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Numerical Analysis COT4501 Fall 2007 Midterm I October 9, 2007 1. Taylor series approximation [35 points overall] [10 points] What is the second order Taylor series approximation p2 (x) and the remainder R2 (x) for f (x) = x log x when the approximation is carried out at x0 = 1 and in the interval [1, 2]? [Here log(x) denotes the natural logarithm.] [25 points] 1. [5 points] Write down the nth term in the Taylor series approximation of f (x) = x log x when the approximation is carried out at an arbitrary x0 and is valid in the interval [x0 , 2x0 ]. 2. [10 points] Bound the nth term in the Taylor series approximation using the fact that x ≥ x0 for all points in the interval [x0 , 2x0 ]. Let us denote the bound by g (n) (x, x0 ). ∞ (x−x0 )2 3. [10 points] Sum up n=2 g (n) (x, x0 ) to show that f (x) − p1 (x) ≤ 2(2x0 −x) or in other x (x−x0 )2 words, show that x log x0 − x + x0 ≤ 2(2x0 −x) for all x ∈ [x0 , 2x0 ]. [Hint: Use a geometric series summation. Here p1 (x) is the rst order Taylor series approximation of f (x) = x log x.] 2. Derivative approximation [35 points overall] f (x+ h )−f (x− h ) [10 points] Is the approximation f (x) ≈ 2 h 2 a valid rst derivative approxima- tion? Discuss. [25 points] Begin with f (x) ≈ Af (x + ah) + Bf (x + bh). 1 1. [10 points] Expand f (x + ah) and f (x + bh) up to second order using a Taylor series approximation. 2. [15 points] Construct an approximation to f (x) by ensuring that i) the term involving f (x) is zero, ii) the term involving f (x) is zero, and that iii) the coecient of f (x) is one. Write down the resulting constraints involving A, B, a, b, and h. Pick a set of possible values for (A, B, a, b) that satisfy the constraints. 3. Linear Interpolation [30 points overall] [10 points] Rolle's Theorem: Given a function f (x) which is continuous and dierentiable in an interval [a, b] and with f (a) = f (b) = 0, show that there exists a point ξ ∈ [a, b] such that f (ξ) = 0. [20 points] Extension of the linear interpolation formula to quadratic interpolation. Given three points (x0 , f (x0 )), (x1 , f (x1 )), and (x2 , f (x2 )): 1. [10 points] Construct a single second order polynomial approximation p2 (x) (by ex- tending the linear approximation formula) of the form p2 (x) = ax2 + bx + c such that p2 (x0 ) = f (x0 ), p2 (x1 ) = f (x1 ) and p2 (x2 ) = f (x2 ). [Hint: We are looking for a single quadratic polynomial here. Do not combine two piecewise linear approximations.] 2. [10 points] What are the values of a, b, and c? 2 List of Useful Formulae (x−x0 )i f (i) (x0 ) (x−x0 )(n+1) f (n+1) (ξ[x0 ,x] ) Taylor series approximation: f (x) = n i=0 i! + (n+1)! with ξ[x0 ,x] h(n+1) f (n+1) (ξ [x,h] ) hi f (i) (x) n in the interval [x0 , x]. Also, f (x + h) = i=0 i! + (n+1)! where ξ[x,h] depends on both x and h. Euler's method: Approximation of y = f (t, y ) is yn+1 = yn + hf (tn , yn ) with y0 = y(t0 ). Dierence Approximations of Derivatives: f (x) = f (x+h)−f (x) − 1 hf (ξx,h ), f (x) = h 2 f (x+h)−2f (x)+f (x−h) 1 2 h2 − 12 hf (ξx,h ). Linear Interpolation formula: p1 (x) = x11−x0 f (x0 ) + xx−x00 f (x1 ). x −x 1 −x n Bisection method: |α − xn | ≤ 1 (b − a) with the midpoint of the interval chosen at 2 each step. Newton's method: Update rule for nding a root to f (x) = 0 is xn+1 = xn − f (xn )) with f (xn initial condition x0 . √ Quadratic formula: Solution of ax2 + bx + c = 0 is x = −b± 2a −4ac . b 2 Geometric series summation: ∞ arn = 1−r where r < 1. n=0 a Mean Value Theorem: Let f be a given function continuous on [a, b] and dierentiable f (b)−f (a) on (a, b). Then there exists a point ξ ∈ [a, b] such that f (ξ) = b−a . Derivatives: d dx x log x = (1 + log x), d dx 1 log x = x , d 1 dx x 1 = − x2 , d 1 dx xn n = − xn+1 . 3

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