Linear approximation ?Write an equation of the line that represents the linear approximation To the following functions at a. Then (?? ?graph the function and the linear approximation near a: (?? ?use the linear approximation to estimate the given quantity; and (?c? ?compute the percent error in your approximation. f?? ?) = 1? ? + 1);? ? = 0; 1/1.1
Solution 41E STEP 1 .The linear approximation formula is given by y = L(x) = f(a) + f a)(x a) 1 Given f(x) = x+1 and a = 0. 1 f(a) = 0+1 = 1 Then f () = 1 2 f(a) = 1 2 = 1 (x+1) (0+1) Therefore y = L(x) = 1 1(x 0) = 1 x Thus we get L(x) = 1 x Thus we get the equation of the line as y = 1 x STEP 2 (a).graph the function and the linear approximation near a: STEP 3 (b). use the linear approximation to estimate the given quantity; We have to estimate the value of 1 using the linear approximation 1.1 Thus we have 1 1 1.1= 1+0.1 L(0.1) = 1 0.1 = 0.9 Thus 1 0.9 1.1 STEP 4 (c) .compute the percent error in your approximation. 1 The exact value of 1.1 = 0.909 |approxexact| Therefore percent error = 100 × |exact| = 100 × ||0.909|0= 100 × 0.0099 = 0.99% percent error = 0.99%
