ws09 - WORKSHEET 9 - Fall 1995 1. Recall that sin2 (x) +...

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WORKSHEET 9 - Fall 1995 1. Recall that sin 2 ( x )+cos 2 ( x ) = 1. Write an expression for each trig function squared in terms of one of the other trig functions. a) sin 2 ( x )b ) c o s 2 ( x )c ) t a n 2 ( x )d ) c o t 2 ( x )e ) s e c 2 ( x )f ) c s c 2 ( x ) 2. Compute the limits. a) lim x 0 sin 2 2 x x 2 d) lim x 0 1+cot 2 3 x 4 x 2 b) lim x 0 1 cos 3 x x 2 e) lim x π/ 6 [1 sec( x + π 3 )]( x π 6 ) 2 c) lim x 0 x 1 sec x f) lim x 2 cos x 2 x π 3. For each of the following, de±ne a function satisfying the conditions. Graph your function. a) g ( x ) is a rational function which is de±ned for all x except x =1 , 2. b) lim x 0 f ( x )doesnotex istand | f ( x ) | < 2 for all x . c) f ( x ) is discontinuous at every point. 4. Sketch graphs of the following functions, determine the points of discontinuity, and state whether it’s a removable or nonremovable (i.e. essential) discontinuity. a) f ( x )= x 2 1 x +1 b) f ( x x 2 ,x < 2 2 = 2 x, 2 <x 0 2 , 0 <x< 2 3 x , 2 x c) f ( x | x +2 | . 5. a) Compute g 0 (1), where g ( x x + f ( x ) x f ( x ) , f (1) = 4, and f 0 (1) = 2. b) Suppose f ( x xf 0 ( x ) for all x in the domain of f ( x ). What can we say about the function g ( x )? 6. Suppose that f is di²erentiable at
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