Find {{MATH_9}}, the linear approximation to {{MATH_10}} at {{MATH_11}}. {{MATH_12}} {{CMD_13}} Use the linear approximation {{MATH_14}} to estimate the value of {{MATH_15}}. {{MATH_16}} {{CMD_17}} Find the average rate of change of {{MATH_18}} on the interval {{MATH_19}}. {{MATH_20}} {{CMD_21}} Choose the correct statement. {{CMD_22}} {{CMD_23}}There exists a point {{MATH_24}} in {{MATH_25}} such that {{MATH_26}}, although the function {{MATH_27}} does not satisfy the conditions on the mean value theorem on {{MATH_28}}.{{CMD_29}} {{CMD_30}}There exists a point {{MATH_31}} in {{MATH_32}} such that {{MATH_33}}, since the function {{MATH_34}} satisfies the conditions on the mean value theorem on {{MATH_35}}.{{CMD_36}} {{CMD_37}}There exists a point {{MATH_38}} in {{MATH_39}} such that {{MATH_40}}, but we don't have enough information to find such a point.{{CMD_41}} {{CMD_42}}There exists no point {{MATH_43}} in {{MATH_44}} such that {{MATH_45}}.{{CMD_46}} {{CMD_47}} {{CMD_48}} {{CMD_49}} Again, choose the correct statement. {{CMD_50}} {{CMD_51}}For each {{MATH_52}} in {{MATH_53}}, there exists a point {{MATH_54}} in {{MATH_55}} such that {{MATH_56}}, although the function {{MATH_57}} does not satisfy the conditions on the mean value theorem on {{MATH_58}}.{{CMD_59}} {{CMD_60}}For each {{MATH_61}} in {{MATH_62}}, there exists a point {{MATH_63}} in {{MATH_64}} such that {{MATH_65}}, since the function {{MATH_66}} satisfies the conditions on the mean value theorem on {{MATH_67}}.{{CMD_68}} {{CMD_69}}There may be {{MATH_70}} in {{MATH_71}}, for which there exists a point {{MATH_72}} in {{MATH_73}} such that {{MATH_74}}, but not necessarily every {{MATH_75}} in {{MATH_76}}.{{CMD_77}} {{CMD_78}}For each {{MATH_79}} in {{MATH_80}}, there does not exist a point {{MATH_81}} in {{MATH_82}} such that {{MATH_83}}.{{CMD_84}} {{CMD_85}} {{CMD_86}} Suppose we know that {{MATH_87}} for {{MATH_88}} in {{MATH_89}}. By the mean value theorem, for each {{MATH_90}} in{{MATH_91}}, there exists {{MATH_92}} in {{MATH_93}} such that {{MATH_94}} {{CMD_95}} Recall that we assumed {{MATH_96}} for each {{MATH_97}} in {{MATH_98}}. Use your answer above to bound {{MATH_99}} for {{MATH_100}} in {{MATH_101}} from above and from below. {{MATH_102}} {{CMD_103}} Use the bounds {{MATH_104}} for {{MATH_105}} in {{MATH_106}} you computed in the previous part to approximate {{MATH_107}} from above and below. {{MATH_108}} {{CMD_109}} Let {{MATH_110}} be the linear approximation to {{MATH_111}} at {{MATH_112}} you computed before. We used {{MATH_113}} to approximate {{MATH_114}} and found that {{MATH_115}} is approximately equal to {{MATH_116}}. Using our previous answers, we can deduce that this approximation to {{MATH_117}} is a {{CMD_118}}{{CMD_119}}over approximation{{CMD_120}}{{CMD_121}}under approximation{{CMD_122}}{{CMD_123}}. {{CMD_124}} {{CMD_125}} {{CMD_126}} {{CMD_127}} {{CMD_128}} {{CMD_129}} {{CMD_130}} {{CMD_131}} {{CMD_132}} {{CMD_133}}