Question

Find the acute angles between the curves at their points of intersection. (The angle between two curves is the angle between
their tangent lines at the point of inter section.)
y = x2, y = x3

Answer 1

Answer:

Step-by-step explanation:

Given

curve $f(x)=x^2$

and $g(x)=x^3$

Intersection of two curves

$x^3=x^2$

$x^3-x^2=0$

$x^2(x-1)=0$

i.e. at $x=0\ and\ x=1$

Slope at $=1$

$f'(x)=2x$

at $x=1$

$f'(1)=2$

$g'(x)=3x^2$

at $x=1$

$g'(x)=3$

$a=<1,2>\ and\ b=<1,3>$

where a and b are vector with their slope at x=1

a=<1,2> i.e. passes through x=1 and slope =2

angle between them

$\cos \theta =\frac{a\cdot b}{|a||b|}$

where $\theta$ is the angle between them

$\cos \theta =\frac{1\times 1+2\times 3}{\sqrt{5}\cdot \sqrt{10}}$

$\cos \theta =\frac{7}{\sqrt{50}}$

$\theta =cos^{-1}(\frac{7}{\sqrt{50}})$

$\theta =8.15^{\circ}$