(a) Given a function f that satisfies f(-1) = 3, f(0) = -4, f(1) = 5 and f(2)=-6. Find the unique Lagrange interpolating polynomial of degree 3 that agrees with all given function values. Show your steps. (b) Given a closed and bounded interval [c, d], and suppose this interval consists of (n + 1) regularly spaced nodes (including the two endpoints

Question

(a) Given a function f that satisfies f(-1) = 3, f(0) = -4, f(1) = 5 and f(2)=-6. Find the unique Lagrange interpolating polynomial of degree 3 that agrees with all given function values. Show your steps. (b) Given a closed and bounded interval [c, d], and suppose this interval consists of (n + 1) regularly spaced nodes (including the two endpoints, c and d). Denote the nodes as {X1, X2, X3,..., Xn+1}. n+1 Prove that for all x € [c, d], [(x − x₁)(x − x₂) ... (x − Xn+1)| ≤n! ⋅ (d = c) (c) Let f(x) = sinx, and 4 nodes are given (including the two endpoints), at x = 0,7, and An interpolating polynomial of degree 3, P3 (x) is used to approximate f(x) on [0,1]. Show that when P3(x) is used to "approximate" f(x) at x = 2, the maximum possible error is less than 0.003. Show your steps. 12

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(a) Given a function

f

that satisfies

f (- 1) = 3, f (0) = - 4, f (1) = 5

and

f (2) = - 6

. Find the unique Lagrange interpolating polynomial of degree 3 that agrees with all given function values. Show your steps. (b) Given a closed and bounded interval

[c, d]

, and suppose this interval consists of

(n + 1)

regularly spaced nodes (including the two endpoints,

c

and

d

). Denote the nodes as

{x_{1}, x_{2}, x_{3}, \dots, x_{n + 1}}

. Prove that for all

x \in [c, d], ∣ (x - x_{1}) (x - x_{2}) \dots (x - x_{n + 1}) ∣ \leq n! \cdot (\frac{d - c}{n})^{n + 1}

(c) Let

f (x) = sin x

, and 4 nodes are given (including the two endpoints), at

x = 0, \frac{π}{6}, \frac{π}{3}

and

\frac{π}{2}

. An interpolating polynomial of degree

3, P_{3} (x)

is used to approximate

f (x)

on

[0, \frac{π}{2}]

. Show that when

P_{3} (x)

is used to "approximate"

f (x)

at

x = \frac{π}{12}

, the maximum possible error is less than 0.003 . Show your steps.

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Expert Answer to - (a) Given a function f that satisfies f(-1) = 3, f(0) = -4, f(1) = 5 and f(2)=-6. Find the unique La

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Solution for - (a) Given a function f that satisfies f(-1) = 3, f(0) = -4, f(1) = 5 and f(2)=-6. Find the unique La

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This an additional answer to - (a) Given a function f that satisfies f(-1) = 3, f(0) = -4, f(1) = 5 and f(2)=-6. Find the unique La

(Solved): (a) Given a function f that satisfies f(-1) = 3, f(0) = -4, f(1) = 5 and f(2)=-6. Find the unique La ...

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