Final answer:
To compute the tangent vector of the cycloid r(t) at t = π / 4, calculate the derivative r'(t) = (1 - cos t, sin t) and evaluate it at t = π / 4, giving the tangent vector (1 - √2/2, √2/2).
Step-by-step explanation:
To compute the tangent vector of the cycloid r(t) at t = π / 4, calculate the derivative r'(t) = (1 - cos t, sin t) and evaluate it at t = π / 4, giving the tangent vector (1 - √2/2, √2/2).
The student is asking how to compute the tangent vector of a cycloid at a specific point in time. The cycloid is given by the parametric equation r(t) = (t − sin t, 1 − cos t) and the time at which the tangent vector needs to be computed is t = π / 4.
To find the tangent vector, we need to calculate the derivative of the position vector r(t). The derivative r'(t) gives us the velocity vector, which is tangent to the path of the cycloid at any point t.
The derivative of the position vector r(t) is r'(t) = (1 − cos t, sin t). Substituting t = π / 4 into the derivative, we get r'(π / 4) = (1 − cos(π / 4), sin(π / 4)) which simplifies to r'(π / 4) = (1 − √2/2, √2/2). Therefore, the tangent vector of the cycloid at t = π / 4 is (1 − √2/2, √2/2).