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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.
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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