Write the equation of a tangent to the curve `x=t, y=t^2 and z=t^3` at its point `M(1, 1, 1): (t=1)`.

Answer

To find the equation of the tangent to the curve defined by the parametric equations \( x = t \), \( y = t^2 \), and \( z = t^3 \) at the point \( M(1, 1, 1) \) where \( t = 1 \), we can follow these steps:

### Step 1: Find the derivatives of the parametric equations.
We start by differentiating the equations with respect to \( t \):

1. \( \frac{dx}{dt} = 1 \)
2. \( \frac{dy}{dt} = 2t \)
3. \( \frac{dz}{dt} = 3t^2 \)