Exam1 EXAMI Sec 1 1 THE TAYLOR SERIES

Exam-1 EXAM-I

Sec: 1. 1 THE TAYLOR SERIES Taylor series ( center is a ) If the function f and its first n + 1 derivatives are continuous on an interval containing a and x, truncation error n-th Taylor polynomial Polynomial with degree n there exists a point ξ between a and x such that the reminder

Some Comments on Root finding Formula Error Equation guess p 0 suff close to p* Newton’s p 0 suff close to p* Secant Any Bisection Any Fixed-point False position Secant +test Any

Sec: 2. 2 Fixed- Point Iteration Theorem 2. 4 (Fixed-Point Theorem) Let g ∈ C[a, b] be such that g(x) ∈ [a, b], for all x in [a, b]. Suppose, in addition, that g’ exists on (a, b) and that a constant 0 < k < 1 exists with |g(x)| ≤ k, for all x ∈ (a, b). Then for any number p 0 in [a, b], the sequence defined by pn = g( pn− 1), n ≥ 1, converges to the unique fixed point p in [a, b]. Two important error equation

Sec: 3. 1 Interpolation and the Lagrange Polynomials Given the set of data Theorem 3. 2 If x 0, x 1, . . . , xn are n + 1 distinct numbers and f is a function whose values are given at these numbers, then a unique polynomial p(x) of degree at most n exists with f (xk) = p(xk), for each k = 0, 1, . . . , n. This polynomial is given by Remark

Sec: 3. 3 Divided Differences n-th divided difference Example First 1 3 2 6 3 19 5 99 Second Third

Sec: 3. 3 Divided Differences General Form of Newton’s Interpolating Polynomials