Let {{MATH_72}} be a function defined by {{MATH_73}} Approximate {{MATH_74}}, using {{MATH_75}}, a linear approximation to the function {{MATH_76}} at {{MATH_77}}. {{CMD_78}} To start, write {{MATH_79}} {{ENV_80}} {{CMD_81}} {{COMMENT_82}} {{ENV_83}} {{COMMENT_84}} {{COMMENT_85}} {{COMMENT_86}} {{CMD_87}} Now we evaluate {{MATH_88}}. This was easy to compute, since {{MATH_89}} is a linear function! Using a calculator we can see that {{MATH_90}} and that {{MATH_91}}. Therefore, {{MATH_92}} is not only easy to evaluate, but also gives us a close approximation of {{MATH_93}}. {{CMD_94}} What would happen if we chose {{MATH_95}} instead? Then we would use {{MATH_96}}, the linear approximation to the function {{MATH_97}} at {{MATH_98}}. In that case, {{MATH_99}}. The graph of {{MATH_100}}, together with the graphs of {{MATH_101}} and {{MATH_102}} is given in the figure below. {{CMD_103}} {{COMMENT_104}} {{ENV_105}} {{COMMENT_106}} {{COMMENT_107}} {{COMMENT_108}} {{CMD_109}} From the picture we can see that {{MATH_110}}. So, our choice, {{MATH_111}}, was better! {{CMD_112}}