Q.Find the acute angles between the curves at their points of intersection: $y=x^2$, $y=x^3$.

$y=x^2$ and $y=x^3$ intersect where $x^2=x^3\Rightarrow x^2(x-1)=0\Rightarrow x=0$ or $x=1$, giving points

$(0,0)$ and $(1,1)$.

Slopes: for $y=x^2$, $dy/dx=2x$; for $y=x^3$, $dy/dx=3x^2$.

At $(0,0)$: both slopes are $0$, so both curves have the same tangent line there (the X-axis) -- they meet

tangentially, with angle $0$ between them.

At $(1,1)$: slope of $y=x^2$ is $2$; slope of $y=x^3$ is $3$. Using …

Find the intersection points by equating the two curve equations, differentiate each curve to get the slope at each intersection point, and apply the …

Reporting only one of the two intersection points and missing that (0,0) is a special (tangential, zer …