Unit 5 test: Parametric, Polar and Vector-Valued Functions

A curve is given by $x = t^2 + 2t$ and $y = t^3 - 4$. Find $\dfrac{dy}{dx}$ at $t = 1$.

For $x = \sin t$ and $y = t^2$, find $\dfrac{dy}{dx}$ in terms of $t$.

The curve $x = t^2 - 6t$, $y = t^3$ has exactly one vertical tangent line. At what point $(x, y)$ does it occur?

For $x = t^2$ and $y = \ln t$ with $t > 0$, find $\dfrac{d^2y}{dx^2}$ at $t = 1$.

For the curve $x = t^3$, $y = t^2$, which statement about concavity is true for $t \ne 0$?

Find the length of the curve $x = 3t^2$, $y = 2t^3$ for $0 \le t \le \sqrt{3}$.

Which integral gives the length of the curve $x = \ln t$, $y = t^2$ for $1 \le t \le 3$?

Let $\vec{r}(t) = \langle e^{2t},\; t^3 - t \rangle$. Find $\vec{r}\,'(0)$.

A ball's acceleration is $\vec{a}(t) = \langle 0, -10 \rangle$. Its velocity at $t = 0$ is $\langle 3, 4 \rangle$ and its position at $t = 0$ is $\langle 0, 0 \rangle$. Find its position at $t = 1$.

A particle moves with velocity $\vec{v}(t) = \langle 3t,\; t^2 + 2 \rangle$. What is its speed at $t = 1$? Give an exact answer or a decimal to three places.

Calculator allowed. A particle moves with velocity $\vec{v}(t) = \langle \sin(t^2),\; \cos t \rangle$. Find the total distance it travels from $t = 0$ to $t = 2$, to three decimal places.

Find the slope of the polar curve $r = 4\cos(2\theta)$ at $\theta = \dfrac{\pi}{6}$. Give an exact answer or a decimal to three places.

For the polar curve $r = 1 + 3\sin\theta$, is the distance from the origin increasing or decreasing at $\theta = \dfrac{7\pi}{6}$?

Find the area of one petal of the rose $r = 2\sin(3\theta)$.

Find the area of the region inside $r = 2 + 2\cos\theta$ and outside $r = 2$. Give an exact answer or a decimal to three places.