Linear approximation ?Write an equation of the line that represents the linear approximation To the following functions al?a.?Then (a ? ) ?graph the function and the linear approximation ? n?ear a: (b? ) ?use the Hnear approximation to estimate the given qua ? nt?ity; and ? ) ?compute the percent error in your approximation. ? ? ? ?f(?x)=cos? ;? =?/4; cos (0.8)

Solution 42E STEP 1 .The linear approximation formula is given by y = L(x) = f(a)+f (a)(xa) Given f(x) = cos x and a = . 4 f(a) = cos( ) = 1 4 2 1 Then f (x) =sin x f(a) =sin 4 2 Therefore y = L(x) = 1(x )4= 1(1(x ) 4 2 2 2 1 Thus we get L(x) = 2(1x+ ) 4 Thus we get the equation of the line as y = 1(1x+ ) 2 4 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 cos (0.8)using the linear approximation Thus we have 1 1 cos(0.8) L(0.8) = 2(10.8+ ) =4 2(0.2+ )4= 0.69678 Thus cos(0.8) 0.697 STEP 4 (c) .compute the percent error in your approximation. The exact value of cos(0.8) = 0.6967 Therefore percent error = 100× |ap|exact|ac= 100× |0.|0.6967|6= 100×0.00043 = 0.043% percent error = 0.043%