Given:
[tex]\begin{gathered} y=tan(x) \\ x=\frac{\pi}{4} \end{gathered}[/tex]
To Determine: The Linearization of the given
Solution
The formula for finding the linearization of a function is given below
[tex]L(x)=f(x_0)+f^{\prime}(x_0)(x-x_0)[/tex][tex]\begin{gathered} x_0=\frac{\pi}{4}(given) \\ f(x_0)=tan(\frac{\pi}{4})=1 \\ f^{\prime}(x_0)=f^{\prime}(x)=sec^2x \\ f^{\prime}(x_0)=\frac{1}{cos^2x} \end{gathered}[/tex][tex]\begin{gathered} f^{\prime}(x_0)=(cos^2x_0)^{-1} \\ f^{\prime}(\frac{\pi}{4})=(cos\frac{\pi}{4})^{-2} \\ =(\frac{1}{\sqrt{2}})^{-2} \\ =(\sqrt{2})^2 \\ =2 \end{gathered}[/tex]
Therefore
[tex]L(x)=1+2(x-\frac{\pi}{4})[/tex][tex]L(x)=2x+1-\frac{\pi}{2}[/tex]
