i + e Find the open interval(s) on which the curve given by the vector-valued function is smooth. r(t) = e'i - e-tj + 2tk Consider the vector-valued function. r(t) = e'i - e-tj + 2tk Recall that a curve is smooth on an interval if r' is continuous and nonzero on the interval. Differentiate r(t) with respect to t. r'(t) = el k
i + e Find the open interval(s) on which the curve given by the vector-valued function is smooth. r(t) = e'i - e-tj + 2tk Consider the vector-valued function. r(t) = e'i - e-tj + 2tk Recall that a curve is smooth on an interval if r' is continuous and nonzero on the interval. Differentiate r(t) with respect to t. r'(t) = el k
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![**Title: Analyzing the Smoothness of a Vector-Valued Function**
### Problem Statement
Find the open interval(s) on which the curve given by the vector-valued function is smooth.
\[ \mathbf{r}(t) = e^{t} \mathbf{i} - e^{-t} \mathbf{j} + 2t \mathbf{k} \]
### Explanation
Consider the vector-valued function:
\[ \mathbf{r}(t) = e^{t} \mathbf{i} - e^{-t} \mathbf{j} + 2t \mathbf{k} \]
Recall that a curve is smooth on an interval if \(\mathbf{r}'\) is continuous and nonzero on the interval.
### Differentiation Process
Differentiate \(\mathbf{r}(t)\) with respect to \(t\).
\[ \mathbf{r}'(t) = e^{t} \mathbf{i} + e^{-t} \mathbf{j} + 2 \mathbf{k} \]
### Interpretation
In this problem, you need to differentiate the given vector-valued function with respect to \(t\) and determine the continuity and non-zero nature of the derivative \(\mathbf{r}'(t)\) to identify smooth intervals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc08bab33-dfc3-4475-87c9-eef4dfb69a19%2F1d53f566-e891-4547-b1ad-e0d16c6a7ad5%2Flnvisqm_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Analyzing the Smoothness of a Vector-Valued Function**
### Problem Statement
Find the open interval(s) on which the curve given by the vector-valued function is smooth.
\[ \mathbf{r}(t) = e^{t} \mathbf{i} - e^{-t} \mathbf{j} + 2t \mathbf{k} \]
### Explanation
Consider the vector-valued function:
\[ \mathbf{r}(t) = e^{t} \mathbf{i} - e^{-t} \mathbf{j} + 2t \mathbf{k} \]
Recall that a curve is smooth on an interval if \(\mathbf{r}'\) is continuous and nonzero on the interval.
### Differentiation Process
Differentiate \(\mathbf{r}(t)\) with respect to \(t\).
\[ \mathbf{r}'(t) = e^{t} \mathbf{i} + e^{-t} \mathbf{j} + 2 \mathbf{k} \]
### Interpretation
In this problem, you need to differentiate the given vector-valued function with respect to \(t\) and determine the continuity and non-zero nature of the derivative \(\mathbf{r}'(t)\) to identify smooth intervals.
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