4. (a) Calculate the Euler number correct to 5 decimals, using the remainder term of a...
4. (a) Calculate the Euler number correct to 7 decimals, using the remainder term of a Taylor series. (b) Compute the first seven non-zero terms in the Taylor expansion of f(x) = cos x about the point Xo = .
Please help me solve this. thanks 5. Using Taylor series, derive the error term for the approximation f' (x) ~ -3 f(x) + 4 f(x + b)-f(x + 2h)]. 2h 5. Using Taylor series, derive the error term for the approximation f' (x) ~ -3 f(x) + 4 f(x + b)-f(x + 2h)]. 2h
PLEASE ANSWER ALL! SHOWS STEPS 2. (a) Prove by using the definition of convergence only, without using limit theo- (b) Prove by using the definition of continuity, or by using the є_ó property, that 3. Let f be a twice differentiable function defined on the closed interval [0, 1]. Suppose rems, that if (S) is a sequence converging to s, then lim, 10 2 f (x) is a continuous function on R r,s,t e [0,1] are defined so that r...
a tinctlon of series y I Taylor The 6. Taylor's Remainder Theorem. fn)(0) where fw) is the n-th derivative of f, and the remainder term Ry is given by NN+1 for some point c between 0 and z. (Note. You do not need to prove Taylor's Remainder Theorem.) Problems (a) (5%) write this series for the function ez for a general N (b) (10%) Apply Taylor's Remainder Theorem to show that the Taylor series of function f = ez converges...
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....
(3.5) Summing the Euler S-function (n): The Euler 6-function 6(n) counts the number of positive integers less than or equal to n, which are relatively prime with n. Evaluate 4(d), and prove that your answer is correct. (3.4) Relatively Prime Numbers and the Chinese Re- mainder Theorem: Give an example of three positive integers m, n, and r, and three integers a, b, and c such that the GCD of m, n, and r is 1, but there is no...
I. Let f : R2 → R be defined by f(x)l cos (122) 211 Compute the second order Taylor polynomial of f near the point xo - 0. A Road Map to Glory (On your way to glory, please keep in mind that f is class C) a) Fill in the blanks: The second order Taylor's polynomial at h E R2 is given by T2 (h) = 2! b) Compute the numbers, vectors and matrices that went into the blanks...
4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is differentiable on (0,1) and f(x) < 1 for all x € (0,1) and n e N. (a) If (fn (0))nen converges to a number A, prove that lim sup|fn(x) = 1+|A| for all x € [0, 1]. n-too : (b) Suppose that (fr) converges uniformly on [0, 1] to a function F : [0, 1] + R. Is F necessarily differentiable on (0,1)? If...
got the first one but the last 4 are giving me trouble. i) Calculate the first four Taylor polynomials Po(r), P(), P2(), P3(1) ii) Sketch the function together with the first four Taylor polynomials Po(x), P (r), P(x), P3(1) around the basis point iii) Find the general form of the n-th Taylor polynomials P,(x) and write down the Taylor series for the function about the basis point a iv) Calculate the interval of convergence of the Taylor series (calculate first...
5. (a) (10) Write down the Taylor series for3) and find the 6th Taylor polynomial p() (b) (10) Find the Taylor series about 0 for f(a) 3 cos, and use the Lagrange Remainder Formula toshow that for any z, nlim。m(z) = 0. em t 5. (a) (10) Write down the Taylor series for3) and find the 6th Taylor polynomial p() (b) (10) Find the Taylor series about 0 for f(a) 3 cos, and use the Lagrange Remainder Formula toshow that...