Let x1 = 1.5. Using Newton\'s Method, state the formula for X2 and then find X2.
ID: 3287936 • Letter: L
Question
Let x1 = 1.5. Using Newton's Method, state the formula for X2 and then find X2. Show all of your work. Illustrate the procedure on the diagram below. State the formula for x3 and X4, and then find X3 and X4. Show all of your work. Illustrate the procedure on the diagram below. Do all of your calculated values for X2, X3 and X4 seem to be approaching the value of "c" where f(x) crosses the x-axis (i.e., the root)? What is Newton's Method used for? Assuming f(x) has a root, list three circumstances where Newton's Method will fail. Explain the meaning of the indefinite integral f(x) dx. How does an indefinite integral f(x) dx differ from a definite integral b and a f(x) dx ? What do each of these integrals represent? In your own words, explain the difference between the Fundamental Theorem of Calculus Part 1 and the Fundamental Theorem of Calculus part 2.Explanation / Answer
2a.) In numerical analysis, Newton's method is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function. circumstances when it can fail are: i)Bad starting points In some cases the conditions on the function that are necessary for convergence are satisfied, but the point chosen as the initial point is not in the interval where the method converges. This can happen, for example, if the function whose root is sought approaches zero asymptotically as x goes to or . In such cases a different method, such as bisection, should be used to obtain a better estimate for the zero to use as an initial point. ii)Derivative does not exist at root A simple example of a function where Newton's method diverges is the cube root, which is continuous and infinitely differentiable, except for x = 0, where its derivative is undefined. iii)Non-quadratic convergence In some cases the iterates converge but do not converge as quickly as promised. In these cases simpler methods converge just as quickly as Newton's method. 2b) A definite integral is an integral with upper and lower limits. If is restricted to lie on the real line, the definite integral is known as a Riemann integral. However, a general definite integral is taken in the complex plane, resulting in the contour integral with, and in general being complex numbers and the path of integration from to known as a contour. In calculus, an antiderivative, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to f, i.e., F ? = f. The process of solving for antiderivatives is called antidifferentiation (or indefinite integration) and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval. 2c) http://www.emathhelp.net/notes?nid=114 refer the above link for the best answer.
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