Unit 2 test: Vector-Valued Functions

Let $\mathbf{r}(t) = \langle t^2 + 1,\; e^{t - 1},\; \ln t \rangle$. Find $\mathbf{r}(1)$.

Find the domain of $\mathbf{r}(t) = \langle \sqrt{9 - t^2},\; \ln t,\; e^t \rangle$. Write it as an inequality in $t$.

Find $\displaystyle\lim_{t \to 1} \left\langle \frac{t^2 - 1}{t - 1},\; t^3,\; \frac{\ln t}{t - 1} \right\rangle$.

Which vector function traces the curve where the cylinder $x^2 + z^2 = 9$ meets the plane $y = 2x$?

Let $\mathbf{r}(t) = \langle \sin 2t,\; e^{-t},\; t^2 + 4t \rangle$. Find $\mathbf{r}'(0)$.

Find the unit tangent vector $\mathbf{T}(1)$ for $\mathbf{r}(t) = \langle t,\; t^2,\; \tfrac{4}{3}t^{3/2} \rangle$.

Evaluate $\displaystyle\int_0^{\pi} \langle \sin t,\; \cos t,\; t \rangle\,dt$.

Suppose $\mathbf{r}'(t) = \langle 4t,\; \cos t,\; 3t^2 \rangle$ and $\mathbf{r}(0) = \langle 3, 1, -3 \rangle$. Find $\mathbf{r}(\pi)$. Use pi for $\pi$.

Find the length of $\mathbf{r}(t) = \langle 6t,\; 3\sqrt{2}\,t^2,\; 2t^3 \rangle$ for $0 \le t \le 1$.

Find the curvature of the helix $\mathbf{r}(t) = \langle 2\cos t,\; 2\sin t,\; t \rangle$.

Find the curvature of $y = x^3$ at $x = 1$. Give an exact answer or a decimal to 4 places.

A particle has position $\mathbf{r}(t) = \langle e^t,\; e^{-t},\; \sqrt{2}\,t \rangle$. Find its speed at $t = \ln 2$.

A ball is launched from the origin with initial velocity $\langle 30, 40, 96 \rangle$ ft/s and moves under gravity alone, $\mathbf{a} = \langle 0, 0, -32 \rangle$ ft/s². Where does it land (return to $z = 0$)?

For $\mathbf{r}(t) = \left\langle 2t,\; t^2,\; \tfrac{1}{3}t^3 \right\rangle$, find the normal component of acceleration $a_N$ at $t = 1$.

For a particle with position $\mathbf{r}(t)$, which quantity equals the rate of change of its speed?