document.write( "Question 1203748: Please help me with this calculus problem about derivatives of Inverse functions.\r
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Algebra.Com's Answer #839562 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Calculate g(b) and g'(b) where g is the inverse function of f(x).
\n" ); document.write( "f(x) = x + sin(x), b= 1.
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document.write( "You are given function f(x) = x + sin(x).\r\n" );
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document.write( "They want you find g(1), where g(x) is the inverse function of f(x).\r\n" );
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document.write( "First of all, in order you understand better everything from the very beginning, \r\n" );
document.write( "f(x) = x + sin(x) is a MONOTONIC function on the entire domain, which is the set of all real numbers.\r\n" );
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document.write( "The range of f(x) is the entire set of real numbers. \r\n" );
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document.write( "It is monotonic function, since its derivative f'(x) = 1 + cos(x) is non-negative everywhere\r\n" );
document.write( "excluding discrete points x = , k = 0, +/-1, +/-2, . . . \r\n" );
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document.write( "g(1) is the real number x such that f(x) = x + sin(x) = 1.     (1)\r\n" );
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document.write( "Since f(x) is monotonic, such number x (the root of the equation (1), \r\n" );
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document.write( "     (a)  exists, and \r\n" );
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document.write( "     (b)  is UNIQUE.\r\n" );
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document.write( "To find this root x, the algebraic methods do not work. We should use numerical methods\r\n" );
document.write( "and special computer solvers.\r\n" );
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document.write( "One such solver is Wolfram Mathworld / Transcendent solver\r\n" );
document.write( "https://mathworld.wolfram.com/TranscendentalEquation.html\r\n" );
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document.write( "It is free of charge.\r\n" );
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document.write( "Using it, we find the root of equation (1).  It is  x = 0.510973.\r\n" );
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document.write( "So, g(1) = 0.510973  (approximate value).\r\n" );
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document.write( "Now we should find g'(1).\r\n" );
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document.write( "Use the identity for inverse function  g'(b) = 1/f'(x).  It gives\r\n" );
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document.write( "g'(1) = 1/f'(0.510973) =  =  = 0.534111  (approximate value).\r\n" );
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document.write( "ANSWER.  g(1) = 0.510973  (approximate value).  g'(1) = 0.534111  (approximate value).\r\n" );
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\n" ); document.write( "\n" ); document.write( "The problem is solved, in full,
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\n" ); document.write( "\n" ); document.write( "Ignore the post by @MathLover1, for your safety,
\n" ); document.write( "since madam simply doesn't know the subject.\r
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\n" ); document.write( "\n" ); document.write( "In order for your understanding would be even better,
\n" ); document.write( "I advise you to go to web-site www.desmos.com/calculator\r
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\n" ); document.write( "\n" ); document.write( "Find there free of charge plotting tool and make plots of functions
\n" ); document.write( "f(x) = x + sin(x) and h(x) = 1.\r
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\n" ); document.write( "\n" ); document.write( "You will see monotonicity of f(x) and will see the root of equation f(x) = 1,
\n" ); document.write( "where the intersection point of these two plots is.\r
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