Suppose f(): R² → R² and g(): R³ → R² are given by: ƒ = :)-(²) and ()-(7) = g 3y² + Find (f o g)'(x) at x = (₁) 2 y2 X3 = + X3
Suppose f(): R² → R² and g(): R³ → R² are given by: ƒ = :)-(²) and ()-(7) = g 3y² + Find (f o g)'(x) at x = (₁) 2 y2 X3 = + X3
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Suppose \( f(\cdot) : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) and \( g(\cdot) : \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) are given by:
\[ f\begin{pmatrix} y_1 \\ y_2 \end{pmatrix} = \begin{pmatrix} y_1 y_2 - y_2^2 \\ 3y_2 + y_2 \end{pmatrix} \]
and
\[ g\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 2x_1 + x_2 - x_3 \\ x_1^2 x_2 x_3 + x_3^2 \end{pmatrix} \]
Find \((f \circ g)'(x)\) at \( x = \begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix} \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadd5b56d-97e3-4d3c-9aa9-5b563ad77377%2Fd4366229-97d2-4040-b781-257db9892988%2Fs0nwvkh_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose \( f(\cdot) : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) and \( g(\cdot) : \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) are given by:
\[ f\begin{pmatrix} y_1 \\ y_2 \end{pmatrix} = \begin{pmatrix} y_1 y_2 - y_2^2 \\ 3y_2 + y_2 \end{pmatrix} \]
and
\[ g\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 2x_1 + x_2 - x_3 \\ x_1^2 x_2 x_3 + x_3^2 \end{pmatrix} \]
Find \((f \circ g)'(x)\) at \( x = \begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix} \)
Expert Solution
Step 1
Similar calculations can perform if we change y2 to y1 in the 2nd component where only y is written. I hope I produced something that is useful for your purpose of asking. Thank you.
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