Question

find the gradient of the tangent to the parabola y=4x-x^2 at (0,0)
hence calculate the size of the angle between the line y=x and this tangent.
(as i cant show you the diagram all that it shows is the line going through (0,0) and having one point of contact)

Answer 1

This is a Calculus question.
dy/dx = 4 - 2x
at (0,) dy/dx = 4

so the slope or gradient of the tangent at (0,0) is 4
and the slope of y = x is 1

Did you know that the slope of a line is the same as the tangent of the angle that line makes with the x-axis?

so the line with slope of 4 makes an angle of 75.96º with the x-axis
the line y=x makes and angle of 45º with the x-axis

so the angle between them is (75.96-45)º
= 30.96º