Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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**Question:** Find a parametric equation for the tangent line to the curve defined by \(\mathbf{r}(t) = \langle t^2, 4\sqrt{t}, e^{2t} \rangle\) when \(t = 1\). **Solution:** To find the parametric equation for the tangent line, follow these steps: 1. **Find \(\mathbf{r}'(t)\), the derivative of the vector function:** \(\mathbf{r}(t) = \langle t^2, 4\sqrt{t}, e^{2t} \rangle\). - Differentiate each component with respect to \(t\): - \(\frac{d}{dt}(t^2) = 2t\) - \(\frac{d}{dt}(4\sqrt{t}) = \frac{d}{dt}(4t^{1/2}) = 2t^{-1/2}\) - \(\frac{d}{dt}(e^{2t}) = 2e^{2t}\) Thus, \(\mathbf{r}'(t) = \langle 2t, 2t^{-1/2}, 2e^{2t} \rangle\). 2. **Evaluate \(\mathbf{r}(t)\) and \(\mathbf{r}'(t)\) at \(t = 1\):** \(\mathbf{r}(1) = \langle 1^2, 4\sqrt{1}, e^{2\cdot1} \rangle = \langle 1, 4, e^2 \rangle\). \(\mathbf{r}'(1) = \langle 2\cdot1, 2\cdot1^{-1/2}, 2e^{2\cdot1} \rangle = \langle 2, 2, 2e^2 \rangle\). 3. **Write the equation of the tangent line:** The parametric equation of the tangent line at \(t = 1\) is given by: \(\mathbf{T}(t) = \mathbf{r}(1) + \mathbf{r}'(1) \cdot (t - 1)\). Plug in the values:
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Transcribed Image Text:**Question:** Find a parametric equation for the tangent line to the curve defined by \(\mathbf{r}(t) = \langle t^2, 4\sqrt{t}, e^{2t} \rangle\) when \(t = 1\). **Solution:** To find the parametric equation for the tangent line, follow these steps: 1. **Find \(\mathbf{r}'(t)\), the derivative of the vector function:** \(\mathbf{r}(t) = \langle t^2, 4\sqrt{t}, e^{2t} \rangle\). - Differentiate each component with respect to \(t\): - \(\frac{d}{dt}(t^2) = 2t\) - \(\frac{d}{dt}(4\sqrt{t}) = \frac{d}{dt}(4t^{1/2}) = 2t^{-1/2}\) - \(\frac{d}{dt}(e^{2t}) = 2e^{2t}\) Thus, \(\mathbf{r}'(t) = \langle 2t, 2t^{-1/2}, 2e^{2t} \rangle\). 2. **Evaluate \(\mathbf{r}(t)\) and \(\mathbf{r}'(t)\) at \(t = 1\):** \(\mathbf{r}(1) = \langle 1^2, 4\sqrt{1}, e^{2\cdot1} \rangle = \langle 1, 4, e^2 \rangle\). \(\mathbf{r}'(1) = \langle 2\cdot1, 2\cdot1^{-1/2}, 2e^{2\cdot1} \rangle = \langle 2, 2, 2e^2 \rangle\). 3. **Write the equation of the tangent line:** The parametric equation of the tangent line at \(t = 1\) is given by: \(\mathbf{T}(t) = \mathbf{r}(1) + \mathbf{r}'(1) \cdot (t - 1)\). Plug in the values:
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