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- 3t 3t 2e-³t). Let r(t) = (2t — 4, 5e¯ Find the line (L) tangent to r(t) at t = = 1.

- 3t 3t 2e-³t). Let r(t) = (2t — 4, 5e¯ Find the line (L) tangent to r(t) at t = = 1.

Calculus: Early Transcendentals
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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Question

3.2.5

**Problem Statement:** Let \(\vec{r}(t) = \langle 2t - 4, \, 5e^{-3t}, \, -2e^{-3t} \rangle\). Find the line (\(L\)) tangent to \(\vec{r}(t)\) at \(t = 1\). **Tangent Line Equation:** \[ L: \langle x, y, z \rangle = \text{[Box for input]} + t \, \text{[Box for input]} \] **Explanation:** We need to find the line tangent to the vector function \(\vec{r}(t)\) at the specific point where \(t = 1\). The tangent line at \(t = 1\) can be derived by determining the derivative \(\vec{r}'(t)\) and evaluating it at \(t = 1\), as well as finding the point \(\vec{r}(1)\) on the curve. The general form of the tangent line \(L\) is: \[ L(t) = \vec{r}(1) + t \cdot \vec{r}'(1) \] Substitute \(\vec{r}(1)\) and \(\vec{r}'(1)\) into this form to get the complete equation of the tangent line.
Transcribed Image Text:**Problem Statement:** Let \(\vec{r}(t) = \langle 2t - 4, \, 5e^{-3t}, \, -2e^{-3t} \rangle\). Find the line (\(L\)) tangent to \(\vec{r}(t)\) at \(t = 1\). **Tangent Line Equation:** \[ L: \langle x, y, z \rangle = \text{[Box for input]} + t \, \text{[Box for input]} \] **Explanation:** We need to find the line tangent to the vector function \(\vec{r}(t)\) at the specific point where \(t = 1\). The tangent line at \(t = 1\) can be derived by determining the derivative \(\vec{r}'(t)\) and evaluating it at \(t = 1\), as well as finding the point \(\vec{r}(1)\) on the curve. The general form of the tangent line \(L\) is: \[ L(t) = \vec{r}(1) + t \cdot \vec{r}'(1) \] Substitute \(\vec{r}(1)\) and \(\vec{r}'(1)\) into this form to get the complete equation of the tangent line.
Let \(\vec{r}(t) = \langle -2t^4 + 5, 5e^{2t}, -3\sin(5t) \rangle\). Find the unit tangent vector \(\vec{T}(t)\) at the point \(t = 0\). Round to 4 decimal places. \(\vec{T}(0) =\) [Input Box]
Transcribed Image Text:Let \(\vec{r}(t) = \langle -2t^4 + 5, 5e^{2t}, -3\sin(5t) \rangle\). Find the unit tangent vector \(\vec{T}(t)\) at the point \(t = 0\). Round to 4 decimal places. \(\vec{T}(0) =\) [Input Box]
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