Exercise 37

Consider the two vector-valued functions given by $$\begin{gathered} \vr(t) = \left\langle t + 1, \cos\left(\frac{\pi}{2} t\right), \frac{1}{1+t} \right\rangle \end{gathered}$$ and $$\begin{gathered} \vw(s) = \left\langle s^2, \sin\left(\frac{\pi}{2}s\right), s \right\rangle. \end{gathered}$$

1. Determine the point of intersection of the curves generated by $\vr(t)$ and $\vw(s)$. To do so, you will have to find values of $a$ and $b$ that result in $\vr(a)$ and $\vw(b)$ being the same vector.

2. Use the value of $a$ you determined in (a) to find a vector form of the tangent line to $\vr(t)$ at the point where $t = a$.

3. Use the value of $b$ you determined in (a) to find a vector form of the tangent line to $\vw(s)$ at the point where $s = b$.

4. Suppose that $z = f(x,y)$ is a function that generates a surface in three-dimensional space, and that the curves generated by $\vr(t)$ and $\vw(s)$ both lie on this surface. Note particularly that the point of intersection you found in (a) lies on this surface. In addition, observe that the two tangent lines found in (b) and (c) both lie in the tangent plane to the surface at the point of intersection. Use your preceding work to determine the equation of this tangent plane.