Answer
$$\left( {\bf{a}} \right){g_L}\left( 9 \right) = 26.4 - 1.6x{\text{ }}\,\,\,\,\left( {\bf{b}} \right){g_L}\left( {9.5} \right) = 11.2$$
Work Step by Step
$$\eqalign{
& {\text{We have}}: \cr
& g\left( 9 \right) = 12,{\text{ }}g'\left( 9 \right) = 1.6;{\text{ }}x = 9.5 \cr
& \cr
& \left( {\bf{a}} \right){\text{Using the formula }}\left( {{\text{see page 253}}} \right){\text{ }} \cr
& {g_L}\left( x \right) = g\left( c \right) + g'\left( c \right)\left( {x - c} \right) \cr
& {\text{Let }}c = 9 \cr
& {g_L}\left( x \right) = g\left( 9 \right) + g'\left( 9 \right)\left( {x - 9} \right) \cr
& {\text{Replace }}g\left( 0 \right){\text{ and }}g'\left( 0 \right) \cr
& {g_L}\left( 9 \right) = 12 - 1.6\left( {x - 9} \right) \cr
& {\text{multiply}} \cr
& {g_L}\left( 9 \right) = 12 - 1.6x + 14.4 \cr
& {g_L}\left( 9 \right) = 26.4 - 1.6x \cr
& \cr
& \left( {\bf{b}} \right){\text{use the linearization to estimate }}g\left( x \right){\text{ at }}x = 9.5 \cr
& {g_L}\left( {9.5} \right) = 26.4 - 1.6\left( {9.5} \right) \cr
& {\text{simplifying}} \cr
& {g_L}\left( {9.5} \right) = 11. \cr} $$
