Decreasing Functions

Duration: 9 min

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  1. About the Course: About the Course
  2. Section 1 | Language (Hindi): भाषा, लिपि, ध्वनि, वर्ण, अक्षर, स्वर, व्यंजन, मात्रा और संयुक्ताक्षर, वर्तनी, वर्ण-विन्यास, अनुस्वार, अनुनासिक, चंद्रबिंदु और ‘र’ के रूप, शब्द, पद और शब्द-भेद, संज्ञा और सर्वनाम, विशेषण और क्रिया, लिंग, वचन, पुरुष और कारक, काल, वाच्य, वाक्य और वाक्य-भेद, संधि, समास, विलोम शब्द, पर्यायवाची शब्द, सामान्य अशुद्धियाँ, अनेक शब्दों के लिए एक शब्द, मुहावरे और लोकोक्तियाँ, अपठित गद्यांश और अपठित पद्यांश, Multi-Topic Content (Cross-Concept Material), वर्णमाला
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  5. Section 1 | Language (English): Vocabulary, Noun, Subject Verb Agreement (Verb Noun Agreement), Adjectives, Tenses, Pronoun, Verb, Adverb, Preposition, Articles, Determiner, Conjunction, Modals, Active Passive Voice (Voice), Direct and Indirect Speech, Sentence Correction (Error Correction), Sentence Completion (Fill in the blanks), Sentence Construction, Sentence Re-arrangements (Para Jumbles) (Narrative Sequencing), Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Cloze Test (Cloze Passage), Verbal Analogy (Word Based Analogy), Facts, Inference and Judgements
  6. Section 2 | Primary Maths & Reasoning: Ratio and Proportion (Ratios), Divisibility Rules, Mixture and Alligation, Permutation and Combination, Number System, HCF LCM, Sequence and Series (Series), Speed Time and Distance, Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Percentage, Average, Profit and Loss, Simple Interest and Compound Interest, Time and Work, Mensuration and Geometry, Age Problems, Arithmetic, Powers and Exponents (Surds and Indices), Algebra, Probability, Trigonometry, Syllogisms, Seating Arrangements, Blood Relations, Directions (Direction Test), Analytical Reasoning (Counting Figures Reasoning), Puzzle Solving (Puzzles), Calendar, Clock, Cubes & Dices, Venn Diagram, Data Arrangement, Inequality, Analogy, Ranking, Order and Sequence
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  8. Section 3 | Computer System and Org. (CSO): Basics of COA, Floating Point Rep, Cache Memory Organization, Instr Formats & Modes, Programming the Basic Computer, Input Output Organisation, Register Transfer and Microoperations, Control Unit Design
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  10. Section 3 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Threads & Process Creation, Memory Management, Virtual Memory, Disc Scheduling, File Management, Process Synchronization
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  12. Section 3 | Data Structures: Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Minimum Spanning Trees, Shortest Path Algos
  13. Section 3 | Theory Of Computation & Compiler Design: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization
  14. Section 3 | Emerging Trends (AI, Societal Ethics & MS Office): Artificial Intelligence, Emerging Tech Trends, Society, Law & Ethics, MS Office Applications
  15. Section 3 | Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Maintenance and Configuration Management, Software Testing
  16. Section 3 | C Programming: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling
  17. Section 3 | OOPS, C++, JAVA, HTML: OOPS with C++, Java, HTML Basics, CSS
  18. Section 3 | Java: Java Fundamentals, Array, String, IO & References
  19. Section 3 | Python: Introduction to Python, Tokens, Variables, Data Types, I/O, Operators & Expressions, Control Flow & Looping, Collections & Strings, Functions & Modules, File Handling & Data Analysis
  20. Section 3 | Digital Electronics: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation
  21. Section 3 | Multimedia: Multimedia
  22. BPSC Previous Year Papers:
  23. BPSC Full Mock Tests:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This lecture introduces the concept of decreasing functions in algorithm analysis, defining them as functions where the value decreases as the input size n increases. The instructor explains that for decreasing functions, running time gets smaller as the problem size grows, which is counterintuitive compared to standard complexity analysis. The core educational objective is to arrange specific reciprocal and logarithmic functions in increasing order of growth as n approaches infinity. The video utilizes concrete examples, including the set {1/n, 1/n^2, 1/log n, 1/sqrt(n)}, to demonstrate how to compare these rates. The instructor employs tables to evaluate function values at specific integers (n=1, 2, 3, 4) and sketches graphs to visualize the asymptotic behavior where time approaches zero as n grows large. Key distinctions are made between terms like 1/n and 1/n^2, establishing that 1/n decreases slower than 1/n^2. The lecture emphasizes the hierarchy of decreasing functions, noting that (1/n) is asymptotically bigger than (1/n^2), and explores the relationship between polynomial reciprocals and logarithmic terms.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video begins by defining decreasing (decrement) functions, stating on-screen that a function f(n) is called decreasing if its value decreases as n increases. The instructor notes that for these functions, 'As input size grows, the running time keeps getting smaller.' An example problem is immediately presented to arrange four functions in increasing order of growth as n approaches infinity: 1/n, 1/n^2, 1/log n, and 1/sqrt(n). The instructor writes calculations on the right side of the screen, including arithmetic sums like '2 + 30 + 10 = 32 -> 1 sec' and '5 + 83 + 67 + 34 + 12 + 18 = 0 sec', likely illustrating timing concepts. Annotations such as 'n=3' and 'n=6' appear briefly to support the explanation of variable substitution.

  2. 2:00 – 5:00 02:00-05:00

    The instructor elaborates on the definition of decreasing functions by constructing a table for f(n) = 1/n. The visible text shows the relationship between n and 1/n, listing values: 'n | 1/1 = 1', '2 | 1/2', and '3 | 1/3'. This table demonstrates that as n increases from 1 to 4, the value of 1/n decreases. The instructor sketches a graph showing time decreasing as n approaches infinity, visually reinforcing the concept that larger inputs yield smaller running times. The focus remains on the inverse relationship between n and 1/n, preparing students to compare this behavior against other functions like 1/n^2.

  3. 5:00 – 9:02 05:00-09:02

    The lecture progresses to comparing the growth rates of reciprocal terms. The instructor fills a comparison table for 1/n and 1/n^2, calculating values for n=1 (where 1/1 = 1) and showing that for n >= 2, '1/n > 1/n^2'. This establishes that 1/n decreases slower than 1/n^2. The instructor writes the inequality '(1/n) Asymptotically bigger than (1/n^2)' on screen. The analysis extends to other terms, listing 'n', '1/n', '1/n^a', 'log n', and 'sqrt(n)'. The instructor writes '(1/n) < (1/log n)' and notes that 'log n > 1/n^0.5', boxing the final ordered list of functions to solve the initial problem.

The lecture systematically builds understanding of decreasing functions by first defining the concept, then applying it to specific examples. The teaching flow moves from abstract definition to concrete tabular evaluation, and finally to asymptotic comparison. Key evidence includes the on-screen text defining f(n) as decreasing if its value drops as n increases, and the specific inequality '(1/n) < (1/log n)' derived from comparing growth rates. The instructor uses the table method to prove that 1/n is greater than 1/n^2 for n > 1, which is crucial for ordering the functions. The visual graph of time versus n provides a geometric interpretation of the algebraic relationships. Students should note that for decreasing functions, 'asymptotically bigger' means the function value remains larger (closer to zero from above) compared to functions that decrease faster. The final ordering relies on recognizing that 1/n^2 decreases fastest, followed by 1/sqrt(n), then 1/log n, and finally 1/n decreases slowest among the given set.

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