Help please?

Question

Use a differential to approximate sqrt 63.6 . Give your answer to at least 8 significant figures or use fractions to give the exact answer.

Hint: let f(x)=sqrt x and compute dy with x=64 and dx=-0.4

Answer 1

Use the taylor approximation
f(x) = f(a) +(x-a) f´(a) +(x-a) ^2/2! f´´(a) +(x-a) ^3/3!f´´´(a)++
a=64
f(x) =x^1/2,f´(x)=1/2 x^-1/2,f´´(x) =-1/4 x^-3/2 ,f´´´(x)=3/8 x^-5/2
f´´´´(x)=-15/16x^-7/2
f(63.6)= 8-0.4*1/16-0.16/8^4-0.4^3/6*8^5
-0.4^4*15/(16*24`8^7)
=7.9749608
with the differential approximate you can´t asuure 8 figures
dy= 1/(2sqrt x)*dx =1/16*(-0.4) =-1/40 = -0.025
and sqrt63.6) = 8-0.025=7.975