Solve \(\frac{dy}{dt}\) , when \(t =\frac{\pi}{4}\):
\(z = 5x + 2y,\)
$x = 2 cos t + 1, $
\(y = sin t - 3,\)
$f_x(x,y) = 5 $
$f_y(x,y) = 2 $
\(\frac{dx}{dt} = -2 sin\ t\)
\(\frac{dy}{dt} = cos\ t\)
\(\frac{dz}{dt} = 5(-2 sin\ t) + 2(cos\ t) = -10 sin\ t + 2 cos\ t\)
when \(\frac{\pi}{4}\):
t <- pi/4
dz <- -10*sin(t) + 2*cos(t)
dz## [1] -5.656854