How do you estimate the quantity using the Linear Approximation of #(3.9)^(1/2)#?

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Answer 1 · 2016-12-20T00:18:40

Function #f(x) = x^(1/2) = sqrtx#

Find an #a# near #3.9# with #f(a)# easy to find.

Use #a=4#, so #f(a) = f(4) = sqrt4 = 2#

The linear approximation of #f# at #(a,f(a))# is a form of the equation for the tangent line at #(a,f(a))#

#L(x) = f(a) + f'(a)(x-a)#

For #f(x) = sqrtx#, we have #f'(x) = 1/(2sqrtx)#.

So, #f'(a) = f'(4) = 1/(2sqrt4) = 1/4#

#L(x) = 2+1/4(x-4)#

#L(3.9) = 2+0.25(-0.1) = 2-0.025 = 1.975#

(Note that: #1.975^2 = 3.900625# an

#sqrt3.9 ~~ 1.974842#.)