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2.(10 pts) Suppose that P5 (2) interpolates function f(0) = -at 6 evenly distributed points 0,...
Given the data points (-3,5),(-2,5),(-1,3), (0, 1) (a) Find the interpolating polynomial passing through these points. (b) Using your polynomial from (a), evaluate P(1). (c) This polynomial interpolates the function f(x) = 24. Find an upper bound for the approximation in part (b).
Question 2 6 pts Let T2(x) be the Taylor polynomial for f(x) = 2x + 2 centered at c = 1. Fill in the blanks in the paragraph below. Use exact values. The Error Notice that 4.2 = f(1.1) T2(1.1) = Bound says that the maximum possible value of the error is Tonal x-c"+1 1V 4.2 -T2(1.1) < (n + 1)! where K = and 2 - 1+1 (n+1)! Question 3 4 pts Fill in the blank. Use exact values...
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...
3. [10 pts.] Suppose r.v. y is uniformly distributed over (0,27) i.e. f*() = 1/2, for 0 <o<27 and 0 elsewhere. Consider the following r.v.'s: X = cos y and Y = siny. a. Prove that X and Y are orthogonal. b. Prove that X and Y are uncorrelated.
Suppose f is a continuous and differentiable function on [0,1] and f(0)= f(1). Let a E (0, 1). Suppose Vr,y(0,1) IF f'(x) 0 and f'(y) ±0 THEN f'(x) af'(y) Show that there is exactly f(ax) and f'(x) 0 such that f(x) one Hint: Suppose f(x) is a continuous function on [0, 1] and f(0) x € (0, 1) such that f(x) = f(ax) f(1). Let a e (0,1), there exists an Suppose f is a continuous and differentiable function on...
6. (10 points) The graph of a function fis given below. 6 2 0 2 6 a. For which values of x in the interval (-4,4) is f not continuous? Give the type of discontinuity (ump, infinite or removable) in each case. b. For which values of x in the interval (-4,4) is fnot differentiable? c. Give the value of the following: i lim f(x) ii. lim f(x) iii. lim f(x) 22+
1. (a) We want to develop a method for calculating the function sint dt f)-inf t 0 for small or moderately small values of x. This is a special function called the "sine integral", and it is related to another special function called the "exponential integral". It arises in diffraction problems. Derive a Taylor-series expression for f(x), and give an upper bound for the error when the series is terminated after the n-th order term. [HINT: (-1)"*z ? + R...
- Question 2 3 points Consider the function f (x) = ln (1+2). (a) Enter the degree-n term in the Taylor Series around x = 0. ((-1)^(n-1)*x^n)/n (b) Enter the error term En (2) which will also be a function of x and n. ((-1)^n*x^(n+1))/((n+1)*(1+z)^(n+1) (c) Find an upper bound for the absolute value of the error term when x > 0. It may help to remember that z is between x and 0. x^(n+1)/(n+1) 90 (d) Use this formula...
We want to produce an evenly spaced table of values for the function f(x) sin(x) for x E [0,Tt/2] such that, with cubic interpolation, we can give the values of the function at any point in the interval with an error less than 5 10-12. That means finding a number n such that with h = π/2n and Xk-kh, k-0, , n the cubic interpolation polynomial with the interpolation points XK-1,XK, X+1 XK+2 for x has an error less than...
Suppose f : B(0.1) C is holomorphic, with irg:) 1 for every z є B(0,1). Suppose also that f(0)-0, so f(z)g(2) for some holomorphic function g: B(0,1)C. (a) By applying the Maximum Principle to g on B(0, r) where 0 < r < 1 , deduce that If( S for every 2E (0, 1) . (b) Show also that |f'(0) S1 (c) Show that if lf(z)- for some z B(0,1)\(0), or if If,(0)| = 1 , then there is a...