Skip to main content

Section 9.5 Parametric Curves

Exercises Exercises

1.

Which is a parametric equation for the curve \(16 = \left(x-4\right)^{2}+\left(y+8\right)^{2}\text{?}\)
  • \(\displaystyle c(t) = \left(4\cos\mathopen{}\left(t\right)-8,4+4\sin\mathopen{}\left(t\right)\right)\)
  • \(\displaystyle c(t) = \left(16\cos\mathopen{}\left(t\right)-8,4+16\sin\mathopen{}\left(t\right)\right)\)
  • \(\displaystyle c(t) = \left(4+4\cos\mathopen{}\left(t\right),4\sin\mathopen{}\left(t\right)-8\right)\)
  • \(\displaystyle c(t) = \left(4+16\cos\mathopen{}\left(t\right),16\sin\mathopen{}\left(t\right)-8\right)\)

2.

Each set of parametric equations below describes the path of a particle that moves along the circle \(x^2+(y-1)^2=4\) in some manner. Match each set of parametric equations to the path that it describes.

3.

Consider the curve given by the parametric equations
\begin{equation*} x = t (t^2-147) , \quad y = 2 (t^2-147) \end{equation*}
a.) Determine the point on the curve where the tangent is horizontal.
\(t=\)
b.) Determine the points \(t_1\text{,}\) \(t_2\) where the tangent is vertical and \(t_1 \lt t_2\) .
\(t_1=\)
\(t_2=\)

4.

Find an equation for each line that passes through the point (4, 3) and is tangent to the parametric curve
\begin{equation*} x=3t^2+1,\;\;y=2t^3+1. \end{equation*}
If there are multiple answers then separate distinct answers with commas.

5.

The functions \(f(t)\) and \(g(t)\) are shown below.
\(f(t)\) \(g(t)\)
If the motion of a particle whose position at time \(t\) is given by \(x=f(t)\text{,}\) \(y=g(t)\text{,}\) sketch a graph of the resulting motion and use your graph to answer the following questions:
(a) The slope of the graph at \(\left(1.75,0.5\right)\) is
(enter undef if the slope is not defined)
(b) At this point the particle is moving
and
.
(c) The slope of the graph at \(\left(0.25,0.5\right)\) is
(enter undef if the slope is not defined)
(d) At this point the particle is moving
and
.

6.

Consider the parametric curve given by
\begin{equation*} x=t^3-12t, \qquad y= 1 t^2 - 1 \end{equation*}
(a) Find \(dy/dx\) and \(d^2y/dx^2\) in terms of \(t\text{.}\)
\(dy/dx\) =
\(d^2y/dx^2\) =
(b) Using "less than" and "greater than" notation, list the \(t\)-interval where the curve is concave upward.
Use upper-case "INF" for positive infinity and upper-case "NINF" for negative infinity. If the curve is never concave upward, type an upper-case "N" in the answer field.
\(t\)-interval: \(\lt t \lt \)

7.

Calculate the length of the path over the given interval.
\begin{equation*} (\sin 3 t, \cos 3 t), \, 0 \le t \le \pi \end{equation*}

8.

Find the length of the curve \(x=1+3t^2,\;\;y=4+2t^3,\;\;0 \le t \le 1.\)
Length =

9.

Use the parametric equations of an ellipse
\begin{equation*} x=a\cos(\theta),\;\;y=b\sin(\theta),\;\;0 \le \theta \le 2\pi, \end{equation*}
to find the area that it encloses.
Area =

10.

Find the area of the region enclosed by the parametric equation
\begin{equation*} x = t^3 - 5 t \end{equation*}
\begin{equation*} y = 6 t^2 \end{equation*}
You have attempted of activities on this page.
PrevTopNext
Feedback