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

A parametric curve is defined by a pair of equations of the form:

- **A)** \(r=f(\theta)\) alone
- **B)** \(y=f(x)\) alone
- **C)** \(x=f(y)\) alone
- **D)** \(x=f(t)\), \(y=g(t)\), where both \(x\) and \(y\) are expressed as functions of a third variable \(t\)

For a parametric curve, \(\dfrac{dy}{dx}\) is computed as:

- **A)** \(\dfrac{dy}{dx} = \dfrac{dx/dt}{dy/dt}\)
- **B)** \(\dfrac{dy}{dx} = dy/dt\) alone
- **C)** \(\dfrac{dy}{dx} = \dfrac{dy}{dt} \cdot \dfrac{dx}{dt}\)
- **D)** \(\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}\), provided \(dx/dt \ne 0\)

For \(x=t^2\) and \(y=t^3\), find \(\dfrac{dy}{dx}\) in terms of \(t\).

- **A)** \(\dfrac{3t^2}{2t}\) unsimplified
- **B)** \(6t\)
- **C)** \(\dfrac{2t}{3}\)
- **D)** \(\dfrac{3t}{2}\)

The second derivative \(\dfrac{d^2y}{dx^2}\) for a parametric curve is computed as:

- **A)** \(\dfrac{d^2y}{dx^2} = \dfrac{d}{dt}\left[\dfrac{dy}{dx}\right] \div \dfrac{dx}{dt}\)
- **B)** \(\dfrac{d^2y}{dx^2} = \dfrac{d}{dt}\left[\dfrac{dy}{dx}\right]\)
- **C)** \(\dfrac{d^2y}{dx^2} = \dfrac{dy}{dx} \cdot \dfrac{dx}{dt}\)
- **D)** \(\dfrac{d^2y}{dx^2} = \dfrac{d^2y/dt^2}{d^2x/dt^2}\)

For \(x=t^2, y=t^3\) (with \(dy/dx = 3t/2\)), find \(\dfrac{d^2y}{dx^2}\).

- **A)** \(\dfrac{3}{2}\)
- **B)** \(\dfrac{3t}{4}\)
- **C)** \(\dfrac{3}{4t}\)
- **D)** \(\dfrac{3}{4}\)

The arc length of a parametric curve from \(t=a\) to \(t=b\) is given by:

- **A)** \(L = \sqrt{(b-a)^2}\)
- **B)** \(L = \int_a^b \dfrac{dy}{dx}\,dt\)
- **C)** \(L = \int_a^b \left(\dfrac{dx}{dt} + \dfrac{dy}{dt}\right)dt\)
- **D)** \(L = \int_a^b \sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2}\,dt\)

Find the arc length of \(x=3t, y=4t\) from \(t=0\) to \(t=2\).

- **A)** \(5\)
- **B)** \(8\)
- **C)** \(10\)
- **D)** \(6\)

A vector-valued function is typically written as:

- **A)** \(\vec{r}(t) = x(t) \cdot y(t)\)
- **B)** \(\vec{r}(t)\) always has exactly one component
- **C)** \(\vec{r}(t) = \langle x(t), y(t) \rangle\), combining two component functions into a single vector output
- **D)** \(\vec{r}(t) = x(t) + y(t)\)