sols5 - SOLUTIONS TO HOMEWORK ASSIGNMENT #5 1. Each of the...

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SOLUTIONS TO HOMEWORK ASSIGNMENT #5 1. Each of the following questions can be done with little computation. Enter your answers in the boxes and show any work in the spaces provided. Find derivatives of the following functions and simplify as much as possible: (a) f ( x )= x tan(sin x ) (b) f ( x )= e x + e - x . (c) f ( x )=ln(ln x ). (d) y = x sin x . (e) y = ± 1+ 1 x ² x . Solution: (a) f 0 ( x )= 1 2 x tan(sin x )+ x sec 2 (sin x )cos x. (b) f 0 ( x )= 1 2 x e x - 1 2 x e - x = 1 2 x ³ e x - e - x ´ . (c) f 0 ( x )= 1 x ln x . (d) ln y =s in x ln x = y 0 y =cos x ln x + sin x x = y 0 = y ± cos x ln x + sin x x ² = x sin x ± cos x ln x + sin x x ² (e) ln y = x ln ± 1+ 1 x ² = y 0 y =ln ± 1+ 1 x ² + x 1+1 /x × - 1 x 2 =l n ( 1+ 1 /x ) - 1 x +1 = y 0 =(1+1 /x ) x ± ln(1 + 1 /x ) - 1 x +1 ² . 2. Show that d dx ln ³ x + 1+ x 2 ´ = 1 1+ x 2 . Solution: d dx ln ³ x + 1+ x 2 ´ = 1 x + 1+ x 2 × 1+ x 1+ x 2 ! = 1 x + 1+ x 2 × 1+ x 2 + x 1+ x 2 ! = 1
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3. Find the equations of the tangent lines to the graphs of y = f ( x )at x = a . (a) f ( x )= e x 2 - x ,a =1 (b) f ( x )= ln x x 2 ,a = e. (c) f ( x )= e x 2 cos πx ,a =1 . Solution: (a) f (1)=1and f 0 (1)=(2 x - 1) e x 2 - x | x =1 = 1. Therefore the tangent line has the equation y =1+( x - 1) = x. (b)
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sols5 - SOLUTIONS TO HOMEWORK ASSIGNMENT #5 1. Each of the...

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