Chapter 4 - Applications of the Derivative - 4.5 Linear Approximation and Differentials - 4.5 Exercises - Page 289: 22

$$a = 0,{\text{ }}\tan \left( {3^\circ } \right) \approx 0.05235$$

$$\eqalign{ & \tan \left( {3^\circ } \right) \cr & {\text{Using the function }}f\left( x \right) = \tan x,{\text{ }}\left( {{\text{Use the example 3 as a guide}}} \right) \cr & f\left( x \right) = \tan x \cr & {\text{we have that }}3^\circ = 3^\circ + 3^\circ ,{\text{ let }}a = 0^\circ \cr & {\text{* Evaluate }}f\left( 0 \right) \cr & f\left( 0 \right) = \tan \left( 0 \right) \cr & f\left( 0 \right) = 0 \cr & \cr & {\text{Differentiating we obtain}} \cr & f'\left( x \right) = {\sec ^2}x \cr & {\text{* Evaluate at }}a = 0,{\text{ }}f'\left( 0 \right) \cr & f'\left( 0 \right) = {\sec ^2}\left( 0 \right) = 1 \cr & \cr & {\text{Using the definition of a linear approximation to }}f{\text{ at }}a{\text{ }}\left( {page{\text{ 282}}} \right) \cr & L\left( x \right) = f\left( a \right) + f'\left( a \right)\left( {x - a} \right),{\text{ for an interval }}I{\text{ containing }}a \cr & \cr & {\text{The linear approximation }}L\left( x \right){\text{ at }}a{\text{ is:}} \cr & L\left( x \right) = 0 + 1\left( {x - 0} \right) \cr & L\left( x \right) = x \cr & \cr & {\text{Now}}{\text{, before applying }}L\left( x \right){\text{ to approximate tan}}\left( {{\text{3}}^\circ } \right){\text{ we must}} \cr & {\text{convert to radian mesure}}{\text{, because the derivative formulas for}} \cr & {\text{trigonometric functions require angles in radians}}{\text{.}} \cr & 3^\circ = 3^\circ \left( {\frac{\pi }{{180^\circ }}} \right){\text{rad}} = \frac{\pi }{{60}}{\text{rad}} \cr & 3^\circ \approx 0.0523598{\text{rad}} \cr & L\left( x \right) = x \cr & L\left( {0.0523598} \right) = 0.0523598 \cr & \cr & {\text{Therefore}}{\text{,}} \cr & \tan \left( {3^\circ } \right) \approx L\left( {0.523598} \right) \cr & \tan \left( {3^\circ } \right) \approx 0.0523598{\text{rad}} \cr & {\text{The result into a calculator is 0}}{\text{.0524077}} \cr} $$