 # Active Calculus

## Section9.4Parametric Curves

### ExercisesExercises

#### 1.

Which is a parametric equation for the curve $$64 = \left(x+8\right)^{2}+\left(y-6\right)^{2}\text{?}$$
• $$\displaystyle c(t) = \left(64\cos\!\left(t\right)-8,6+64\sin\!\left(t\right)\right)$$
• $$\displaystyle c(t) = \left(6+64\cos\!\left(t\right),64\sin\!\left(t\right)-8\right)$$
• $$\displaystyle c(t) = \left(6+8\cos\!\left(t\right),8\sin\!\left(t\right)-8\right)$$
• $$\displaystyle c(t) = \left(8\cos\!\left(t\right)-8,6+8\sin\!\left(t\right)\right)$$

#### 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^2-27) , \quad y = 2 (t^2-27) \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 is more than one answer then order your answers consecutively so lines with smaller slopes appear before lines with larger slopes. If some answer fields are not used enter None in the unused answer blanks and list them last.
Tangent line 1: $$y$$ =
Tangent line 2: $$y$$ =
Tangent line 3: $$y$$ =

#### 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
• neither left nor right
• to the left
• to the right
and
• neither up nor down
• up
• down
.
(c) The slope of the graph at $$\left(0.25,0.5\right)$$ 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= 7 t^2 - 7 \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 8 t, \cos 8 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 - 2 t \end{equation*}
\begin{equation*} y = 7 t^2 \end{equation*}