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)
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
To find the gradient of the tangent to the parabola, we need to find the derivative of the given parabolic equation y = 4x - x^2 with respect to x.
Differentiating y = 4x - x^2 using the power rule for derivatives, we get:
dy/dx = 4 - 2x
Now, to find the gradient of the tangent at the point (0,0), substitute x = 0 into the derivative:
dy/dx = 4 - 2(0)
dy/dx = 4
Therefore, the gradient of the tangent to the parabola y = 4x - x^2 at the point (0,0) is 4.
To calculate the size of the angle between the line y = x and the tangent, we can use the fact that the angle between two lines is equal to the arctan of the difference in their slopes.
The slope of the line y = x is 1, and the slope of the tangent we found is 4. Therefore, the difference in slopes is 4 - 1 = 3.
Now, we can calculate the size of the angle using the arctan function:
angle = arctan(3)
Using a calculator, we find that angle ≈ 71.57 degrees.
Hence, the size of the angle between the line y = x and the tangent to the parabola at (0,0) is approximately 71.57 degrees.