Estimations with linear approximation ?Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error. 3 ?65

Solution 16E STEP 1 Linear approximation is a method of estimating the value of a function,f(x),near a point x=a,using the formula L(x) = f(a)+f (a)(xa) STEP 2 Here,let f(x) = x;and choose a=64 1 We get f (x) = 3x3 f(a) = 64 = 4 and f(a) = 1 2= 1 = 1 3×643 3×16 48 Therefore the Linear approximation of the function,f(x),near a point x=a, is L(x) = f(a)+f (a)(xa) L(x) = 4+ (x64) 48 STEP 3 According to the question 65 = f(65) f(65) L(65) 1 = 4+ (4864) = 4.02 3 Therefore 65 4.02