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Q. 29

Expert-verifiedFound in: Page 860

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

Find parametric equations for each of the vector-valued functions in Exercises 26–34, and sketch the graphs of the functions, indicating the direction for increasing values of *t*.

$r\left(t\right)=(1+\mathrm{sin}t,3-\mathrm{cos}2t),fort\in [0,2\mathrm{\pi}]$

The parametric equation of the vector valued functions $r\left(t\right)=(1+\mathrm{sin}t,3-\mathrm{cos}2t),fort\in [0,2\mathrm{\pi}]$ is $y=2{x}^{2}-4x+4$.

And the graph of the function is:

The function:

$r\left(t\right)=(1+\mathrm{sin}t,3-\mathrm{cos}2t),fort\in [0,2\mathrm{\pi}]$

The parametric equations for the given function is:

$x=1+\mathrm{sin}t\phantom{\rule{0ex}{0ex}}y=3-\mathrm{cos}2t$

We know that,

$\mathrm{cos}2t=1-2{\mathrm{sin}}^{2}t$

From the given *x, y,*

$x-1=\mathrm{sin}t\phantom{\rule{0ex}{0ex}}\mathrm{cos}2t=3-y$

So,

$\mathrm{cos}2t=1-2{\mathrm{sin}}^{2}t\phantom{\rule{0ex}{0ex}}3-y=1-2{(x-1)}^{2}\phantom{\rule{0ex}{0ex}}y=2{x}^{2}-4x+4$

The ordered pairs of the function is:

t | x=1+sin t | y=3-cos2t | (x, y) |

$0$ | $1$ | $2$ | $(1,2)$ |

$\frac{\mathrm{\pi}}{2}$ | $2$ | $4$ | $(2,4)$ |

$\mathrm{\pi}$ | $1$ | $2$ | $(1,2)$ |

$\frac{3\mathrm{\pi}}{2}$ | $0$ | $4$ | $(0,4)$ |

$2\mathrm{\pi}$ | $1$ | $2$ | $(1,2)$ |

The graph of the function is:

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