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Unformatted text preview: , 3], and has values f (1 . 0) = 0 . 1860, f (1 . 5) = 0 . 9411, f (2 . 0) = 1 . 1550, f (2 . 5) = 1 . 4511, and f (3 . 0) = 1 . 2144. Find the best possible Simpson’s rule approximation to I = R 3 1 f ( x ) dx , based on this data. Give a bound for the size of the error, and specify the smallest interval you can that must contain the value of I . MATA26Y page 2 6. Calculate each of the following integrals. [4] (a) I = Z π cos ± 1 2 θ ² dθ [4] (b) I = Z xe x dx [4] (c) I = Z p 1 2 x dx [4] (d) I = Z dx x 2 4 7. Calculate each of the following integrals. [4] (a) I = Z √ 3 2 x p 1 x 2 dx [6] (b) I = Z √ 3 2 p 1 x 2 dx [4] (c) I = Z e x sin xdx...
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2000test2 - 3 and has values f(1 0 = 0 1860 f(1 5 = 0 9411...

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