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 = 7x2, y = 7x3

Answer 1

The angles between the curves at the points of intersection are:

0º, 1.3º

Explanation:

The intersections points are found by setting the equations equal to each other and solving the resulting equation:

`7x^2=7x^3\\x^3-x^2=0\\x^2(x-1)=0\\x=0,x=1`

The angles of the tangent lines can be found by stating their slopes.

To find the slope we differentiate the equations:

`y'_1=14x,y'_2=21x^2`

Then we plug the x-coordinates of the intersections:

For x=0 we get the slopes are both 0:

`y'_1=14(0)=0,y'_2=21(0)^2=0`

So the angles of inclination of the lines are the same their difference is 0. Hence the angle between the tangent curves is also 0º at the point of intersection at x=0

For x=1 we get the following slopes:

`y'_1=14(1)=14,y'_2=21(1)^2=21`

The slopes are the tangents of the angles. Therefore, to get the angle between the lines we do:

`arctan(21)-arctan(14)\approx87.2737\º-85.9144\º\approx1.3\º`

So, 1.3º is the angle between the curves at the second point of intersection at x=1.