Using the definition of e z by its power series [ ( 8.1 ) of Chapter 2 ] , and the theorem (Chapters 1 and 2 ) that power series may be differentiated term by term (within the disk of convergence), and the result of Problem $30,$ show that ( d / d z ) e z = e z .
Using the definition of e z by its power series [ ( 8.1 ) of Chapter 2 ] , and the theorem (Chapters 1 and 2 ) that power series may be differentiated term by term (within the disk of convergence), and the result of Problem $30,$ show that ( d / d z ) e z = e z .
Using the definition of
e
z
by its power series
[
(
8.1
)
of Chapter
2
]
,
and the theorem (Chapters 1 and 2 ) that power series may be differentiated term by term (within the disk of convergence), and the result of Problem $30,$ show that
(
d
/
d
z
)
e
z
=
e
z
.
Refer to page 140 for problems on infinite sets.
Instructions:
• Compare the cardinalities of given sets and classify them as finite, countable, or uncountable.
•
Prove or disprove the equivalence of two sets using bijections.
• Discuss the implications of Cantor's theorem on real-world computation.
Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]
Refer to page 120 for problems on numerical computation.
Instructions:
• Analyze the sources of error in a given numerical method (e.g., round-off, truncation).
• Compute the error bounds for approximating the solution of an equation.
•
Discuss strategies to minimize error in iterative methods like Newton-Raphson.
Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]
Refer to page 145 for problems on constrained optimization.
Instructions:
•
Solve an optimization problem with constraints using the method of Lagrange multipliers.
•
•
Interpret the significance of the Lagrange multipliers in the given context.
Discuss the applications of this method in machine learning or operations research.
Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440 AZF/view?usp=sharing]
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