Let ℓ be the tangent line to curve y = 4x^3 at point (1, 4). The angle of inclination of ℓ is angle ϕ that ℓ makes with the positive direction of the x-axis. Calculate ϕ correct to the nearest degree.

Answer 1

Remember that the derivative of a function defines the slope of the line tangent to it at that point.

If you picture the right triangle formed by ℓ (the hypotenuse) with the x-axis and angle ϕ at their intersection, its slope can be characterized by the ratio of the opposite side over the adjacent side. This is the definition of the tan function.

The derivative of y = 4x³ is dy/dx = 12x²

Therefore,

tanϕ = 12x², or for x = 1, tanϕ = 12, so ϕ = the inverse tangent of 12.