Errors in approximations Suppose ?f?(?x?) = 1/(1 + ?x?) is to be approximated near x? = 0. Find the linear approximation ?t? of a? ? = 0. Then, complete the following table showing the errors in various approximations. Use a calculator to obtain the exact values. The percent error is 100 • | approximation - exact | / | exact |. Comment on the behavior of the errors as ?x approaches 0.

SOLUTION STEP 1 .The linear approximation formula is given by L(x) = f(a) + f(a)(x a) Given f(x) = 1+x and a = 0. f(a) = 1 = 1 1+0 1 1 Then f x) = (1+x) f(a) = 1 = 1 Therefore L(x) = 1 1(x 0) = 1 x Thus we get L(x) = 1 x x Linear Exact Percent approx value error 0.1 0.9 0.9090 0.99009 9 9 0.01...