Chapter 4 - Applications of the Derivative - 4.5 Linear Approximation and Differentials - 4.5 Exercises - Page 289: 54

$$\eqalign{ & \left( a \right)L\left( x \right) = x \cr & \left( b \right){\text{graph}} \cr & \left( c \right)0.0523598 \cr & \left( d \right)0.0915\% {\text{ error}} \cr} $$

$$\eqalign{ & f\left( x \right) = \tan x;{\text{ }}a = 0;{\text{ }}\tan \left( {3^\circ } \right) \cr & {\text{Differentiate }}f\left( x \right) \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {\tan x} \right] \cr & f'\left( x \right) = {\sec ^2}x \cr & {\text{Evaluate }}f\left( x \right){\text{ and }}f'\left( x \right){\text{ at }}a = 0 \cr & f\left( 0 \right) = \tan \left( 0 \right) = 0 \cr & f'\left( 0 \right) = {\sec ^2}\left( 0 \right) = 1 \cr & \cr & \left( a \right){\text{Use the linear approximation formula }}\left( {{\text{See page 287}}} \right) \cr & f\left( x \right) = L\left( x \right) = f\left( a \right) + f'\left( a \right)\left( {x - a} \right){\text{ }}\left( {\bf{1}} \right) \cr & {\text{Substitute }}f\left( a \right){\text{ and }}f'\left( a \right){\text{ into }}\left( {\bf{1}} \right) \cr & L\left( x \right) = 0 + 1\left( {x - 0} \right) \cr & L\left( x \right) = x \cr & \cr & \left( b \right){\text{The graph of the function and the linear approximation }} \cr & {\text{at }}x = 0{\text{ is shown below}}{\text{.}} \cr & \cr & \left( c \right){\text{ Estimating the given value function at }}f\left( {3^\circ } \right) \cr & {\text{Now}}{\text{, before applying }}L\left( x \right){\text{ to approximate tan}}\left( {{\text{3}}^\circ } \right){\text{ we must}} \cr & {\text{convert to radian mesure}}{\text{, because the derivative formulas for}} \cr & {\text{trigonometric functions require angles in radians}}{\text{.}} \cr & 3^\circ = 3^\circ \left( {\frac{\pi }{{180^\circ }}} \right){\text{rad}} = \frac{\pi }{{60}}{\text{rad}} \cr & 3^\circ \approx 0.0523598{\text{rad}} \cr & L\left( x \right) = x \cr & L\left( {0.0523598} \right) = 0.0523598 \cr & \cr & {\text{Therefore}}{\text{,}} \cr & \tan \left( {3^\circ } \right) \approx L\left( {0.0523598} \right) \cr & \tan \left( {3^\circ } \right) \approx 0.0523598 \cr & \cr & \left( d \right){\text{ The percent error is:}} \cr & \frac{{\left| {{\text{approximation}} - {\text{exact}}} \right|}}{{{\text{exact}}}} \times 100\% \cr & {\text{The exact value given by a calculator is }} \cr & \tan \left( {3^\circ } \right) = 0.05240777928 \cr & \cr & \frac{{\left| {{\text{0}}{\text{.0524077}} - 0.05240777928} \right|}}{{0.05240777928}} \times 100\% = 0.091\% {\text{ error}} \cr} $$