Compute the derivative: a. r(t) =< et²-2,8 - sec 4t, 7 > b. r(t) = sint cos ti - t4 In t²j
Compute the derivative: a. r(t) =< et²-2,8 - sec 4t, 7 > b. r(t) = sint cos ti - t4 In t²j
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please compute the derivatives for both parts provided in the photo below
![## Calculus Exercise: Compute the Derivative
### Problem Statement:
Compute the derivative of the following vector-valued functions:
#### a.
\[ \mathbf{r}(t) = \langle e^{t^2 - 2}, 8 - \sec(4t), 7 \rangle \]
#### b.
\[ \mathbf{r}(t) = \sin(t) \cos(t) \mathbf{i} - t^4 \ln(t^2) \mathbf{j} \]
### Explanation:
In part (a), you have a vector-valued function \(\mathbf{r}(t)\) with three components: \( e^{t^2 - 2} \), \( 8 - \sec(4t) \), and \( 7 \).
In part (b), the vector-valued function \(\mathbf{r}(t)\) has two components: \( \sin(t) \cos(t)i \) and \( -t^4 \ln(t^2)j \).
To solve for the derivatives, we need to differentiate each component with respect to \( t \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fc33cc1-e822-44c8-a303-0b5fd2ff9f49%2Fffa38d72-0b5b-49aa-8146-0ddd0fd1d7ef%2Fsdehlxo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Calculus Exercise: Compute the Derivative
### Problem Statement:
Compute the derivative of the following vector-valued functions:
#### a.
\[ \mathbf{r}(t) = \langle e^{t^2 - 2}, 8 - \sec(4t), 7 \rangle \]
#### b.
\[ \mathbf{r}(t) = \sin(t) \cos(t) \mathbf{i} - t^4 \ln(t^2) \mathbf{j} \]
### Explanation:
In part (a), you have a vector-valued function \(\mathbf{r}(t)\) with three components: \( e^{t^2 - 2} \), \( 8 - \sec(4t) \), and \( 7 \).
In part (b), the vector-valued function \(\mathbf{r}(t)\) has two components: \( \sin(t) \cos(t)i \) and \( -t^4 \ln(t^2)j \).
To solve for the derivatives, we need to differentiate each component with respect to \( t \).
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