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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![**Vector Calculus: Computing Derivatives of Vector-Valued Functions**
Given the vector-valued function \( \mathbf{r}(t) = \langle t, t^8, t^{12} \rangle \), we are tasked with computing its derivative with respect to \( t \).
The general form for the derivative of a vector-valued function is
\[ \frac{d\mathbf{r}}{dt} = \langle \frac{d}{dt} (t), \frac{d}{dt} (t^8), \frac{d}{dt} (t^{12}) \rangle. \]
Compute the derivative of \( \mathbf{r}(t) \):
\[ \frac{d\mathbf{r}}{dt} = \langle \_, \_, \_ \rangle. \]
Please fill in the blanks with the appropriate derivatives of each component:
\[ \frac{d\mathbf{r}}{dt} = \langle \frac{d}{dt} (t), \frac{d}{dt} (t^8), \frac{d}{dt} (t^{12}) \rangle. \]
To solve this, apply the power rule for differentiation to each component.
1. For the first component \( t \):
\[ \frac{d}{dt} (t) = 1. \]
2. For the second component \( t^8 \):
\[ \frac{d}{dt} (t^8) = 8t^7. \]
3. For the third component \( t^{12} \):
\[ \frac{d}{dt} (t^{12}) = 12t^{11}. \]
Therefore, the derivative of \( \mathbf{r}(t) \) is:
\[ \frac{d\mathbf{r}}{dt} = \langle 1, 8t^7, 12t^{11} \rangle. \]
The complete solution can be checked as follows:
\[ \frac{d\mathbf{r}}{dt} = \langle 1, 8t^7, 12t^{11} \rangle. \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe393088-5eda-4ff6-aaf1-5c339da8a389%2Fa89e984c-b50b-48e5-a24d-6c866232612b%2Fzi7n1o_processed.png&w=3840&q=75)
Transcribed Image Text:**Vector Calculus: Computing Derivatives of Vector-Valued Functions**
Given the vector-valued function \( \mathbf{r}(t) = \langle t, t^8, t^{12} \rangle \), we are tasked with computing its derivative with respect to \( t \).
The general form for the derivative of a vector-valued function is
\[ \frac{d\mathbf{r}}{dt} = \langle \frac{d}{dt} (t), \frac{d}{dt} (t^8), \frac{d}{dt} (t^{12}) \rangle. \]
Compute the derivative of \( \mathbf{r}(t) \):
\[ \frac{d\mathbf{r}}{dt} = \langle \_, \_, \_ \rangle. \]
Please fill in the blanks with the appropriate derivatives of each component:
\[ \frac{d\mathbf{r}}{dt} = \langle \frac{d}{dt} (t), \frac{d}{dt} (t^8), \frac{d}{dt} (t^{12}) \rangle. \]
To solve this, apply the power rule for differentiation to each component.
1. For the first component \( t \):
\[ \frac{d}{dt} (t) = 1. \]
2. For the second component \( t^8 \):
\[ \frac{d}{dt} (t^8) = 8t^7. \]
3. For the third component \( t^{12} \):
\[ \frac{d}{dt} (t^{12}) = 12t^{11}. \]
Therefore, the derivative of \( \mathbf{r}(t) \) is:
\[ \frac{d\mathbf{r}}{dt} = \langle 1, 8t^7, 12t^{11} \rangle. \]
The complete solution can be checked as follows:
\[ \frac{d\mathbf{r}}{dt} = \langle 1, 8t^7, 12t^{11} \rangle. \]
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