### Problem Statement**Objective:** Find the open interval(s) on which the curve given by the vector-valued function is smooth. (Enter your answer using interval notation.)**Function Given:**\[ \mathbf{r}(t) = 6t^2 \mathbf{i} + 9t^3 \mathbf{j} \]**Input:**A text box is provided to enter the interval, accompanied by a red check mark (indicating an incorrect or incomplete input).### Step-by-Step ExplanationTo solve this problem, follow these steps:1. **Understand the Vector-Valued Function:** The given function is: \[ \mathbf{r}(t) = 6t^2 \mathbf{i} + 9t^3 \mathbf{j} \] This represents a parametric curve in two-dimensional space defined by the component functions \[ x(t) = 6t^2 \] and \[ y(t) = 9t^3. \]2. **Definition of Smoothness:** A curve is smooth if its derivative is continuous and non-zero for every point in the interval. 3. **Find the Derivative:** Differentiate the vector-valued function with respect to $ t $: \[ \mathbf{r}'(t) = \frac{d}{dt}(6t^2) \mathbf{i} + \frac{d}{dt}(9t^3) \mathbf{j} \] \[ \mathbf{r}'(t) = 12t \mathbf{i} + 27t^2 \mathbf{j} \]4. **Determine Continuity and Non-Zero Condition:** The curve is smooth if $\mathbf{r}'(t) \neq \mathbf{0}$: \[ \mathbf{r}'(t) = \langle 12t, 27t^2 \rangle \neq \mathbf{0} \] Solve: \[ 12t \neq 0 \quad \text{and} \quad 27t^2 \neq 0 \] These conditions are satisfied for all $ t $ except $ t = 0 $.5. **Determine the Interval:** The curve is smooth on two open intervals: \[ (-\infty,

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Find the open interval(s) on which the curve given by the vector-valued function is smooth. (Enter your answer using interval notation.) r(t) = 6t2i + 9t³j

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Concept explainers

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.

Question

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### Problem Statement

**Objective:**
Find the open interval(s) on which the curve given by the vector-valued function is smooth. (Enter your answer using interval notation.)

**Function Given:**
\[ \mathbf{r}(t) = 6t^2 \mathbf{i} + 9t^3 \mathbf{j} \]

**Input:**
A text box is provided to enter the interval, accompanied by a red check mark (indicating an incorrect or incomplete input).

### Step-by-Step Explanation

To solve this problem, follow these steps:

1. **Understand the Vector-Valued Function:**
The given function is:
\[ \mathbf{r}(t) = 6t^2 \mathbf{i} + 9t^3 \mathbf{j} \]
This represents a parametric curve in two-dimensional space defined by the component functions
\[ x(t) = 6t^2 \]
and
\[ y(t) = 9t^3. \]

2. **Definition of Smoothness:**
A curve is smooth if its derivative is continuous and non-zero for every point in the interval.

3. **Find the Derivative:**
Differentiate the vector-valued function with respect to $ t $:
\[ \mathbf{r}'(t) = \frac{d}{dt}(6t^2) \mathbf{i} + \frac{d}{dt}(9t^3) \mathbf{j} \]
\[ \mathbf{r}'(t) = 12t \mathbf{i} + 27t^2 \mathbf{j} \]

4. **Determine Continuity and Non-Zero Condition:**
The curve is smooth if $\mathbf{r}'(t) \neq \mathbf{0}$:
\[ \mathbf{r}'(t) = \langle 12t, 27t^2 \rangle \neq \mathbf{0} \]
Solve:
\[ 12t \neq 0 \quad \text{and} \quad 27t^2 \neq 0 \]
These conditions are satisfied for all $ t $ except $ t = 0 $.

5. **Determine the Interval:**
The curve is smooth on two open intervals:
\[ (-\infty,

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