Rolle's Theorem Solved Numericals Explained in Hindi l Engineering Mathematics

5 Minutes Engineering

17 Jun 202304:10

Summary

TLDRIn this five-minute engineering video, the host dives into numerical problems, aiming to enhance viewers' understanding of solving polynomial functions. The video focuses on the function f(x) = x² - 5x + 4, explaining the importance of continuity and differentiability within the closed interval [1, 4]. It demonstrates how to find the function's value at specific points and its derivatives, ultimately applying the Rolle's Theorem to deduce the existence of a critical point between the given interval. The host encourages viewers to like, share, and watch the video for a deeper dive into numerical concepts.

Takeaways

🔧 The video introduces a tutorial on solving numerical problems related to engineering concepts.
📝 The function f(x) = x² - 5x + 4 is provided, and it is continuous over the closed interval [1, 4].
✅ The function is also differentiable over the open interval (1, 4), as the derivative f'(x) = 2x - 5 exists and is continuous.
🔍 The video checks the condition f(a) = f(b), where a = 1 and b = 4, confirming that f(1) = 0 and f(4) = 0, thus they are equal.
🎯 According to Rolle's Theorem, there exists a value c in (1, 4) such that f'(c) = 0. The video calculates c as 2.5, which lies within the interval.
🔄 The video repeats the process for another function, f(x) = x², over the interval [-1, 1].
💡 It is confirmed that f(x) = x² is continuous and differentiable over the interval, with f(-1) = 1 and f(1) = 1, satisfying the conditions of Rolle's Theorem.
🧮 The value of c is found to be 0 in this case, which also lies within the interval [-1, 1].
📜 The video provides a clear explanation of how to apply Rolle's Theorem to specific polynomial functions.
👍 The video concludes by encouraging viewers to like and share the content if they found it helpful.

Q & A

What is the main topic of the video?
-The main topic of the video is to demonstrate and explain the process of solving a quadratic equation and understanding the properties of a polynomial function within a closed interval.
What is the given polynomial function in the video?
-The given polynomial function is f(x) = x^2 - 5x + 4.
What are the values of 'a' and 'b' in the context of the video?
-In the context of the video, 'a' is given as 1 and 'b' is given as 4, referring to the closed interval [1, 4].
What is the first condition to check for the function in the video?
-The first condition to check is whether the function is continuous, which means it should exist and have a value at every point in the closed interval [1, 4].
Is the function in the video differentiable within the open interval (a, b)?
-Yes, the function is differentiable within the open interval (1, 4) as its derivative is 2x - 5, which exists and is continuous in the given interval.
How to find the value of 'a' or 'b' where the function equals zero?
-To find the value where the function equals zero, set the function equal to zero and solve for 'x'. In this case, x^2 - 5x + 4 = 0 can be solved to find the roots.
What is the derivative of the given polynomial function?
-The derivative of the polynomial function f(x) = x^2 - 5x + 4 is 2x - 5.
What does the video suggest to do after finding the derivative of the function?
-After finding the derivative, the video suggests to set it equal to zero to find the critical points, which are potential points where the function could have a local maximum or minimum.
How does the video explain the relationship between the derivative and the function's extremum?
-The video explains that if the derivative equals zero, it indicates a potential extremum. By solving 2x - 5 = 0, we find the critical point 'c' which is 2.5, which lies between 'a' and 'b'.
What is the conclusion of the video regarding the function's extremum?
-The conclusion is that the function has an extremum at 'c' = 2.5, which is within the interval [1, 4], and this extremum is a minimum since the derivative changes sign around this point.
What is the final message of the video to the viewers?
-The final message is to encourage viewers to like the video if they found it helpful and to share it with friends, thanking them for watching.

Outlines

00:00

📚 Introduction to Numerical Problem Solving

The script begins with an introduction to a video on engineering, promising a comprehensive look at numerical problem solving. The presenter introduces a quadratic function, x² - 5x + 4, and sets the stage for an in-depth exploration of its properties. The focus is on understanding the function's continuity and differentiability within the closed interval [1, 4], highlighting the importance of these mathematical concepts in real-world applications.

Mindmap

Keywords

💡Continuity

Continuity in mathematics, particularly in calculus, refers to the property of a function where the function's value at a certain point is the same as the limit of the function as it approaches that point. In the video, the concept of continuity is discussed in the context of a polynomial function, emphasizing that the function is continuous over the closed interval [1, 4].

💡Derivative

A derivative in calculus is a measure of how a function changes as its input changes. It is the slope of the tangent line to the function at a certain point. The video script mentions finding the derivative of the function 2x - 5, which is essential for understanding the function's behavior and determining critical points.

💡Polynomial Function

A polynomial function is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The script discusses a polynomial function, x² - 5x + 4, and its properties, such as continuity and differentiability.

💡Interval

In mathematics, an interval refers to a set of numbers with a specific range. The video script specifies a closed interval [1, 4], which includes all the numbers from 1 to 4, and an open interval (1, 4), which excludes the endpoints. These intervals are used to define the domain where the function is being analyzed.

💡Limit

The limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. The script mentions checking if the function's limit exists and is equal to the function's value at a point, which is a condition for continuity.

💡Critical Point

A critical point is a point on the graph of a function where the derivative is zero or undefined, indicating a potential maximum or minimum of the function. The script calculates the derivative of the function and sets it to zero to find the critical point.

💡Root Theorem

The Root Theorem, also known as the Intermediate Value Theorem, states that if a continuous function has values of opposite signs at two points in its domain, then it has at least one root between those points. The script does not explicitly mention this theorem, but the discussion of finding roots of the function is related to this concept.

💡Differentiable

A function is differentiable at a point if it has a derivative at that point, meaning the function's rate of change at that point is well-defined. The script checks if the function is differentiable over the open interval (1, 4), which is a requirement for the function to have a well-defined tangent line at every point in that interval.

💡Closed Interval

A closed interval in mathematics includes all the numbers between its endpoints, including the endpoints themselves. The script discusses the function's continuity over the closed interval [1, 4], which is important for understanding the function's behavior at the endpoints.

💡Open Interval

An open interval excludes its endpoints, meaning it contains all the numbers between its endpoints but not the endpoints themselves. The script mentions the open interval (1, 4) in the context of checking the function's differentiability, which is crucial for analyzing the function's behavior within the interval without considering the endpoints.

💡Value of a Function

The value of a function at a certain point is the output of the function when a specific input is provided. The script calculates the value of the function at x = 1 and x = 4, which is necessary for determining if the function satisfies certain conditions, such as being equal to its limit at those points.

Highlights

Introduction to a new educational video on numerical methods in engineering.

The video will cover the solution of a quadratic function x² - 5x + 4.

Explanation of the closed interval [1, 4] and the point of division.

Checking the first condition for the function to be continuous.

Deriving the function's derivative and checking for differentiability in the open interval.

Calculation of the function's value at x = 1 and x = 4 to check for equality.

Finding the function's value at x = 2.5 using the derivative and solving for x.

Discussion on the function's continuity and differentiability within the interval.

Exploring the function's behavior at the endpoints and within the interval.

Demonstration of the function's value at x = -1 and x = 1 using the derivative.

Verification of the function's values at x = -1 and x = 1 for equality.

Explanation of the function's derivative and its implications for the function's behavior.

Solution of the derivative equation 2x - 5 = 0 to find the critical point.

Conclusion of the video with an invitation to like and share for more numerical methods content.

The video aims to improve understanding of numerical methods in engineering.

Encouragement for viewers to apply the concepts learned in the video to their studies or work.

Thanking the viewers for watching and looking forward to their continued engagement.