Engineering Mathematics II - Lecture Notes

Introduction

• मुझसे नाम गुलशन कुमार है और आज हम इंजीनियरिंग मैथमेटिक्स II का एक शॉर्ट रिवीजन करेंगे।
• पहले यूनिट्स: यूनिट नंबर 1, 2, 4, 5 को पूरा कर चुके हैं।
• आज का विषय: यूनिट नंबर 3 का रिविजन।

Topics in Unit 3

1. Definition of Sequence:
   • Definition of sequences and series with examples.
2. Convergence of Series:
   • Tests for convergence of series.

Important Topics for Revision

• Ratio Test (7 Marks Question):
  • एक बहुत महत्वपूर्ण टेस्ट है जो अकडू के एग्जाम में रेगुलरली पूछा जाता है।
• Remaining topics contributing to 22 marks questions.

Key Topics for Today's Revision

• 4-Year Series
• Half Range
• Sine and Cosine Series
• Important concepts related to the above topics.

Fourier Series

• Definition:
  • Fourier Series is defined for periodic functions.
• Periodic Function:
  • A periodic function repeats its value after a fixed interval (e.g., sine and cosine functions).

Important Formulas

• Trigonometric Identities:
  • Simple identities that need to be memorized.
• Properties of Definite Integral:
  • Important for calculating integrals, particularly in Fourier series.

Integration by Parts

• Importance:
  • Integral calculation is crucial in Mathematics II; it appears in every unit.
• Formula:
  • Integration by parts is used to simplify complex integrals.
  • It is essential to understand and apply correctly for success in exam questions.

Odd and Even Functions

• Even Function:
  • f(-x) = f(x), graph is symmetric about the y-axis.
• Odd Function:
  • f(-x) = -f(x), graph is symmetric about the origin.
• Properties:
  • The graph of even functions will integrate to twice the area from 0 to a positive limit.
  • The graph of odd functions will integrate to zero over symmetric limits.

Practical Application - Example Problems

1. Find the Fourier Series:
   • Example: Given function f(x) and its interval; apply the defined formulas to find the series.

Conclusion

• Review important definitions, tests, integration techniques, and properties.
• Practice problems related to Fourier series to solidify understanding.
• Ensure to understand the relationship between periodic functions and their Fourier expansions.

Study Tips

• Memorize key formulas and definitions.
• Solve previous years' exam questions on Fourier series and integration.
• Practice drawing graphs for odd and even functions.