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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![Certainly! Here's a transcription of the information in the image as it would appear on an educational website:
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**Concept 6: Properties of Vector Functions (continued)**
(c) Find \( \int \mathbf{G}(t) \, dt \)
---
**Derivative Properties of Vector-Valued Functions**
For \(\mathbf{F}(t) = \langle f(t), g(t), h(t) \rangle\) where \(f(t), g(t), h(t)\) are differentiable functions,
1. \(\mathbf{F}'(t) = \langle f'(t), g'(t), h'(t) \rangle\)
2. \(\int \mathbf{F}(t) \, dt = \langle \int f(t) \, dt, \int g(t) \, dt, \int h(t) \, dt \rangle\)
3. \(\lim_{t \to a} \mathbf{F}(t) = \langle \lim_{t \to a} f(t), \lim_{t \to a} g(t), \lim_{t \to a} h(t) \rangle\)
4. \(\frac{d}{dt}[\mathbf{F}(t) + \mathbf{G}(t)] = \mathbf{F}'(t) + \mathbf{G}'(t)\)
5. \(\frac{d}{dt}[c\mathbf{F}(t)] = c\mathbf{F}'(t)\)
6. \(\frac{d}{dt}[p(t)\mathbf{F}(t)] = p'(t)\mathbf{F}(t) + p(t)\mathbf{F}'(t)\)
7. \(\frac{d}{dt}[\mathbf{F}(t) \cdot \mathbf{G}(t)] = \mathbf{F}'(t) \cdot \mathbf{G}(t) + \mathbf{F}(t) \cdot \mathbf{G}'(t)\)
8. \(\frac{d}{dt}[\mathbf{F}(t) \times \mathbf{G}(t)] = \mathbf{F}'(t) \times \mathbf{G}(t) + \mathbf{F}(t) \times \mathbf{G}'(t)\)
9. \(\frac{d}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fddb72669-0680-48fe-8aa6-56d86c073f3c%2F3e04863d-0d69-47d9-b65d-eda9608c4fb4%2Fwnsk93g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Here's a transcription of the information in the image as it would appear on an educational website:
---
**Concept 6: Properties of Vector Functions (continued)**
(c) Find \( \int \mathbf{G}(t) \, dt \)
---
**Derivative Properties of Vector-Valued Functions**
For \(\mathbf{F}(t) = \langle f(t), g(t), h(t) \rangle\) where \(f(t), g(t), h(t)\) are differentiable functions,
1. \(\mathbf{F}'(t) = \langle f'(t), g'(t), h'(t) \rangle\)
2. \(\int \mathbf{F}(t) \, dt = \langle \int f(t) \, dt, \int g(t) \, dt, \int h(t) \, dt \rangle\)
3. \(\lim_{t \to a} \mathbf{F}(t) = \langle \lim_{t \to a} f(t), \lim_{t \to a} g(t), \lim_{t \to a} h(t) \rangle\)
4. \(\frac{d}{dt}[\mathbf{F}(t) + \mathbf{G}(t)] = \mathbf{F}'(t) + \mathbf{G}'(t)\)
5. \(\frac{d}{dt}[c\mathbf{F}(t)] = c\mathbf{F}'(t)\)
6. \(\frac{d}{dt}[p(t)\mathbf{F}(t)] = p'(t)\mathbf{F}(t) + p(t)\mathbf{F}'(t)\)
7. \(\frac{d}{dt}[\mathbf{F}(t) \cdot \mathbf{G}(t)] = \mathbf{F}'(t) \cdot \mathbf{G}(t) + \mathbf{F}(t) \cdot \mathbf{G}'(t)\)
8. \(\frac{d}{dt}[\mathbf{F}(t) \times \mathbf{G}(t)] = \mathbf{F}'(t) \times \mathbf{G}(t) + \mathbf{F}(t) \times \mathbf{G}'(t)\)
9. \(\frac{d}{
![**[M R N] Concept 6: Properties of Vector Functions**
Let \( \mathbf{F}(t) = t^{-4} \mathbf{i} - 5 \mathbf{j} + (2t - 4t^3) \mathbf{k} \) and \( \mathbf{G}(t) = 2 \cos t \, \mathbf{i} + e^{-5t} \mathbf{j} - \sin 4t \, \mathbf{k} \).
(a) Find \( \dfrac{d}{dt} [\mathbf{F}(t) \cdot \mathbf{G}(t)] \)
(b) Given \( p(t) = 3\sqrt{t} \), evaluate \( \dfrac{d}{dt} [p(t) \mathbf{F}(t)] \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fddb72669-0680-48fe-8aa6-56d86c073f3c%2F3e04863d-0d69-47d9-b65d-eda9608c4fb4%2Fx99fct_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**[M R N] Concept 6: Properties of Vector Functions**
Let \( \mathbf{F}(t) = t^{-4} \mathbf{i} - 5 \mathbf{j} + (2t - 4t^3) \mathbf{k} \) and \( \mathbf{G}(t) = 2 \cos t \, \mathbf{i} + e^{-5t} \mathbf{j} - \sin 4t \, \mathbf{k} \).
(a) Find \( \dfrac{d}{dt} [\mathbf{F}(t) \cdot \mathbf{G}(t)] \)
(b) Given \( p(t) = 3\sqrt{t} \), evaluate \( \dfrac{d}{dt} [p(t) \mathbf{F}(t)] \)
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