What is the square root of 123.478?

There are three ways to answer this question:

• Use of a calculator. All calculators and even the cheapest one’s have a square root key.

sqrt. 123478 = 11.11206551

• Use of natural logs. A calculator is needed.

a^x = e^(xlna)

123.478^(1/2) = e^(.5*ln 123.478) = 11.11206551

• Use of derivatives. Pencil and paper is required.

dy = f’(x)*( delta x)

delta y = f(x + (delta x)) - f(x)

Looking at the problem a guess of what number would be close to the square root of 123. Not really a guess it”s 11 and 11 squared is 121. Close enough.

Let x = 121 the difference is 2.478 , let D(x) equal this number.

sqrt. 123.478 = sqrt. 121 + (delta sqrt. x)

f(x) = y = sqrt. x = x^(1/2)

f’(x) = 1/(2* sqrt. x)

dy = f’(x) * (delta x)

** dy = 1/(2* sqrt. 121) x (2.478)

dy = .1126363…

delta x = delta y = dy

sqrt. 121 + dy = 11 + .1126363636... = 11.1126363...

>P>Cutem esta resposta está perto, mas poderia ser melhor. Você provavelmente notou que como D(x) se torna menor (< 1) a resposta parecerá ser igual àquela obtida de uma calculadora.

Vamos fazê-lo novamente:

Começar com ** (anotado acima) 11.11 ^2 = x = 123.4321 . Seremos tomados a raiz quadrada de 123,4321 que é 11,11. D(x) será .0459

dy = 1/(2 x 11,11) x .0459 = .0020657

sqrt. 123.4321 + dy = 11.11 + .0020657 = 11.1120657

Bom!