# Active Calculus

## Section9.5Parametric Curves

### ExercisesExercises

#### 1.

Which is a parametric equation for the curve $$16 = \left(x-9\right)^{2}+\left(y-2\right)^{2}\text{?}$$
• $$\displaystyle c(t) = \left(9+4\cos\mathopen{}\left(t\right),2+4\sin\mathopen{}\left(t\right)\right)$$
• $$\displaystyle c(t) = \left(9+16\cos\mathopen{}\left(t\right),2+16\sin\mathopen{}\left(t\right)\right)$$
• $$\displaystyle c(t) = \left(2+4\cos\mathopen{}\left(t\right),9+4\sin\mathopen{}\left(t\right)\right)$$
• $$\displaystyle c(t) = \left(2+16\cos\mathopen{}\left(t\right),9+16\sin\mathopen{}\left(t\right)\right)$$

#### 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^2-12) , \quad y = 9 (t^2-12) \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*}

#### 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 $$\left(0.25,0.5\right)$$ is
(enter undef if the slope is not defined)
(b) At this point the particle is moving
• neither left nor right
• to the left
• to the right
and
• neither up nor down
• up
• down
.
(c) The slope of the graph at $$\left(1.75,0.5\right)$$ is
(enter undef if the slope is not defined)
(d) At this point the particle is moving
• neither left nor right
• to the left
• to the right
and
• neither up nor down
• up
• down
.

#### 6.

Consider the parametric curve given by
\begin{equation*} x=t^3-12t, \qquad y= 5 t^2 - 5 \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 6 t, \cos 6 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 - 8 t \end{equation*}
\begin{equation*} y = 5 t^2 \end{equation*}