Find the least squares quadratic 𝑥
2 + 𝑏𝑥 + 𝑐 = 0, which best fits the curve 𝑦 = √𝑥 over
the interval 0 ≤ 𝑥 ≤ 1.
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We need to find the quadratic function x squared plus b x plus c that best fits the square root function over the interval from zero to one. We'll use the least squares method, which minimizes the integral of the squared difference between the two functions.
To find the optimal quadratic, we set up an optimization problem. The error function E of b and c is the integral of the squared difference between square root of x and our quadratic. To minimize this error, we take partial derivatives with respect to b and c and set them equal to zero.
Now we need to evaluate the definite integrals that appear in our partial derivative equations. We compute the integral of x to the three halves power, x cubed, x squared, x, square root of x, and the constant one, all from zero to one. These standard integrals give us the numerical values we need for our system of equations.
Using the integral values, we substitute into our partial derivative equations to get a system of linear equations. After simplification, we get twenty b plus thirty c equals nine, and three b plus six c equals two. Solving this system gives us b equals negative one fifth and c equals thirteen thirtieths. The optimal quadratic is shown in red, fitting closely to the square root function.
To summarize what we've learned: We successfully found the least squares quadratic that best fits the square root function. The optimal quadratic is x squared minus one fifth x plus thirteen thirtieths. We used calculus to minimize the error function and solved a linear system to find the coefficients. This powerful technique can be applied to fit polynomials to any continuous function.