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 intersection. Give your answers in degrees, rounding to one decimal place. Enter your answers as a comma-separated list.) y = 2x2, y = 2x3

Answer 1

Answer:

4.6  degrees or zero degree

Step-by-step explanation:

We are given that two curves

$y=2x^2,y=2x^3$

We have to find the angle between two curves at the point of intersection.

At the point of intersection the value of both curves equal

Therefore, $2x^2=2x^3$

[tex2x^2-2x^3=0[/tex]

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

$x=0, x=1$

Let $g(x)=2x^2$

$f(x)=2x^3$

$g'(x)_1=4x$

$g(1)=m_1=4,g(0)=0$

$f'(x)=6x^2$

$f'(1)=m_2=6$

$f'(0)=0$

The angle between two curves is given by

$tan\alpha=\mid \frac{m_1-m_2}{1+m_1m_2}\mid$

Substitute the values then we get

$tan\alpha=\mid \frac{4-6}{1+(4)(6)}\mid$

$tan\alpha=\frac{2}{25}$

$\apha=tan^{-1}(\frac{2}{25})=tan^{-1}(0.08)=4.6 degree$

If we substitute $m_1=m_2=0$

$tan\alpha=\mid \frac{0-0}{1+0}\mid=0$

Hence, the acute angles between two given curves=4.6 degrees or 0 degree.