Find the intersection:
\(y^2 = \sqrt{y}\)
\(y^4 = y\)
\(y^4 - y = 0\)
\(y=0 ~ and ~ y=1\)
Intersection \((0,0), (1,1)\)
At \((0,0)\)
Slope\((m_1)\) of \(y^2 =x\).
\(m_1 = \frac{dy}{dx} = \frac{1}{2y}\)
\(m_1 = \frac{dy}{dx} = \frac{1}{0}\)
\(m_1 = \frac{dy}{dx} = ~undefined\), the tangent line is a horizontal line.
Slope\((m_2)\) of \(y =x^2\).
\(m_2 = \frac{dy}{dx} = 2x\)
\(m_2 = \frac{dy}{dx} = 0\) a vertical line
The angle between line is \(90^o\).
At \((1,1)\)
Slope\((m_1)\) of \(y^2 =x\).
\(m_1 = \frac{dy}{dx} = \frac{1}{2y}\)
\(m_1 = \frac{dy}{dx} = \frac{1}{2}\)
Slope\((m_2)\) of \(y =x^2\).
\(m_2 = \frac{dy}{dx} = 2x\)
\(m_2 = \frac{dy}{dx} = 2(1)\)
\(m_2 = \frac{dy}{dx} = 2\)
The angle between line is \(\tan \theta= \frac{m_2 - m_1}{1+ m_2m_1}\).
\(\theta = ~arctan (\frac{2-0.5}{1+2(0.5)})\)
\(\theta = 36.87^o\)