Problems

01

Tangent lines to parametric curves

For the following parametric curves, find a parametrization of the tangent line to the curve at the given time:

• (a) $(t^2,t^4,0)$ at $t=-2$
• (b) $(1-t^2,5t,2t^3)$ at $t=2$
• (c) $(t,t,\sqrt{4-t^2})$ at $t=1$

02

Natural vectors and curvature

For the following parametric space curves, compute $\mathbf{a}(t)$, $\mathbf{T}(t)$, $\mathbf{N}(t)$, and $\kappa (t)$. Decompose $\mathbf{a}$ in terms of $\mathbf{T}$ and $\mathbf{N}$.

• (a) $(t^2,\ln (t),t\ln (t))$
• (b) $(e^t,e^{-t},t)$

03

Quadratic curve is planar

The component functions for a parametric space curve are quadratic polynomials in the parameter $t$. Show that this space curve lies in a plane.