Let f(x) = sin(2x) and let [a, b] = [0, 1]. = (a) Construct a piecewise-linear polynomial that interpolates f at {x0, x1, x2} : {0, 1/2, 1}. Let's call this object P1,2. (b) Construct a piecewise linear polynomial that interpolates ƒ at {x0, x1, x2, X3, X4} : {0, 1/4, 1/2, 3/4, 1}. Let's call this object P₁,4. (c) For x = [0, 1], draw a graph (by hand, or, if you'd like, with MATLAB) of: (i) f(x), (ii) the answer to part (a), (iii) the answer to part (b), and (iv) a piecewise-linear polynomial that interpolates f at x = {0, 1/8, 1/4, 3/8, 1/2,5/8, 3/4, 7/8, 1} (no need to derive a formula). Let's call this last object P1,8. For this example, can we intuitively conclude that the pointwise error |f(x) = P₁,n| → 0 as n → ∞?
Let f(x) = sin(2x) and let [a, b] = [0, 1]. = (a) Construct a piecewise-linear polynomial that interpolates f at {x0, x1, x2} : {0, 1/2, 1}. Let's call this object P1,2. (b) Construct a piecewise linear polynomial that interpolates ƒ at {x0, x1, x2, X3, X4} : {0, 1/4, 1/2, 3/4, 1}. Let's call this object P₁,4. (c) For x = [0, 1], draw a graph (by hand, or, if you'd like, with MATLAB) of: (i) f(x), (ii) the answer to part (a), (iii) the answer to part (b), and (iv) a piecewise-linear polynomial that interpolates f at x = {0, 1/8, 1/4, 3/8, 1/2,5/8, 3/4, 7/8, 1} (no need to derive a formula). Let's call this last object P1,8. For this example, can we intuitively conclude that the pointwise error |f(x) = P₁,n| → 0 as n → ∞?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Piecewise linear polynomials
![[1] [Approximation by piecewise linear polynomials]
Let f(x) = sin(2x) and let [a, b] = [0, 1].
(a) Construct a piecewise-linear polynomial that interpolates f at {xo, x1, x₂} =
{0, 1/2, 1}. Let's call this object P₁,2.
=
(b) Construct a piecewise linear polynomial that interpolates f at {xo, X1, X2, X3, X4} =
{0, 1/4, 1/2, 3/4, 1}. Let's call this object P₁,4.
(c) For x = [0, 1], draw a graph (by hand, or, if you'd like, with MATLAB)
of: (i) f(x), (ii) the answer to part (a), (iii) the answer to part (b), and (iv) a
piecewise-linear polynomial that interpolates f at x = {0, 1/8, 1/4, 3/8, 1/2,5/8, 3/4,7/8, 1}
(no need to derive a formula). Let's call this last object P1,8.
For this example, can we intuitively conclude that the pointwise error
|ƒ (x) − P1,n| → 0 as n → ∞?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2F70b5b480-bf49-4c50-9d2a-c88893cb519c%2Fj4adnxv_processed.png&w=3840&q=75)
Transcribed Image Text:[1] [Approximation by piecewise linear polynomials]
Let f(x) = sin(2x) and let [a, b] = [0, 1].
(a) Construct a piecewise-linear polynomial that interpolates f at {xo, x1, x₂} =
{0, 1/2, 1}. Let's call this object P₁,2.
=
(b) Construct a piecewise linear polynomial that interpolates f at {xo, X1, X2, X3, X4} =
{0, 1/4, 1/2, 3/4, 1}. Let's call this object P₁,4.
(c) For x = [0, 1], draw a graph (by hand, or, if you'd like, with MATLAB)
of: (i) f(x), (ii) the answer to part (a), (iii) the answer to part (b), and (iv) a
piecewise-linear polynomial that interpolates f at x = {0, 1/8, 1/4, 3/8, 1/2,5/8, 3/4,7/8, 1}
(no need to derive a formula). Let's call this last object P1,8.
For this example, can we intuitively conclude that the pointwise error
|ƒ (x) − P1,n| → 0 as n → ∞?
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