AC Parametric Curves

Section 9.5 Parametric Curves

Exercises Exercises

1.

Which is a parametric equation for the curve \(81 = \left(x+7\right)^{2}+\left(y-5\right)^{2}\text{?}\)

\(\displaystyle c(t) = \left(81\cos\mathopen{}\left(t\right)-7,5+81\sin\mathopen{}\left(t\right)\right)\)
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\(\displaystyle c(t) = \left(5+9\cos\mathopen{}\left(t\right),9\sin\mathopen{}\left(t\right)-7\right)\)
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\(\displaystyle c(t) = \left(9\cos\mathopen{}\left(t\right)-7,5+9\sin\mathopen{}\left(t\right)\right)\)
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\(\displaystyle c(t) = \left(5+81\cos\mathopen{}\left(t\right),81\sin\mathopen{}\left(t\right)-7\right)\)
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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.

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3.

Consider the curve given by the parametric equations

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\begin{equation*} x = t (t^2-48) , \quad y = 2 (t^2-48) \end{equation*}

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a.) Determine the point on the curve where the tangent is horizontal.

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\(t=\)

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b.) Determine the points \(t_1\text{,}\) \(t_2\) where the tangent is vertical and \(t_1 \lt t_2\) .

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\(t_1=\)

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\(t_2=\)

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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.

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5.

The functions \(f(t)\) and \(g(t)\) are shown below.

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\(f(t)\)	\(g(t)\)

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:

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(a) The slope of the graph at \(\left(1.75,0.5\right)\) is

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(enter undef if the slope is not defined)

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(b) At this point the particle is moving

neither left nor right
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to the left
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to the right
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and

neither up nor down
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up
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down
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(c) The slope of the graph at \(\left(0.25,0.5\right)\) is

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(enter undef if the slope is not defined)

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(d) At this point the particle is moving

neither left nor right
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to the left
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to the right
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and

neither up nor down
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up
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down
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6.

Consider the parametric curve given by

\begin{equation*} x=t^3-12t, \qquad y= 8 t^2 - 8 \end{equation*}

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(a) Find \(dy/dx\) and \(d^2y/dx^2\) in terms of \(t\text{.}\)

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\(dy/dx\) =

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\(d^2y/dx^2\) =

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(b) Using "less than" and "greater than" notation, list the \(t\)-interval where the curve is concave upward.

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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.

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\(t\)-interval: \(\lt t \lt \)

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