Chapter 11: Q. 29 (page 860)

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

Short Answer

Expert verified

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:

Step by step solution

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

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the parametric equations.

Step 3. Find the ordered pairs.

Step 4. Graph the function.

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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+sint,3-cos2t),fort∈[0,2π]

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the parametric equations.

Step 3. Find the ordered pairs.

Step 4. Graph the function.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Logic and Functions

Decision Maths

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

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