Using Newton's Method | MIT 18.01SC Single Variable Calculus, Fall 2010
TLDRIn this educational video, the professor introduces Newton's Method for solving equations, specifically focusing on the equation 2cosine(x) = 3x. Starting with an initial guess of pi/6, the video demonstrates how to approximate the solution using Newton's iterative formula. It also provides a graphical approach to understand the uniqueness of the solution and the rationale behind the initial guess. The process involves calculating the function and its derivative, applying Newton's formula to refine the guess, and emphasizing the method's effectiveness in finding roots of equations.
Takeaways
- π The video is a recitation focused on practicing Newton's Method to find solutions to equations.
- π The specific equation to be solved is \(2 \cos(x) = 3x\), using Newton's Method to approximate the solution.
- π The initial value \(x_0\) chosen for the method is \(\pi / 6\), which is considered a reasonable starting point.
- π The video includes a graphical analysis to show that the equation has only one solution by sketching the curves of \(y = \cos(x)\) and \(y = 3x/2\).
- π The intersection of the two curves on the graph represents the solution to the equation, indicating a single point of intersection.
- π€ The choice of \(\pi / 6\) as the starting point is justified by the fact that the solution must lie between 0 and 1, and \(\pi / 6\) fits within this range.
- π Newton's Method is applied to find the zero of the function \(2 \cos(x) - 3x\), which equates to solving the original equation.
- π The derivative of the function needed for Newton's Method is \(-2 \sin(x) - 3\).
- π’ Newton's Method formula is used to calculate the next x-value: \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\).
- π The script demonstrates the calculation from \(x_0\) to \(x_1\) using the function and its derivative evaluated at \(\pi / 6\).
- π The final takeaway is that the viewer should be able to approximate \(x_2\) using the method demonstrated, aiming for a solution accurate to at least two decimal places.
Q & A
What is the main topic of the video?
-The main topic of the video is the application of Newton's Method to approximate a solution to the equation 2 cosine x equals 3x.
What is the initial value (x_0) suggested for starting Newton's Method in the video?
-The initial value (x_0) suggested for starting Newton's Method is pi over 6.
What is the purpose of sketching the curves y = cosine x and y = 3/2 x on the same xy-plane?
-The purpose of sketching these curves is to visually identify the intersection point, which represents the solution to the equation 2 cosine x equals 3x.
Why is pi over 6 a reasonable first value to choose for Newton's Method in this context?
-Pi over 6 is a reasonable first value because the solution to the equation is known to be between 0 and 1, and pi over 6 falls within this range.
What is the derivative of the function 2 cosine x - 3x?
-The derivative of the function 2 cosine x - 3x is -2 sine x - 3.
How is Newton's Method defined in the context of this video?
-Newton's Method is defined as the iterative process where the next x-value is the previous x-value minus the function evaluated at the previous value divided by the derivative evaluated at the previous value.
What is the function that needs to be solved to find the solution to the equation 2 cosine x equals 3x?
-The function to be solved is 2 cosine x - 3x, as finding its zero will solve the original equation.
How does the video demonstrate that the equation 2 cosine x equals 3x has only one solution?
-The video demonstrates this by sketching the curves y = cosine x and y = 3/2 x and showing that they intersect at only one point.
What is the value of cosine (pi over 6) used in the video?
-The value of cosine (pi over 6) used in the video is root 3 over 2.
What is the approximate value of x_1 obtained after applying Newton's Method with the initial value x_0 as pi over 6?
-The approximate value of x_1 obtained is around 0.564, depending on the precision of the calculations.
What is the next step after finding x_1 using Newton's Method in the video?
-The next step is to find x_2 by applying Newton's Method again, using x_1 and evaluating the function and its derivative at x_1.
Outlines
π Introduction to Newton's Method
The professor begins by welcoming students back to recitation and introduces the topic of using Newton's Method to find solutions to equations. Specifically, the equation in focus is 2 cosine x equals 3x. The professor suggests starting with an initial value of x_0 as pi over 6 and aims to find the second approximation, x_2. The students are encouraged to pause the video to work on the problem before the solution is revealed. The professor also explains why pi over 6 is a reasonable starting point and confirms that the equation has only one solution by sketching the curves of y equals cosine x and y equals 3/2 x on the same xy-plane.
π Applying Newton's Method to the Equation
The professor proceeds with the application of Newton's Method to the function derived from the equation 2 cosine x minus 3x. The derivative of the function is identified as negative 2 sine x minus 3. Newton's Method formula is reiterated, which involves updating the x-value by subtracting the function value at the current x divided by its derivative at that x. Using pi over 6 as the initial guess, the professor calculates the first approximation, x_1, and suggests that the second approximation, x_2, should be similar, depending on the precision maintained. The summary concludes by reminding the purpose of the exercise: to find a solution to the given equation using Newton's Method and to understand the graphical representation of the problem.
Mindmap
Keywords
π‘Newton's Method
π‘Roots
π‘Derivative
π‘Function
π‘Cosine Function
π‘Sine Function
π‘Initial Value
π‘Approximation
π‘Intersection
π‘Graphical Method
π‘Iterative Process
Highlights
Introduction to Newton's Method for solving equations.
Specific application of Newton's Method to approximate the solution to 2cosine(x) = 3x.
Initial value for Newton's Method is set to pi over 6 (Ο/6).
Explanation of why Ο/6 is a reasonable starting point.
Demonstration of the uniqueness of the solution through a graphical analysis.
Sketching the curves y = cosine(x) and y = 3/2x to find their intersection.
Confirmation that there is only one solution to the equation.
Graphical justification for the choice of Ο/6 as an initial guess.
Explanation of the process to find zeros of the function 2cosine(x) - 3x.
Derivation of the derivative of the function: -2sine(x) - 3.
Formula for Newton's Method and its application to the problem.
Calculation of x_1 from x_0 using Newton's Method.
Evaluation of the function and its derivative at pi over 6.
Approximation of x_1 to be around 0.564.
Process for calculating x_2 from x_1 using Newton's Method.
Estimation of x_2 to be fixed to the first two decimal places.
Discussion on the potential slight variation in x_2 due to rounding.
Summary of the application of Newton's Method to find a solution to 2cosine(x) = 3x.
Transcripts
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