# sols4 - SOLUTIONS TO HOMEWORK ASSIGNMENT#4 1 Find all x...

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SOLUTIONS TO HOMEWORK ASSIGNMENT #4 1. Find all x such that f 0 ( x ) = 0, where: (a) f ( x )=cos( x 2 ). (b) f ( x )=s in2 x . (c) f ( x )=3 x 5 - 5 x 3 . (d) f ( x )= x 3 +5 x 2 +3 x . Solutions: (a) f 0 ( x - 2 x sin( x 2 )=0 ⇐⇒ x =0or x 2 = nπ, for some integer n x = ± where n =0 , 1 , 2 ,... (b) f 0 ( x )=2cos2 x x = π 4 + 2 , where n , ± 1 ± 2 (c) f 0 ( x )=15 x 4 - 15 x 2 x , ± 1 . (d) f 0 ( x x 2 +10 x +3=0 x = - 10 ± 64 6 = - 3 , - 1 / 3 . 2. Find the equations of the tangent lines to the graphs of y = f ( x )at x = a ,where : (a) f ( x sin 2 x cos x ,a = π 3 . (b) f ( x | x 2 - 2 x | = 1 2 . (c) f ( x in 2 πx 3 =5 . (d) f ( x sin x 1+cos x = π 3 . Solutions: In all cases the tangent line is given by y - f ( a f 0 ( a )( x - a ) . (a) f 0 ( a 2cos( x )cos(2 x ) - sin(2 x ) × ( - sin( x )) (cos( x )) 2 ± ± ± x = π/ 3 =1 . By deﬁnition F ( x ) ± ± ± x = a = F ( a ) . Since f ( 3) = 3 the equation is y - 3= x - 3 . (b) Since x 2 - 2 x is negative at x / 2weseethat f ( x )=2 x - x 2 for x suﬃciently close to x / 2 (by continuity of the function x 2 - 2 x ). Therefore f 0 ( x - 2 x near x / 2, and so the equation of the tangent line is y - 3 / 4= x - 1 / 2, that is y = x + 1 4 .

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sols4 - SOLUTIONS TO HOMEWORK ASSIGNMENT#4 1 Find all x...

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