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Section 12.3 Assignment 3

  1. Write these cartesian loci as polar loci. (6)

    1. \begin{equation*} 4x + 5y = 1 \end{equation*}

    2. \begin{equation*} \frac{1}{3x^2 + 3y^2 + 4} = 7 \end{equation*}

    3. \begin{equation*} \frac{y}{x} + x^2 + y^2 = 7 \end{equation*}

  2. Write these cartesian loci as spherical loci. (6)

    1. \begin{equation*} 4x + 3y - 5z = 4 \end{equation*}

    2. \begin{equation*} \frac{1}{x^2 + y^2 + z^2} - \sqrt{x^2 + y^2 + z^2} = 3 \end{equation*}

    3. \begin{equation*} \frac{5z}{\sqrt{x^2 + y^2}} - x^2 - y^2 - z^2 = 4 \end{equation*}

  3. Consider this parametric curve (in cartesian coordinates). (6)

    \begin{equation*} \gamma(t) = (e^{t+1}, e^{t-1}) \end{equation*}

    1. Find \(\gamma(0)\text{.}\)

    2. Determine the length of the curve on \(t \in [0,4]\text{.}\)

    3. Qualitatively describe the behaviour of the curve as \(t \rightarrow \infty\text{.}\)

  4. Consider this parametric curve (in cartesian coordinates). (6)

    \begin{equation*} \gamma(t) = (t-7, 4t+3) \end{equation*}

    1. Find \(\gamma(0)\text{.}\)

    2. Determine the length of the curve on \(t \in [0,4]\text{.}\)

    3. Qualitatively describe the behaviour of the curve as \(t \rightarrow \infty\text{.}\)

  5. Consider this parametric curve. (6)

    \begin{equation*} \gamma(t) = (2 \sin t \cos t, \cos 2t) \end{equation*}

    1. Find \(\gamma(0)\text{.}\)

    2. Determine the length of the curve on \(t \in [0,4]\text{.}\)

    3. Qualitatively describe the behaviour of the curve as \(t \rightarrow \infty\text{.}\)

  6. Consider this parametric curve. (8)

    \begin{align*} \amp \gamma(t) = (t^2, t^2 +1, t^2 +2) \amp \amp t \in [0, \infty] \end{align*}

    1. Calculate \(\gamma(1)\text{.}\)

    2. Determine the intersection (if any) of the curve with the unit sphere.

    3. Determine the intersection (if any) of the curve the the cube with vertices \((\pm 1, \pm 1, \pm 1)\text{.}\)

  7. Consider this parametric curve. (8)

    \begin{align*} \amp \gamma(t) = (\sin (\pi t) + \cos (\pi t), \sin (\pi t) - \cos (\pi t), 2t) \amp \amp t \in [0, \infty] \end{align*}

    1. Calculate \(\gamma(1)\text{.}\)

    2. Determine the intersection (if any) of the curve with the unit sphere.

    3. Determine the intersection (if any )of the curve the the cube with vertices \((\pm 1, \pm 1, \pm 1)\text{.}\)

  8. Consider this parametric curve. (6)

    \begin{align*} \amp \gamma(t) = (t, \frac{1}{t}, 6) \amp \amp t \in (0, \infty] \end{align*}

    1. Calculate \(\gamma(1)\text{.}\)

    2. Determine the intersection (if any) of the curve with the unit sphere.

    3. Determine the intersection (if any )of the curve the the cube with vertices \((\pm 1, \pm 1, \pm 1)\text{.}\)

  9. Parametrize the following curve by arclength for \(t > 0\text{.}\) (8)

    \begin{equation*} \gamma(t) = \left( t \sin t, t \cos t, \frac{2 \sqrt{2} t^{\frac{3}{2}}}{3} \right) \end{equation*}