Section 5.1 Practice

Questions

  1. Take \(f\) to be the function from \([-2,1]\) to \(\mathbb R\) that is given by \[f(x)=2x^2+1.\] Find parameterizations for \(f\) (viewed as a subset of \(\mathbb R^2\)) that have domains equal to the following intervals:

    1. \([-2,1]\)
    2. \([0,3]\)
    3. \([-3,3]\)

  2. Take \(D\) to be the disk of radius \(5\) with center at \((-1,5).\) Find a path that parameterizes the boundary of \(D\) clockwise.

  3. Take \(D\) to be the disk of radius \(2\) with center at \((-2,-2).\) Find a path that parameterizes the boundary of \(D\) counter-clockwise.

  4. Take \(D\) to be the disk of radius \(6\) with center \((2,-6)\).

    1. Determine a parameterization of \(\partial D\), the boundary of \(D\).
    2. Determine an inequality for which \(D\) is a feasible set.
    3. Identify a point in \(D\), a point on \(\partial D\), and a point outside of \(D.\)

  5. Take \(E\) to be the solid ellipse that is given by the equation \[\frac{(x-3)^2}{9}+(y+5)^2 \leq 9.\]

    1. Find a point that is inside \(E.\)
    2. Find a path \(c\) that parameterizes the ellipse \(\partial E.\)

  6. Take \(R\) to be the rectangle with vertices \((1,3)\), \((4,4)\), \((3,7)\), and \((0,6).\)

    1. Find a piecewise linear function \(c\) that parameterizes \(\partial R\) and simulate the motion of a particle whose position at time \(t\) is \(c(t).\)
    2. Describe \(R\) as the feasible set for a system of inequalities.
    3. Determine whether \(p=(2,3.33332)\) is inside or outside \(R.\)

  7. For each triangle \(\Delta p_1p_2p_3\), use the shoelace formula to describe the area of the triangle and determine whether or not the triangle is positively or negatively oriented.

    1. \(p_1=(-1,3),p_2=(1,5),p_3=(0,7)\)
    2. \(p_1=(0,-3),p_2=(3,0),p_3=(-1,3)\)
    3. \(p_1=(3,5),p_2=(2,5),p_3=(2,9)\)

  8. For each parallelogram \(\square p_1p_2p_3p_4\), calculate its area.

    1. \(p_1=(2,4),p_2=(6,7),p_3=(8,12),p_4=(4,9)\)
    2. \(p_1=(1,-1),p_2=(3,2),p_3=(8,4),p_4=(6,1)\)
    3. \(p_1=(3,1),p_2=(4,6),p_3=(9,7),p_4=(8,2)\)

  9. For each ordered vertex set of a polygon path and epoch, identify the given path with a piecewise linear function, \(c\).

    1. ordered vertex set \(((1,4),(3,6),(2,7),(1,5),(-1,3),(-2,6))\), epoch \((0,2,4,5,7,8,10).\)
    2. \(((0,0),(2,3),(2,6),(0,6),(-3,6),(-3,0))\), epoch \((0,3,6,8,10,12,15).\)

  10. For each vertex set given for a triangle \(\Delta\), determine the circumcircle for \(\Delta\).

    1. \(\{(2,5),(2,0),(0,3)\}.\)
    2. \(\{(1,1),(5,2),(3,4)\}.\)

  11. For each triangle given, determine its area and determine whether or not the triangle is positively or negatively oriented:

    1. \(\Delta ((1,3)(-6,1)(-12,-1))\)
    2. \(\Delta ((2,7)(-4,0)(-9,-3))\)

  12. Determine the area of a triangle with side lengths \(4\), \(14\), and \(17\).

Answers

    1. \(c(t)=(t,2t^2+1)\) where \(t\) is in \([-2,1]\)
    2. \((t-2,2(t-2)^2+1)\) where \(t\) is in \([0,3]\)
    3. \(\left(\frac{1}{2}(t-1),2(\frac{1}{2}(t-1))^2+1\right)\) where \(t\) is in \([-3,3]\)

  2. \(c(t)=(5\cos(-t)-1,5\sin(-t)+5)\) where \(t\) is in \([0,2\pi]\)

  3. \(c(t)=(2\cos(t)-2,2\sin(t)-2)\) where \(t\) is in \([0,2\pi]\)

    1. \(c\colon [0,2\pi]\to\mathbb{R}^2\) with \(c(t)=(6\cos(t)+2,6\sin(t)-6)\)
    2. \((x-2)^2+(y+6)^2\leq 36\)
    3. \(p=(2,-5)\) is in \(D\), \(q=(2,-12)\) is in \(\partial D\), and \(u=(2,6)\) is not in \(D\).
    1. Any point in the disk centered at \((0,0)\) with radius \(r=3\) that is transformed by \(\langle 3, -5\rangle+X_3.\) For example, \((1,1)\) is in the disk centered at \((0,0)\) with radius \(r=3\). So the point \(p=\langle 3,-5\rangle +X_3(1,1)\) will be in \(E\). The point is \(p=(6,-4).\)
    2. \((9\cos(t)+3,3\sin(t)-5)\) where \(t\) is in \([0,2\pi]\)
    1. \[c(t)=\begin{cases}\sqrt{10}\left\langle \frac{3}{\sqrt{10}},\frac{1}{\sqrt{10}}\right\rangle t+(1,3) &\text{ if }0\leq t< 1\\ \sqrt{10}\left\langle -\frac{1}{\sqrt{10}},\frac{3}{\sqrt{10}}\right\rangle (t-1)+\sqrt{10}\left\langle \frac{3}{\sqrt{10}},\frac{1}{\sqrt{10}}\right\rangle +(1,3) &\text{ if }1\leq t< 2\\ -\sqrt{10}\left\langle \frac{3}{\sqrt{10}},\frac{1}{\sqrt{10}}\right\rangle (t-2)+\sqrt{10}\left\langle -\frac{1}{\sqrt{10}},\frac{3}{\sqrt{10}}\right\rangle +\sqrt{10}\left\langle \frac{3}{\sqrt{10}},\frac{1}{\sqrt{10}}\right\rangle +(1,3) &\text{ if }2\leq t< 3\\ -\sqrt{10}\left\langle -\frac{1}{\sqrt{10}},\frac{3}{\sqrt{10}} \right\rangle (t-3)+\sqrt{10}\left\langle -\frac{1}{\sqrt{10}},\frac{3}{\sqrt{10}} \right\rangle+(1,3) &\text{ if }3\leq t< 4 \end{cases}\] or \[c(t)=\begin{cases}\left\langle 3,1\right\rangle t+(1,3) &\text{ if }0\leq t< 1\\ \left\langle -1,3\right\rangle (t-1)+(4,7) &\text{ if }1\leq t< 2\\ \left\langle- 3,-1\right\rangle (t-2)+(3,7) &\text{ if }2\leq t< 3\\ \left\langle 1,-3 \right\rangle (t-3)+(0,6) &\text{ if }3\leq t< 4 \end{cases}\]
    2. \[\begin{cases}y\geq\frac{1}{3}(x-1)+3\\y\leq-3(x-4)+4\\ y\leq\frac{1}{3}x+6\\ y\geq-3(x-1)+3\end{cases}\]
    3. Check if it satisfies the above inequalities or translate and rotate the rectangle so one of the edges is on the \(x\)-axis. If you do it the second way, the rotated rectangle has vertices at \((0,0),\left(\frac{10}{\sqrt{10}},0\right),\left(\frac{10}{\sqrt{10}},\frac{10}{\sqrt{10}}\right),\left(0,\frac{10}{\sqrt{10}}\right).\) You can also rewrite these vertices as \((0,0),\left(\sqrt{10},0\right),\left(\sqrt{10},\sqrt{10}\right),\left(0,\sqrt{10}\right).\) Take the point \(p\), and do the same translation and rotation you did to rotate the rectangle. The new point is \(\left(\frac{3.33332}{\sqrt{10}},-\frac{0.00004}{\sqrt{10}}\right).\) For this translated and rotated point to be in the translated and rotated rectangle, its \(x\) and \(y\) coordinates must satisfy \(0\leq x\leq\frac{10}{\sqrt{10}}\) and \(0\leq y\leq \frac{10}{\sqrt{10}}\). The \(y\)-coordinate requirement is not satisfied. So the original \(p\) is not in the rectangle \(R.\)
    1. \(\mathcal{A}(\Delta p_1p_2p_3)=3\), positively oriented.
    2. \(\mathcal{A}(\Delta p_1p_2p_3)=\frac{21}{2}\), positively oriented.
    3. \(\mathcal{A}(\Delta p_1p_2p_3)=2\), negatively oriented.
    1. \(\mathcal{A}(\square p_1p_2p_3p_4)=14\)
    2. \(\mathcal{A}(\square p_1p_2p_3p_4)=11\)
    3. \(\mathcal{A}(\square p_1p_2p_3p_4)=24\)
    1. \(c(t)=\begin{cases}\langle 2,-2 \rangle\frac{t}{2}+(1,4) &\text{ if } 0\leq t<2\\ \langle -1,-1 \rangle\frac{t-2}{2}+(3,6) &\text{ if } 2\leq t<4\\ \langle -1,2 \rangle(t-4)+(2,7) &\text{ if } 4\leq t<5\\\langle -2,2 \rangle\frac{t-5}{2}+(1,5) &\text{ if } 5\leq t<7\\ \langle -1,-3 \rangle(t-7)+(-1,3) &\text{ if } 7\leq t<8\\ \langle 2,6 \rangle\frac{t-8}{2}+(-2,6) &\text{ if } 8\leq t\leq 10\\ \end{cases}\)
    2. \(c(t)=\begin{cases}\langle 2,-3 \rangle\frac{t}{3}+(0,0) &\text{ if } 0\leq t<3\\ \langle 0,-3 \rangle\frac{t-3}{3}+(2,3) &\text{ if } 3\leq t<6\\ \langle -2,0 \rangle\frac{t-6}{2}+(2,6) &\text{ if } 6\leq t<8\\\langle -3,0 \rangle\frac{t-8}{2}+(0,6) &\text{ if } 8\leq t<10\\ \langle 0,6 \rangle\frac{t-10}{2}+(-3,6) &\text{ if } 10\leq t<12\\ \langle 3,0 \rangle\frac{t-12}{3}+(-3,0) &\text{ if } 8\leq t\leq 10\\ \end{cases}\)
    1. \(\left(x-\frac{5}{2}\right)^2+\left(y-\frac{5}{2}\right)^2=\frac{13}{2}\)
    2. \(\left(x-\frac{29}{10}\right)^2+\left(y-\frac{19}{10}\right)^2=\frac{221}{50}.\)
    1. Area is \(1\), triangle is positively oriented
    2. Area is \(8.5\), triangle is negatively oriented

  12. \(\frac{\sqrt{6,615}}{4}\)

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