[10.4] - Reading Assignment

After reading section 10.4...

  1. What is the relationship between the velocity vector and the tangent vector?

    The velocity vector $\myv r'(t)$ is always tangent to the space curve ("trajectory") $\myv r(t)$. So $\myv r'$ is parallel to any tangent vector. It is a scalar multiple of any

  2. A particle moves along the curve defined by the equation $y = \sin (\pi x)$. The $x$-coordinate, $ x (t)$ of the particle satisfies the equation $\frac{dx}{dt}=e^{2t}$. At $t=0$ the curve is at the point $(\frac12,1)$.
    1. Find $x(t)$ in terms of $t$.

      $$\begineq x(t)-x(0)&=&\int_0^t e^{2t'}\,dt'=\left.\frac 12e^{2t'}\right|^{t'=t}_0\\ x(t)-\frac 12&=&\frac 12\left(e^{2t}-e^0\right)=\frac 12e^{2t}-\frac 12\\ x(t)&=&\frac 12e^{2t}\endeq$$

    2. Find $y(t)$ in terms of $t$.

      $$y(x(t))=\sin(\pi x(t))=\sin(\frac12\pi e^{2t}).$$

    3. Find the velocity vector, $\myv v(t)$.

      $$\myv v(t)=\myv r'(t)=\langle x'(t), y'(t) \rangle$$ And we already have $\frac{dx}{dt}=e^{2t}$. And $$y'(t)=\frac{d}{dt}\sin\left(\frac12\pi e^{2t}\right)= \cos\left(\frac12\pi e^{2t}\right) \, \pi e^{2t}$$

  3. Muddy questions? Questions you wonder about?