Final project for Calculus 2 course at Shahid Beheshti University (Spring 2023)
Score: 2 out of 2
This report presents the application of Newton's method to find the root of the function f(x) = arctan(x) - 1/3. Newton's method is an iterative root-finding algorithm that utilizes the derivative of a function to approximate its roots.
- Define the function f(x) = arctan(x) - 1/3.
- Calculate the derivative of f(x) with respect to x, denoted as df(x).
- Set the initial guess x0 = 0.5.
- Set the tolerance (tol) to 0.1e-9 to determine the convergence criteria.
- Set the maximum number of iterations (maxIter) to 100 as a safeguard against infinite loops.
- Initialize the current and previous values of x as x and xPrev, respectively.
- Initialize the iteration counter iter to 0.
- Execute a while loop until the difference between x and xPrev is greater than the tolerance and the maximum number of iterations has not been reached:
- Update xPrev to store the previous value of x.
- Compute df_value as the value of df(x) using the derivative function.
- Compute f_value as the value of f(x) using the function f.
- Update x by subtracting f_value divided by df_value.
- Increment the iteration counter iter.
- Upon exiting the loop, the value of x represents the approximation of the root. Report the obtained root value.
The Newton's method iteration converged to a root approximation of x ≈ 0.5493061443340548 for the given function f(x) = arctan(x) - 1/3.
By utilizing Newton's method, we successfully approximated the root of the function f(x) = arctan(x)
- 1/3. Newton's method is an effective iterative algorithm for finding roots of functions when the initial guess is close enough to the true root.
This report presents the surface and contour plots of the ID function, which is defined as f(x) = x^4 - 4x^2 + 1. The plots provide insights into the behavior and characteristics of the ID function.
- Define the ID function as ID(x) = x^4 - 4x^2 + 1.
- Generate a meshgrid of x and y values in the desired range.
- Calculate the corresponding z values using the ID function.
- Create surface plots with different views to visualize the function in a three-dimensional space.
- Create contour plots with different windows to visualize the function's contours and level curves.
Surface Plots The surface plots provide a three-dimensional representation of the ID function.
- Surface Plot - View 1: The plot shows the ID function from a default viewpoint.
- Surface Plot - View 2: The plot shows the ID function from an elevated viewpoint.
- Surface Plot - View 3: The plot shows the ID function from a different elevated viewpoint. Contour Plots The contour plots display the ID function's contours and level curves.
- Contour Plot - Window 1: The plot shows the contours of the ID function within the range [-2, 2] for both x and y.
- Contour Plot - Window 2: The plot zooms in on the range [-1, 1] for both x and y, providing a closer view of the contours.
- Contour Plot - Window 3: The plot further zooms in on the range [-0.5, 0.5] for both x and y.
- Contour Plot - Window 4: The plot focuses on a small range of [-0.1, 0.1] for both x and y, revealing detailed contours.
The surface and contour plots of the ID function provide a visual representation of its behavior and characteristics. The surface plots demonstrate the overall shape of the function, while the contour plots illustrate the contours and level curves at different zoom levels. These plots help in understanding the critical points, curvature, and overall behavior of the ID function.
This report focuses on finding the first- and second-degree Taylor polynomials for the ID function. The Taylor polynomials provide local approximations of the function around a critical point and help in understanding the behavior of the function in its vicinity.
- Define the ID function as f(x, y) = x^4 - 4x^2 + 1.
- Perform the following tasks:
- Verify the partial derivatives of the quadratic approximation Q at a critical point (a, b).
- Find the first- and second-degree Taylor polynomials L and Q for the ID function at the estimated critical point obtained from Problem 14.94.
- Compare the values of f, L, and Q at a specific point (x0 + 0.1, y0 - 0.1).
- Plot the ID function, the first-degree Taylor polynomial L, and the second-degree Taylor polynomial Q in a 3D space.
Task 1: Verification of Partial Derivatives of Q
The partial derivatives of the quadratic approximation Q at the critical point (a, b) are verified. The first derivative of Q evaluated at (a, b) is constant, while the second derivative of Q evaluated at (a, b) is also constant.
Task 2: First- and Second-Degree Taylor Polynomials
The first-degree Taylor polynomial L and the second-degree Taylor polynomial Q for the ID function at the estimated critical point are calculated. These polynomials provide local approximations of the function in the vicinity of the critical point.
Task 3: Comparison of Function, L, and Q
The values of f, L, and Q are compared at the point (x0 + 0.1, y0 - 0.1). This allows us to observe how well the Taylor polynomials approximate the function at a specific location near the critical point.
Task 4: Plotting Function, L, and Q
A 3D plot is created to visualize the ID function, the first-degree Taylor polynomial L, and the seconddegree Taylor polynomial Q. This plot helps in understanding the similarities and differences between the function and its corresponding Taylor approximations.
The first- and second-degree Taylor polynomials provide local approximations of the ID function around a critical point. By comparing the values and visualizing the function, L, and Q, we can observe the level of approximation and the behavior of the function in the vicinity of the critical point. These Taylor polynomials are useful tools for analyzing and understanding the properties of the ID function near critical points.
The provided code uses Maple to calculate the values of Vn according to the formula Vn = Vn−1 * g(n), where g(n) = n^2 + 1. It then plots the results.
The code initializes the variables m (which represents the number of iterations) and V, a vector to store the calculated values. It sets the initial value of V[1] to 2.
The for loop iterates from n = 2 to n = m and calculates Vn using the formula V[n] = V[n-1] * g(n). This loop populates the vector V with the computed values.
The code also extracts the formula for Vm and assigns it to the variable V_formula.
Lastly, the code creates a plot to visualize the relationship between n and Vn. It creates a point plot (pp) and a line plot (pl) using the values of n and V, and then displays them together. The plot includes labels for the x-axis ("n") and the y-axis ("Vn").
In summary, this code evaluates the values of Vn using the provided formula and plots the results. The derived formula for Vm and the plotted graph provide a visual representation of the relationship between n and Vn.
