Chapter 4 - Applications of the Derivative - 4.1 Linear Approximation and Applications - Exercises - Page 172: 14

-0.08

For small $\Delta x$, $\Delta y\approx dy$. That is, $\Delta y\approx f'(a) dx$. $f'(a)=2\tan a \sec^{2} a=2\tan\frac{\pi}{4}\sec^{2}\frac{\pi}{4}$ $= 2\times1\times2=4$ $dx=-0.02$ $\implies \Delta y\approx 4\times-0.02=-0.08$