Chapter 4 - Applications of the Derivative - Chapter Review Exercises - Page 221: 12

$$L(t)=2\pi (t-1/4) -1 $$

Given $$R(t)=\tan \left(\pi\left(t-\frac{1}{2}\right)\right), \quad a=\frac{1}{4}$$ Since \begin{align*} R'(t)&=\pi \sec^2 \left(\pi\left(t-\frac{1}{2}\right)\right)\\ R(1/4)&= 2\pi \end{align*} Then the linear approximation is given by \begin{align*} L(t)&=R^{\prime}(a)(t-a)+R(a)\\ &=2\pi (t-1/4) -1 \end{align*}