Let f(x)=3+cos(2x+(pi/2))
Find the local linearization of the function at the point a=0.
Use your linearization to find and approximation for f(0.4)
This is the same as finding the tangent line at the point x=0
$\displaystyle f'(x)=-2\sin\left( 2x+ \frac{\pi}{2}\right)$
$\displaystyle f'(0)=-2\sin\left( \frac{\pi}{2}\right)=-2$
note that $\displaystyle f(0)=3$
so the approximation is
$\displaystyle L(x)= -2x+3$
$\displaystyle f(.4) \approx L(.4)=-2(.4)+3=2.2$