Finding Maxima & Minima (Practice Questions Set 2)

Duration: 7 min

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Module outline

  1. About Course: About the Course
  2. Paper-2 | Programming & Data Structures: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing
  3. Paper-2 | Databases / DBMS & SQL: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Relational Calculus, Transaction Management, Concurrency Control
  4. Paper-2 | Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos
  5. Paper-2 | Computer Networks & Security: Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware Basics, Network Access Technologies, Network Security
  6. Paper-2 | Digital Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep
  7. Paper-2 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management
  8. Paper-2 | Theory of Computation: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines
  9. Paper-2 | Computer Organization & Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design, Programming the Basic Computer
  10. Paper-2 | Compiler Design: Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization, Run Time Environment
  11. Paper-1 | General Awareness: Polity, Physics, Biology, Chemistry, Static General Knowledge, GK- Current Affairs, History, Geography, Indian Economy, Environment
  12. Paper-1 | Logical Reasoning: Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Data Interpretation, Blood Relations, Syllogisms, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Selection Decision Table (Decision Making), Clock, Calendar, Cubes & Dices, Directions (Direction Test), Analogy, Seating Arrangements, Analytical Reasoning (Counting Figures Reasoning)
  13. Paper-1 | Quantitative Aptitude: Ratio and Proportion (Ratios), Mixture and Alligation, Permutation and Combination, Number System, HCF LCM, Speed Time and Distance, Average, Age Problems, Percentage, Profit and Loss, Time and Work, Pipe and Cistern, Simple Interest and Compound Interest, Powers and Exponents (Surds and Indices), Mensuration and Geometry, Probability, Statistics, Arithmetic, Algebra
  14. Paper-1 | English: Vocabulary, Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Sentence Re-arrangements (Para Jumbles) (Narrative Sequencing), Sentence Correction (Error Correction), Sentence Completion (Fill in the blanks), Sentence Construction, Cloze Test (Cloze Passage)
  15. Coal India Previous Year Paper:
  16. Engineering Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Linear Algebra, Calculus
  17. Coal India Mock Test:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video features a calculus lecture by Yash Jain Sir from Knowledge Gate, focusing on finding local maxima and minima using derivatives. The lesson begins with a polynomial function example, demonstrating the standard procedure of finding critical points where the first derivative is zero and classifying them using the second derivative test. The instructor then transitions to a trigonometric function to illustrate a case where multiple stationary points exist, highlighting the periodic nature of trigonometric equations and the resulting infinite solutions.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title card displaying the word 'CALCULUS' amidst various mathematical formulas like integrals and series. The scene transitions to Yash Jain Sir, an educator from Knowledge Gate, standing before a whiteboard. He introduces the problem by writing the function $f(x) = rac{x^3}{3} - x$ in black ink. He states the goal is to find the local maxima and minima for this specific cubic function. He begins the solution process by preparing to differentiate the function with respect to $x$.

  2. 2:00 – 5:00 02:00-05:00

    The instructor proceeds to differentiate the function, writing $f'(x) = x^2 - 1$ on the board. He sets the derivative equal to zero, $f'(x) = 0$, to locate the stationary points (SP). Solving $x^2 - 1 = 0$ gives the critical values $x = 1$ and $x = -1$, which he circles. He then computes the second derivative, $f''(x) = 2x$. By substituting $x=1$, he finds $f''(1) = 2$, which is greater than zero, indicating a local minimum. Conversely, for $x=-1$, $f''(-1) = -2$, which is less than zero, indicating a local maximum. He calculates the corresponding function values, writing $-2/3$ for the minimum and $2/3$ for the maximum.

  3. 5:00 – 7:13 05:00-07:13

    A new example is introduced: $f(x) = 2 \cos x - x$. The instructor differentiates this to obtain $f'(x) = -2 \sin x - 1$. Setting the derivative to zero results in the trigonometric equation $\sin x = -1/2$. He draws a rough graph of the function, showing a wave-like pattern superimposed on a linear decline. He circles the equation $\sin x = -1/2$ and explains that this equation has 'many values of x' due to the periodic nature of the sine function. He notes that unlike the previous polynomial example, there are infinite stationary points, making it impossible to list them all.

The lecture effectively demonstrates the application of the first and second derivative tests for optimization. It contrasts a simple polynomial case with a unique solution set against a trigonometric case with an infinite solution set, reinforcing the importance of analyzing the nature of the function when solving for critical points.

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