Question

Define φ:[0,2π]→R² by φ(t)=(sin(t),cos(t)). Compute φ′(t) for all t.

Compute ∥φ′(t)∥ for all t.
Notice that φ′(t) is never zero, yet φ(0)=φ(2π), therefore, Rolle's theorem is not true in more than one dimension.

Answer 1

To compute φ′(t), take the derivative of each component of φ(t) and compute the norm of φ'(t).

Step-by-step explanation:

To compute φ′(t), we need to take the derivative of each component of φ(t). Let's start with the first component:

φ₁(t) = sin(t)

The derivative of sin(t) is cos(t), so φ₁'(t) = cos(t)

Now let's compute the derivative of the second component:

φ₂(t) = cos(t)

The derivative of cos(t) is -sin(t), so φ₂'(t) = -sin(t)

Therefore, the derivative of φ(t) is φ'(t) = (cos(t), -sin(t))

To compute the norm of φ'(t), we use the formula ∥v∥ = sqrt(v₁² + v₂²)

Substituting φ'(t) into the formula, we get ∥φ'(t)∥ = sqrt((cos(t))² + (-sin(t))²) = sqrt(cos²(t) + sin²(t)) = sqrt(1) = 1