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 = 6x2, y = 6x3

Answer 1

At point of intersection the two equations are equal,
hence, 6x³ =6x²
           6x³-6x²=0
           6x²(x-1)=0 , the values of x are 0 and 1
The points of intersection are therefore, (0,0) and (1,6)
To find the slopes of the tangents at the points of intersection we find dy/dx
for curve 1, dy/dx=12x, and the other curve dy/dx=18x²
At x=0, dy/dx=12x =0, dy/dx=18x² = 0, hence the angle between the tangents is 0, because the tangents to the two curves have the same slope which is 0 and pass the same point (0,0) origin.
At x=1, dy/dx =12x = 12, dy/dx= 18x² =18, Hence the angle between the two tangents will be given by arctan 18 -arctan 12 
                                      = 86.8202 - 85.2364 ≈ 1.5838, because the slope of the lines is equal to tan α where α is the angle of inclination of the line.