Problem 4. Consider the inner product space C[-1, 1] with the inner product defined by (f,g) = f(x)g(x) dx and the induced norm. Find the best least squares approximation to the function f(x) = x¹/³ on [-1, 1] by a linear function ((x) = ₁ + ₂x. [Hint: first show that the functions h₁(x) = 1 and h₂(x) = x are orthogonal.]

Q: Find the best-fit quadratic function f(t) = co+ c¡t + czť² to the data points (0, 7), (1, 6), (2,…

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Q: Solve for the orthogonal traysctories x*(3k-x) - ky². = x ? 2

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Q: 11: Given the function f(X) = 8x2 - 4x₁x₂ + 5x and Xo = [5 2]. Then G-¹g at Xo is equal (( G is the…

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Q: In the context of ordinary least square (OLS): Given mịn- ||Y – Xß||? Use derivatives to derive the…

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Q: Suppose L(x) = -5x + 1 is the local linearization to h(x) at x=2. What is h(2) ? not enough info to…

A: Given L(x)=-5x+1. L(x) is local linearization to h(x). We have to find h(2)?

Q: 3. Evaluate f 2xdx + (y + 2)dy along the rectangle with vertices (0, 0), (0,3), (3,3) and (0, 3).

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Q: (ii) Show that any Lagrange multiplier can either take on a value one or zero i.e. L,(2,) = {: # i +…

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Q: 2. Find the equation f(x) = ax + b of the least square line for the points (1,0), (-1,2), (2, 1).

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Q: ) Given y = f(x) and xo = 1, x₁ = 3, x2,= 7.Compute Lagrange multiplier Lo for the data.

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Q: Consider an optimization problem where we want to minimize: f(x1, 12, L3) = 100(x2 – 2)² + (1 – x1)²…

A: Step:-1 Given that fx1, x2, x3 =100x2-x122 +  1-x12 +100x3-x222 +  1-x22 Part (1):- differentiate f…

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Q: (a) What is the 2 × 2 matrix P that projects v = x y onto the x axis, i.e. takes x y to x 0 ? Is P…

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Q: 17. Let y₁ = x₁ + x₂ and y₂ = x² + x² + 2(x₁ + x₂ + x₁x₂). (a) Use the Jacobian to test whether…

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Q: What function should be used to minimize the distance between y = x + 5 and the point (3, 1)? O D(æ)…

A: Explanation of the solution is given below...

Q: Although we used gradient descent to find the minimizer of f in part (a), we can also use calculus…

A: Given: fx=12Ax-b22+γ2x22   To show: The minimizer of the given function is, x^=ATA+γI-1ATb

Q: = The real part for the function f (z) = 4z2 – 2z + 9 إجابتك = The imaginary part for the function f…

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Q: function

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Q: Find the linear least squares polynomial approximation to f(x) on the indicated interval if a. f(x)…

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Q: Prove that (X+Y) (X+Z) =X+YZ

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Q: 3.5. Consider the problem n minimize f (x) = x; log(x; /c;) χεX i=1 subject to Ax = b, where the…

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Q: 4 S X the condition that the error for Trapezoidal Rule will be at most 10-6. 9. Given: dx.…

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Q: Consider the following interpolating points for the function f(x) = x² - 2x: Xi 0 1 23 f(xi) 0 -1 0…

A: xi0123f(xi)0-103We are to solve the first 3 subparts of this problem:We use Matlab code for the…

Q: Suppose a, = (1 -1 1 1) 5. a, = (1 0 1 0) %3D %3D az = (1 1 1 1)'a¸ = (3 2 3 0)" - Please find out…

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Q: 1. Is L = {1 − 2x + 3x^2, x − 2x^2, 2 − 5x + 8x^2} linearly independent? If yes, show why. If not,…

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Q: B- Find the car least squares polynomial approximation (of degree 1) to f(x) = 3x² - 2, on [1,2]

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Q: 5. Find two linearly independent polynomials orthogonal to p(x) = 1 + x² using the inner product…

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