2. Consider the equation f(x) = x4 – 4.x² +4 and the problem of computing the positive root of f(x) = 0. a) will guarantee convergence to the root. If possible, determine starting points a, b in the bisection method that
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- 2. Approximate the root of the following function first using the bisection method and then using method of falsi position with the stopping condition |(x)| <8 x10^-4 .?(x) = x^3+2x^2+10x-20, [1, 2] .Which method converges faster to the solution?For the part highlighted in green, why doesn't negative infinitiy cancel with the negative near the x, and give us infinity? Instead it gives us 0. I know a number to thr negative infinity power is 0, just thought 2 negatives made a positive so was wondering why not here. And also, would thus converge or diverge?Compute the least squares quadratic polynomial p2(x) which gives the best fit toex over the interval [0, 1]. How does p2(x) compare for accuracy with the Taylor polynomialabout x = 0?
- Apply Newton’s method to estimate the solution of x3 − x − 1 = 0 by taking x1 = 1 and finding the least n such that xn and xn+1 agree to three digits after the decimal point.2. Give an example showing that, even if f0(c) = 0, it could happen that f(x) does not have a localmaximum or minimum at x = c. How is this consistent with Fermat’s Theorem in the book?Find the maximum error if the approximation 1 + x + x2/2 is used to approximate ex on the interval [0,2] - using Taylor's inequality.
- find the maximum and minimum values of the func- tion on the given interval y=2x2−4x+2, [0,3]Consider the problem minimize 5x2+5y2−xy−11x+11y+11 (a) Find a point satisfying the first-order necessary conditions for a solution. b) Show that this point is a global minimum. c) What would be the rate of convergence of steepest descent for this problem? d) Starting at x=y=0, how many steepest descent iterations would it take (at most) to reduce the function value to 10−11?Find the third iteration value of an extremum (maximum/minimum value) of if a = 5, b = 0.5, and c = 5 using Newton's Method with an initial guess value of x = - 4.3
- Use the Euler algorithm with a step size h = 0.2 to find an approximate value of y10 for the linear first order initial value problem dy/dx = sinx - y with y(0)=1 in the interval 0 ≤ x ≤ 2 in four decimal places.Find a polynomial of degree n=2 that has the given zero(s) x = −3If y'=x3+y2, where y=1 when x=0, find a degree 3 polynomial that best fits the solution using Taylor Series Polynomial.