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Optimizing complex systems using linear algebra
Optimizing complex systems using linear algebra
Essential Question:
Essential Question:
How can matrices and eigenvalues help us understand and optimize real-life dynamic systems?
The Challenge:
The Challenge:
“Model and optimize a real engineering system (traffic flow / vibration system / population model) using eigenvalues and apply the Cayley-Hamilton theorem to analyze future behavior.”
Guiding Activities:
Guiding Activities:
Identify a real system that can be represented using a 2×2 or 3×3 matrix.
Examples:Traffic flow between 3 junctions
Growth-decay of two interacting chemical species
Vibration of a double-mass-spring system
Form the system matrix A and compute:
Eigenvalues & eigenvectors
Nature (stable/unstable/periodic)
Use the Cayley–Hamilton Theorem to compute Aⁿ (future system state).
Interpret the results—What does the dominant eigenvalue indicate?
Solution Requirements:
Solution Requirements:
Clear explanation of how the system is converted to a matrix
Full calculation of eigenvalues, eigenvectors
CH theorem verification
Interpretation of the results with graphs or tables
Short reflection: “How does linear algebra help predict system behavior?
Expected Learning Outcomes (CO Mapping):
Expected Learning Outcomes (CO Mapping):
CO-1: Application of matrices, eigenvalues, eigenvectors
CO-1: Applying Cayley-Hamilton theorem in engineering context
Engineering solutions often require optimizing multiple variables simultaneously
Engineering solutions often require optimizing multiple variables simultaneously
Essential Question:
Essential Question:
How can multivariable calculus help optimize real engineering systems?
The Challenge:
The Challenge:
“Design and optimize a real-life engineering component (cooling fin, packaging box, storage tank, etc.) using Maxima/Minima and Lagrange multipliers.”
Guiding Activities:
Guiding Activities:
Select an engineering component with two or more variables.
Example:Maximize heat dissipation of a rectangular fin
Minimize material used for a cylindrical tank
Maximize volume of a box with fixed surface area
Form mathematical functions:
Area
Volume
Cost
Find critical points using:
Partial derivatives
Hessian test
Apply Lagrange multipliers to include constraints.
Compare two designs and choose the optimized one.
Solution Requirements:
Solution Requirements:
Full derivation of the function f(x,y)
Calculation of ∂f/∂x, ∂f/∂y
Solving the Lagrange system
Interpretation with diagrams
Short conclusion: “Which design is optimal and why?”
Expected Learning Outcomes (CO Mapping):
Expected Learning Outcomes (CO Mapping):
CO-2: Apply Maxima & Minima
CO-2: Apply Jacobian, Taylor expansion (if used for approximation).
Engineers must ensure that algorithms, models and circuits converge reliably
Engineers must ensure that algorithms, models and circuits converge reliably
Essential Question:
Essential Question:
How can convergence tests help ensure stability in engineering algorithms or circuits?
The Challenge:
The Challenge:
“Analyze whether a real-life engineering series (current series, signal series, iterative algorithm output) converges using three convergence tests.”
Guiding Activities:
Guiding Activities:
Choose any engineering-related series such as:
Fourier signal amplitude series
Current decay series in an RLC circuit
Iterative numerical algorithm output
Represent the values as a mathematical series.
Apply at least three tests:
Ratio test
Root test
Comparison test
Raabe’s test / Logarithmic test
Determine absolute, conditional, or divergent nature.
Explain the consequence on engineering design.
Solution Requirements:
Solution Requirements:
Clear definition of series
Step-by-step application of each test
Summary table of results
Real engineering interpretation
Short justification: Why does convergence matter?
Expected Learning Outcomes (CO Mapping):
Expected Learning Outcomes (CO Mapping):
CO-5: Identify convergence/divergence of series
CO-5: Apply comparison, ratio, root, Raabe, logarithmic tests
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