Angle between two curves

Text

Find the acute angle between the curves $y=x^{2}$ and $y=x^{3}$ at their intersection point with $x>0$.

Solution

Intersection with $x>0$: $x^{2}=x^{3}\Rightarrow x=1$, point $(1,1)$. Slopes: $m_1=2$, $m_2=3$. $\tan\theta=\left|\dfrac{m_2-m_1}{1+m_1m_2}\right|=\dfrac17\approx0.1429$, so $\theta\approx8.13^{\circ}$. $\boxed{\tan\theta=\tfrac17\approx0.143;\quad \theta\approx8.13^{\circ}}$