# 2000test1 - x satisfying j x j R r = order f 9 Let f x =(1 x)sin x[2(a Find the constant approximation f x to f x at x = 1 2 π[6(b Find h> 0

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University of Toronto at Scarborough Physical Sciences Division, Mathematics MATA26Y November 8, 2000 110 minutes Term Test #1 1. Evaluate the limits: [4] (a) lim x 0 x 2(1   p x + 1) [4] (b) lim x 0 1   cos 2 x x sin3 x 2. Simplify the functions: [4] (a) sin(arctan x ) [4] (b) 1   sin 2 ± arccos ± ( x + 1) 2 ²² 3. Let f ( x ) = p 5 x 2   2 x + 3 . [2] (a) Find lim x →∞ f ( x ) [2] (b) Find lim x →-∞ f ( x ) 4. Let f ( x ) = ( x   4)( x 2   4 x   5) ( x 2   2 x   3)(4   x 2 ) . Find [2] (a) the domain of f ( x ). [2] (b) the roots of f ( x ). [4] (c) all x such that f ( x ) > 0. [4] (d) all x such that f ( x ) < 0. [8] 5. Find a nonzero value for the constant k so that the function f ( x ) deﬁned by f ( x ) = tan( kx ) x x < 0 3 x + 2 k 2 x 0 will be continuous at x = 0. 1

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MATA26Y page 2 6. Find the indicated derivatives. [4] (a) f ( x ) = v u u t x 2 + s 2 x ; f 0 ( x ) [6] (b) f ( x ) = sin 2 (arctan(tan x 2 )) ; f 00 ( x ) ( Hint: simplify f ( x ) ﬁrst) [6] (c) ( x + y ) = tan( x + 4 y 2 ) ; dy dx ± ± ± ± (0 , 0) [4] 7. (a) State Rolle’s Theorem. [4] (b) Find exactly how many positive roots the function f ( x ) = 1 ( x + 1) 3   3 x +sin x has. (Do not calculate the root.) [8] 8. For the function f ( x ) = 2 x x 4   1 ﬁnd positive numbers R, m,M such that j x j r m j f ( x ) j j x j r M for all
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Unformatted text preview: x satisfying j x j R . ( r = order( f )) 9. Let f ( x ) = (1 + x )sin x . [2] (a) Find the constant approximation f ( x ) to f ( x ) at x = 1 2 π . [6] (b) Find h > 0 such that ± ± ± x 1 2 π ± ± ± h ) ± ± ± f ( x ) f ( x ) ± ± ± 1 10 . [2] (c) Find the linear approximation f 1 ( x ) to f ( x ) at x = 1 2 π . [6] (d) Find h > 0 such that ± ± ± x 1 2 π ± ± ± h ) ± ± ± f ( x ) f 1 ( x ) ± ± ± 1 10 . 10. Consider the equation tan x = x where x is in radians. [4] (a) Show it has exactly one root r on the interval [4 . 4 , 4 . 6]. [4] (b) Use Newton’s method, starting with r = 4 . 5 to calculate r 1 ,r 2 ,r 3 to ﬁve decimal places. [4] (c) Show that if ¯ r 3 is r 3 rounded to ﬁve decimal places, then j r ¯ r 3 j . 000005....
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## This note was uploaded on 10/04/2011 for the course MATH 16121 taught by Professor Rachelbelinsky during the Spring '11 term at Georgia State University, Atlanta.

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2000test1 - x satisfying j x j R r = order f 9 Let f x =(1 x)sin x[2(a Find the constant approximation f x to f x at x = 1 2 π[6(b Find h> 0

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