Do not FOIL

Calculus Level 3

Given the nested radical equation...

y = x 2 + x 2 3 + x 2 7 + x 2 15 + x 2 31 + x 2 63 + … y = \sqrt{\dfrac {x}{2} + \sqrt{\dfrac {x}{2^3} + \sqrt{\dfrac {x}{2^7} + \sqrt{\dfrac {x}{2^{15}} + \sqrt{\dfrac {x}{2^{31}} + \sqrt{\dfrac {x}{2^{63}} + \ldots }}}}} }

...use the parameter d y d x ≥ 0 \dfrac {\mathrm{d}y }{\mathrm{d}x} \geq 0 to find d y d x \dfrac {\mathrm{d}y }{\mathrm{d}x} at the x x -value given below. (Round your answer to two decimal places).

x = 2 16 2 16 2 16 … x =\sqrt{2\sqrt{16 \sqrt{2\sqrt{16 \sqrt{2\sqrt{16 \ldots }}}}}}

This problem is original.
0.17 0.40 0.25 0

Jonas Katona by Jonas Katona , 22, USA

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1 solution

Ruslan Abdulgani
Mar 10, 2015

x= √(2 √(16 x)) , x4 = 64 x, x=4.
...y = √(x/2+(1/2)√( x/2^1 +√(x/2^3 +√(x/2^7 +√(x/2^15 )+⋯……..)) ) ) = √( x/2+ 1/2 y ),
x=4, y2 – ½ y – 2 = 0, y= 1.686 Y2 = x/2 + ½ y, 2y dy – ½ dy = ½ dx, dy/dx = 1/(4y – 1) = 0.17

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