A robotic arm is programmed to sweep across a linear welding track. Its horizontal position xxx (in cm) relative to a central sensor is modeled by the equation
x=5tan(y−π4)x∈R,−π4<y<3π4 x = 5\tan\left(y - \frac{\pi}{4}\right) \quad x \in \mathbb{R}, \quad -\frac{\pi}{4} < y < \frac{3\pi}{4} x=5tan(y−4π)x∈R,−4π<y<43πwhere y y\,y is the angle of rotation of the arm in radians.
Show that
dydx=ax2+b \frac{dy}{dx} = \frac{a}{x^2 + b} dxdy=x2+bawhere a a\,a and b b\,b are integers to be found.
The point P P\,P on the curve C C\,C has yyy-coordinate π2\displaystyle \frac{\pi}{2}2π. The tangent to C C\,C at P P\,P crosses the xxx-axis at the point QQQ. Find the exact xxx-coordinate of QQQ.
