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Section 4.1 Calculus 2

Planar Parametrizations
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4.1 Planar Parametrizations

4.1.1 Parametric Equations

  • If \(x(t),y(t)\) are both defined as functions of \(t\) over some interval \(I\) of real numbers, then the set of points \(\{(x(t),y(t)):t\in I\}\) is the parametric curve with a system of parametric equations \(x(t),y(t)\). If \(I\) is omitted, then it is assumed that \(t\) belongs to subset of the interval \((-\infty,\infty)\) where both \(x(t),y(t)\) are well-defined.
  • Example Plot the parametric curve \(x=\cos t,y=\sin t\) for \(0\leq t\leq 2\pi\), first by using a chart of \(t,x,y\) values, then by expressing the curve as an equation of \(x,y\).
  • Example Show that the systems of parametric equations \(x_0=t,y_0=t^2\) and \(x_1=2t-2,y_1=4t^2-8t+4\) share the same parametric curve.

4.1.2 Parametrizing Curves Defined by Functions

  • The curve \(y=f(x)\) where \(x\) belongs to the interval \(I\) may be easily parametrized left-to-right by the system of parametric equations \(x=t,y=f(t)\) where \(t\) also belongs to \(I\).
  • Example Give a system of parametric equations for the curve \(y=\ln x\) from \((1,0)\) to \((e^2,2)\).

4.1.3 Parametrizing Line Segments and Circles

  • The line segment joining the points \((x_0,y_0),(x_1,y_1)\) may be parametrized by \(x=x_0+(x_1-x_0)t,y=y_0+(y_1-y_0)t\) where \(0\leq t\leq 1\).
  • Example Give a system of parametric equations for the line segment joining \((2,-3)\) and \((-1,4)\).
  • Example Give two different systems of parametric equations for the portion of the line \(y=3x-2\) between \(x=-1\) and \(x=2\).
  • The full line may be obtained with the same equations by allowing \(t\) to range over all real numbers.
  • The circle with center \((x_0,y_0)\) and radius \(r\) may be parametrized counter-clockwise by \(x=x_0+r\cos\theta,y=y_0+r\sin\theta\) where \(0\leq\theta\leq2\pi\).
  • Example Give a system of parametric equations for the circle \(x^2+y^2=9\).
  • The circle with center \((x_0,y_0)\) and radius \(r\) may be parametrized clockwise by \(x=x_0+r\sin\theta,y=y_0+r\cos\theta\) where \(0\leq\theta\leq2\pi\).
  • Example Give a system of parametric equations for the circle \((x-3)^2+(y+4)^2=25\) beginning at \((3,1)\) and moving clockwise.

Exercises for 4.1

  1. Plot the parametric curve \(x=2-t^2,y=2t^2\) for \(0\leq t\leq 3\), first by using a chart of \(t,x,y\) values, then by expressing the curve as an equation of \(x,y\).
  2. Plot the parametric curve \(x=3^t,y=3^{-t}\) for \(-\infty<t<\infty\), first by using a chart of \(t,x,y\) values, then by expressing the curve as an equation of \(x,y\).
  3. Show that the systems of parametric equations \(x_0=t+2,y_0=e^2e^t\) and \(x_1=\ln t,y_1=t\) share the same parametric curve. Then plot that curve.
  4. Give a system of parametric equations for the curve \(y=\cosh x\) from \((-\ln 2,5/4)\) to \((\ln 2,5/4)\).
  5. Give a system of parametric equations for the line segment joining \((0,-4)\) and \((3,5)\).
  6. Give a system of parametric equations for the line segment joining \((1,2)\) and \((-3,3)\).
  7. Give two different systems of parametric equations for the portion of the line \(y=4-3x\) between \(x=-2\) and \(x=3\).
  8. Let \(a<b\). Find a system of parametric equations which parametrizes the planar curve \(y=f(x)\) right-to-left from \(x=b\) to \(x=a\).
  9. Give a system of parametric equations that parametrize the circle \(x^2+(y+1)^2=9\) counter-clockwise.
  10. Give a system of parametric equations that parametrize the arc of the circle \((x-2)^2+(y+3)^2=4\) clockwise from \((2,-5)\) to \((4,-3)\).

Solutions

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 10.1