How do you estimate the quantity using Linear Approximation and find the error using a calculator of #1/(sqrt(95)) - 1/(sqrt(98))#?

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Answer 1 · 2018-03-26T15:54:03

Let #y = 1/sqrt(x)#. Then #y(100) = 1/sqrt(100) = 1/10#

#y = x^(-1/2) -> y' = -1/2x^(-3/2)#

We get that #y'(100) = -1/2(100)^(-3/2) = -1/(2(10^3)) = -1/2000#

Therefore the equation of the tangent at #x= 100# is

#y - 1/10 = -1/2000(x - 100)#

#y = -1/2000x + 1/20 + 1/10#

#y = -1/2000x + 3/20#

Therefore the linear approximation will be

#y(95) - y(98) = -1/2000(95) + 3/20 - (-1/2000(98) + 3/20)#

#y(95) - y(98) = 0.00150#

With a calculator we get #1/sqrt(95) - 1/sqrt(98) = 0.00158#

This means the percent error is

#"% error" = |(0.00150 - 0.00158)/0.00158| = 5%#

So our approximation is quite precise.

Hopefully this helps!