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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 → ∞?

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.5: Rational Functions
Problem 51E
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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 → ∞?
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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