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(a) Find the interpolating polynomials to the function f(x) = e-* sin(7x) at x = -1,...
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...
this is numerical analysis 2. Consider the function f(x) = -21° +1. (a) Calculate the interpolating...
this is numerical analysis
2. Consider the function f(x) = -21° +1. (a) Calculate the interpolating polynomial pz() for data using the nodes 2o = -1, 11 = 0, 12 = 1. Simplify the polynomial to standard form. Use the error theorem for polynomial interpolation to bound the error f(x) - P2(x) on the interval (-1,2). Is this bound realistic?
Let f(x) = 7x + 1 be the function such that f(x) = 6x2 + 2-1...
Let f(x) = 7x + 1 be the function such that f(x) = 6x2 + 2-1 n2". n=0 Q6.1 10 Points 1 Using the well-known geometric series r" = , |* |< 1, find the formula of Cand n=0 find the domain D of the function f. Please select file(s) Select file(s) Save Answer Q6.2 8 Points Using part Q6.1, find the value of the 102nd derivative of f(x) at x = 0; that is, find f(102)(0). Please select file(s)...
4. Construct the Lagrange interpolating polynomials for the following function, and find a bound for the...
4. Construct the Lagrange interpolating polynomials for the following function, and find a bound for the absolute error on the interval [xo, T, T2 .0,
Consider polynomial interpolation of the function f(x)=1/(1+25x^2) on the interval [-1,1] by (1) ...
Consider polynomial interpolation of the function f(x)=1/(1+25x^2) on the interval [-1,1] by (1) an interpolating polynomial determined by m equidistant interpolation points, (2) an interpolating polynomial determined by interpolation at the m zeros of the Chebyshev polynomial T_m(x), and (3) by interpolating by cubic splines instead of by a polynomial. Estimate the approximation error by evaluation max_i |f(z_i)-p(z_i)| for many points z_i on [-1,1]. For instance, you could use 10m points z_i. The cubic spline interpolant can be determined in...
For the given functions f(x), letxo-1지 = 1.25, and x2 = 1.6. Construct interpolation polynomials of degree at most one and at most two to approximate f (1.4), and find the absolute error. f (x)-sin...
For the given functions f(x), letxo-1지 = 1.25, and x2 = 1.6. Construct interpolation polynomials of degree at most one and at most two to approximate f (1.4), and find the absolute error. f (x)-sin π.x
For the given functions f(x), letxo-1지 = 1.25, and x2 = 1.6. Construct interpolation polynomials of degree at most one and at most two to approximate f (1.4), and find the absolute error. f (x)-sin π.x
EXAMPLE 2 Find sin$(7x) cos”7x) dx. SOLUTION We could convert cos?(7x) to 1 - sin?(7x), but...
EXAMPLE 2 Find sin$(7x) cos”7x) dx. SOLUTION We could convert cos?(7x) to 1 - sin?(7x), but we would be left with an expression in terms of sin(7x) with no extra cos(7x) factor. Instead, we separate a single sine factor and rewrite the remaining sin" (7x) factor in terms of cos(7x): sin'(7x) cos”(7x) = (sinº(7x))2 cos(7x) sin(7x) = (1 - Cos?(7x))2 cos?(7x) sin(7x). in (7x) cos?(7x) and ich is which? Substituting u = cos(7x), we have du = -sin (3x) X...
15. (Taylor Polynomials for sin x) (a) Find the Taylor polynomials about O for f(x) =...
15. (Taylor Polynomials for sin x) (a) Find the Taylor polynomials about O for f(x) = sin for n = 1,2,3,4,5,6,7,8. (b) Based on the pattern in part (a), if n is an odd number what is the relation between T. (x) and Tn+1(x)?
(a) Find the degree 1 interpolating P(x) through the points (a, f(a)) and (b, f(b)) (b) Develop the following formula b...
(a) Find the degree 1 interpolating P(x) through the points (a, f(a)) and (b, f(b)) (b) Develop the following formula by using the interpolating polynomial P1 (x), (c) Find the degree of precision of the approximation, T1 (f) = f(1) + f(-1), for f(r)dr.
(a) Find the degree 1 interpolating P(x) through the points (a, f(a)) and (b, f(b)) (b) Develop the following formula by using the interpolating polynomial P1 (x), (c) Find the degree of precision of the approximation,...
4) (a) Find the function P(x) = a + bcos(m) + c sin(TX), which interpolates the...
4) (a) Find the function P(x) = a + bcos(m) + c sin(TX), which interpolates the data: 0.5 Ly Hint find a, b, and c so that P(O) 2, P(0.5)-5 and P(1)-4] (b) Find a Lagrange's quadratic polynomial interpolating the above data. 1 4
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...
this is numerical analysis
2. Consider the function f(x) = -21° +1. (a) Calculate the interpolating polynomial pz() for data using the nodes 2o = -1, 11 = 0, 12 = 1. Simplify the polynomial to standard form. Use the error theorem for polynomial interpolation to bound the error f(x) - P2(x) on the interval (-1,2). Is this bound realistic?
Let f(x) = 7x + 1 be the function such that f(x) = 6x2 + 2-1 n2". n=0 Q6.1 10 Points 1 Using the well-known geometric series r" = , |* |< 1, find the formula of Cand n=0 find the domain D of the function f. Please select file(s) Select file(s) Save Answer Q6.2 8 Points Using part Q6.1, find the value of the 102nd derivative of f(x) at x = 0; that is, find f(102)(0). Please select file(s)...
4. Construct the Lagrange interpolating polynomials for the following function, and find a bound for the absolute error on the interval [xo, T, T2 .0,
For the given functions f(x), letxo-1지 = 1.25, and x2 = 1.6. Construct interpolation polynomials of degree at most one and at most two to approximate f (1.4), and find the absolute error. f (x)-sin π.x
For the given functions f(x), letxo-1지 = 1.25, and x2 = 1.6. Construct interpolation polynomials of degree at most one and at most two to approximate f (1.4), and find the absolute error. f (x)-sin π.x
EXAMPLE 2 Find sin$(7x) cos”7x) dx. SOLUTION We could convert cos?(7x) to 1 - sin?(7x), but we would be left with an expression in terms of sin(7x) with no extra cos(7x) factor. Instead, we separate a single sine factor and rewrite the remaining sin" (7x) factor in terms of cos(7x): sin'(7x) cos”(7x) = (sinº(7x))2 cos(7x) sin(7x) = (1 - Cos?(7x))2 cos?(7x) sin(7x). in (7x) cos?(7x) and ich is which? Substituting u = cos(7x), we have du = -sin (3x) X...
15. (Taylor Polynomials for sin x) (a) Find the Taylor polynomials about O for f(x) = sin for n = 1,2,3,4,5,6,7,8. (b) Based on the pattern in part (a), if n is an odd number what is the relation between T. (x) and Tn+1(x)?
(a) Find the degree 1 interpolating P(x) through the points (a, f(a)) and (b, f(b)) (b) Develop the following formula by using the interpolating polynomial P1 (x), (c) Find the degree of precision of the approximation, T1 (f) = f(1) + f(-1), for f(r)dr.
(a) Find the degree 1 interpolating P(x) through the points (a, f(a)) and (b, f(b)) (b) Develop the following formula by using the interpolating polynomial P1 (x), (c) Find the degree of precision of the approximation,...
4) (a) Find the function P(x) = a + bcos(m) + c sin(TX), which interpolates the data: 0.5 Ly Hint find a, b, and c so that P(O) 2, P(0.5)-5 and P(1)-4] (b) Find a Lagrange's quadratic polynomial interpolating the above data. 1 4
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