AP Calculus BC Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

For the parametric curve $x=t^3-3t$, $y=t^2$, find $\frac{d^{2}y}{d{x}^2}$ at $t=2$.

For the parametric curve $x=\cos t$, $y=\sin t$, find $\frac{d^{2}y}{d{x}^2}$.

A curve is given parametrically by $x(t)=t^2-1$ and $y(t)=2t$. Find $\frac{dy}{dx}$ in terms of $t$.

Find the area enclosed by the polar curve $r=3\sin \theta$ for $\theta \in [0,\pi ]$.

Find the area of the region inside $r=2$ and outside $r=2-2\cos \theta$.

A particle has acceleration $\mathbf{a}(t)=⟨2,6t⟩$ with $\mathbf{v}(0)=⟨1,0⟩$ and $\mathbf{r}(0)=⟨0,0⟩$. Find $\mathbf{r}(1)$.

A particle moves with position $\mathbf{r}(t)=⟨t^2-1,2t⟩$. Find the speed at $t=2$.

A vector-valued function is $\mathbf{r}(t)=⟨t^2,\sin t⟩$. Find ${\mathbf{r}}^{\prime }(t)$.

For the cardioid $r=1+\cos \theta$, find the total arc length.

For the polar curve $r=2\cos \theta$, find the area enclosed.

Find the slope of the tangent line to the polar curve $r=1+\sin \theta$ at $\theta =\pi /2$.

A particle's position is $\mathbf{r}(t)=⟨e^t,t^2⟩$. Find the total distance traveled from $t=0$ to $t=1$.