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Chapter 10: Problem 29

Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(\theta)=2 \cos ^{3} \theta \mathbf{i}+3 \sin ^{3} \theta \mathbf{j} $$

Short Answer

Expert verified
The curve given by the vector-valued function \( \mathbf{r}(\theta) \) is smooth on the open intervals \( (0, \pi) \) and \( (\pi, 2\pi) \).

Step by step solution

01

Define the Function in Component Form

The vector-valued function \( \mathbf{r}(\theta) \) is given as \(2 \cos ^{3} \theta \mathbf{i}+3 \sin ^{3} \theta \mathbf{j}\).
02

Differentiate the Function Components

Compute the derivative of the function by differentiating each component separately with respect to \( \theta \). The derivative of \(2 \cos ^{3} \theta \) is \( -6 \cos^{2}(\theta) \sin(\theta) \) and the derivative of \(3 \sin ^{3} \theta \) is \(9 \sin^{2}(\theta) \cos(\theta)\). So the derivative function \(\mathbf{r}'(\theta)\) is: \( -6 \cos^{2}(\theta) \sin(\theta) \mathbf{i} + 9 \sin^{2}(\theta) \cos(\theta) \mathbf{j} \).
03

Solve for When the Derivative Equals Zero

Solving for when the derivative is zero, involves setting both \(-6 \cos^{2}(\theta) \sin(\theta) = 0\) and \(9 \sin^{2}(\theta) \cos(\theta) = 0\).These two equations yield that the derivative is equal to zero when \( \theta \) equals \( 0, \pi, 2\pi \). Consequently, between these values the function is smooth.
04

Identifying the Interval of Smoothness

The derivative never equals zero in the open intervals \( (0, \pi) \) and \( (\pi, 2\pi) \). Therefore, these are the intervals on which the function \( \mathbf{r}(\theta) \) is smooth.

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Most popular questions from this chapter

Consider a particle moving on a circular path of radius \(b\) described by $$ \begin{aligned} &\mathbf{r}(t)=b \cos \omega t \mathbf{i}+b \sin \omega t \mathbf{j}\\\ &\text { where } \omega=d \theta / d t \text { is the constant angular velocity. } \end{aligned} $$ Find the acceleration vector and show that its direction is always toward the center of the circle.

Use the model for projectile motion, assuming there is no air resistance. Find the angle at which an object must be thrown to obtain (a) the maximum range and (b) the maximum height.

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Prove that the vector \(\mathbf{T}^{\prime}(t)\) is \(\mathbf{0}\) for an object moving in a straight line.

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In Exercises 59-66, prove the property. In each case, assume that \(\mathbf{r}, \mathbf{u},\) and \(\mathbf{v}\) are differentiable vector-valued functions of \(t,\) \(f\) is a differentiable real-valued function of \(t,\) and \(c\) is a scalar. $$ D_{t}[c \mathbf{r}(t)]=c \mathbf{r}^{\prime}(t) $$
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