Calculate 𝜑 to the nearest degree.

Question

Let ℓ be the tangent line to the curve y = 4x²

 at the point (1, 4). The angle of inclination of ℓ is the angle 𝜑 that ℓ makes with the positive direction of the x-axis. Calculate 𝜑 to the nearest degree.

Answer 1

f(x) = 4x²

The slope of the line is:

f'(x)= 8x

f'(1)=m=8

The tangent line, ℓ, is

y-y₀ = m(x-x₀)

y-4 = 8(x-1)

y = 8x-4

This line crosses the x-axis at x = 1/2.

At this point, tan(φ) = 4/(1/2) = 8

φ = tan^-1(8) ≈ 83°

Answer 2

f(x) = 4x² f'(x) = 8x

Slope of tangent line at (1, 4) = f'(1) = 8

Angle of inclination = φ = Tan^-1(8) = 82.87°