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Q. 29

Expert-verified

Calculus

Found in: Page 860

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

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.
$r (t) = (1 + \sin t, 3 - \cos 2 t), f o r t \in [0, 2 π]$

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

And the graph of the function is:

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information.

The function:

$r (t) = (1 + \sin t, 3 - \cos 2 t), f o r t \in [0, 2 π]$

Step 2. Find the parametric equations.

The parametric equations for the given function is:

$x = 1 + \sin t y = 3 - \cos 2 t$

We know that,

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

From the given x, y,

$x - 1 = \sin t \cos 2 t = 3 - y$

So,

$\cos 2 t = 1 - 2 \sin^{2} t 3 - y = 1 - 2 {(x - 1)}^{2} y = 2 x^{2} - 4 x + 4$

Step 3. Find the ordered pairs.

The ordered pairs of the function is:

t	x=1+sin t	y=3-cos2t	(x, y)
$0$	$1$	$2$	$(1, 2)$
$\frac{π}{2}$	$2$	$4$	$(2, 4)$
$π$	$1$	$2$	$(1, 2)$
$\frac{3 π}{2}$	$0$	$4$	$(0, 4)$
$2 π$	$1$	$2$	$(1, 2)$

Step 4. Graph the function.

The graph of the function is:

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