Errors in approximations Suppose is to be approximated near ? x = 8. Find the linear approximat?ion t?o f ? at ?x = 8. 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 8.

SOLUTION STEP 1 .The linear approximation formula is given by y = L(x) = f(a)+f (a)(xa) Given f(x) = and a = 8. 3 f(a) = 8 = 2 Then f (x) = 12 (a) = 12 = 1 3x3 3(8) 12 Therefore 1 L(x) = 2+ (128) x Linear Exact Percent approx value error 8.1 2.0083 2.0082 0.00497 8.0 2.000833 2.0008 0.00016 1 3 3 8.0 2.000083 2.0000 0.00001 01 33 83 6 8.0 2.000008 2.0000 0.00000 00 333 083 16 1 7.9 1.999991 1.9999 0.00008 99 66 9 3 9 7.9 1.999916 1.9999 0.00083 99 6 7.9 1.999166 1.9991 0.0033 9 7.9 1.9916 1.9916 0.003 We can find that as x approaches to 8 the error is almost equal to 0