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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....
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...
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 e...
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...
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...
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...
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...
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...
- 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...
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...
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...
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...
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