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1. The two-point forward difference quotient with error term is given by where ξ e ll, l + hl. In...
1. Consider the polynonial Pl (z) of degree 4 interpolating the function f(x) sin(x) on the interval n/4,4 at the equidistant points r--r/4, xi =-r/8, x2 = 0, 3 π/8, and x4 = π/4. Estimate the maximum of the interpolation absolute error for x E [-r/4, π/4 , ie, give an upper bound for this absolute error maxsin(x) P(x) s? Remark: you are not asked to give the interpolation polynomial P(r). 1. Consider the polynonial Pl (z) of degree 4...
numerical analysis The integral 1 I = da +4 is to be approximated numerically. (a) Find the least integer M and the appropriate step size h so that the global error for the composite (10) Trapezoidal rule, given by 41,4 ?s"(E), a<<<0, 12 is less than 10-5 for the approximation of I. (b) Use the two-term Gaussian quadrature formula and 6 decimal place arithmetic to approximate I. (Hint: Parameters are ci = 1, i = 1,2;rı = 0.5773502692, r2 =...
The nth-order Taylor polynomial for a function f(x) using the h notation is given as: Pa (x + h) = f(x) + f'(a)h + salt) 12 + () +...+ m (s) n." The remainder of the above nth-order Taylor polynomial is defined as: R( +h) = f(n+1)(C) +1 " hn+1, where c is in between x and c+h (n+1)! A student is using 4 terms in the Taylor series of f(x) = 1/x to approximate f(0.7) around x = 1....
Exercise 6: Given the table of the function f(x)-2" 2 X 0 3 2 f(x) 1 2 4 8 a) Write down the Newton polynomials P1(x), P2(x), Pa(x). b) Evaluate f(2.5) by using Pa(x). c) Obtain a bound for the errors E1(x), E2(x), Es(x) Exercise 7: Consider f(x)- In(x) use the following formula to answer the given questions '(x) +16-30f+16f,- 12h a) Derive the numerical differentiation formula using Taylor Series and find the truncation error b) Approximate f'(1.2) with h-0.05...
4. For f(x) = e-* and h = 0.10 where, C = 1.** a) Use centered approximations to estimate the first and second derivatives of f(x) at x = 2. Use the east accurate formulas available. (10 pts) b) Using the most acurate forward and backward difference formulas, estimate the first derivative of f(x) at x 2. (10 pts) Forward Difference First Derivative 7.) - SD Error OM or) = -1.) + 40..) - 3 ) 2h Second Derivative 'w...
Hello, I am having trouble with part c of this question. Here is my work so far: The solution for part c states that a possible solution is (e^16 * 4^3) / 3! I am having trouble understanding how they got e^16 or why they decided to use e^(4^2) for M in the equation |f(x) - Tn(x)| <= (M / (n + 1)!) * |x - 0|^(n + 1). From my understanding, I have to maximize H^3(x) (i.e. 3rd derivative...
Numerical methods problems 1, 2 and 3 1. Find the area of the region bounded by f(x)-25-x2 , g(x)-V36-x2 . x=2, and (a) right Riemann sum with 8 segments. (b) midpoint rule with 8 segments (e) Simpson's rule with 8 segments. Determine the average of the function f(x)=2x sinyx on the interval [1.8,3.4] using Romberg rule for 1, 2, 4 and 8 segments. 2, A new fuel for recreational boats being developed at the local university was tested at an...
Question l: Consider the function f(x) = sin(parcsinx),-1 < x < 1 and p E R (a) Calculate f(0) in terms of p. Simplify your answer completely fX) sin(p arcsinx) f(o) P The function fand its derivatives satisfy the equation where f(x) denotes the rth derivative of f(x) and f (b) Show thatf0(n2p2)f(m)(o) (x) is f(x). (nt2) (nti) (I-x) (nt 2 e 0 (c) For p E R-仕1, ±3), find the MacLaurin Series for f(x), up to and including the...
real analysis 1,2,3,4,8please 5.1.5a Thus iff: I→R is differentiable on n E N. is differentiable on / with g'(e) ()ain tained from Theorem 5.1.5(b) using mathematical induction, TOu the interal 1i then by the cho 174 Chapter s Differentiation ■ EXERCISES 5.1 the definition to find the derivative of each of the following functions. I. Use r+ 1 2. "Prove that for all integers n, O if n is negative). 3. "a. Prove that (cosx)--sinx. -- b. Find the derivative...
the two codes have an error and I can’t figure out where. First one public class Program { public static void main(String[] args) { int i, j = 1, k = 0; String alphabet = "abcdefghijklmnopqrstuvwxyz"; i = alphabet.length(); while (i>0){ j=(i-(i-j)); System.ou.println(alphabet.charAt(k)+" is the number "+j"letter in the alphabet"); k++; j++; i--; } } } Second one char [] alphabet = {a,b,c,d,e,f,g,h,I,j,k,l,m,n,o,p,a,r,s,t,u,v,w,x,y,z}; for (int i = 0; i < alphabet.length; i++) { System.out.println(alphabet[i] +...