Mathematics Notes - Differential Calculus Problems and Solutions

Find the angle between parabolas $y^2 =x$ and $y =x^2$ at the points of their intersections.

Find the intersection:

$y^2 = \sqrt{y}$

$y^4 = y$

$y^4 - y = 0$

$y=0 ~ and ~ y=1$

Intersection $(0,0), (1,1)$

$the angle of intersection between two parabolas$

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 $\tan \theta= \frac{m_2 - m_1}{1+ m_2m_1}$.

$\theta = ~arctan (\frac{2-0.5}{1+2(0.5)})$

$\theta = 36.87^o$

$the angle of intersection between two parabolas$