Math / Calculus

QuestionObtain the least square approximation for the nommal appoutimation of degree 1 and ? $f(x)=\sqrt[3]{x}$ $O_{n}$ interval $[0,1]$

Studdy Solution

STEP 1

Assumptions
1. We are given the function $f(x) = \sqrt[3]{x}$ .
2. We need to find the least squares approximation of degree 1 and 2.
3. The interval of approximation is $[0, 1]$ .
4. The least squares approximation minimizes the integral of the square of the difference between the function and the approximation over the given interval.

STEP 2

For a polynomial approximation of degree 1, we assume the form $P_1(x) = a_0 + a_1x$ .

STEP 3

The error function $E(a_0, a_1)$ for degree 1 is given by:
$E(a_0, a_1) = \int_{0}^{1} \left( \sqrt[3]{x} - (a_0 + a_1x) \right)^2 \, dx$

STEP 4

To minimize the error, we need to find the partial derivatives of $E(a_0, a_1)$ with respect to $a_0$ and $a_1$ , and set them to zero:
$\frac{\partial E}{\partial a_0} = 0$ $\frac{\partial E}{\partial a_1} = 0$

STEP 5

Calculate the partial derivative with respect to $a_0$ :
$\frac{\partial E}{\partial a_0} = -2 \int_{0}^{1} \left( \sqrt[3]{x} - (a_0 + a_1x) \right) \, dx = 0$

STEP 6

Calculate the partial derivative with respect to $a_1$ :
$\frac{\partial E}{\partial a_1} = -2 \int_{0}^{1} x \left( \sqrt[3]{x} - (a_0 + a_1x) \right) \, dx = 0$

STEP 7

Solve the system of equations obtained from the partial derivatives to find $a_0$ and $a_1$ .

STEP 8

For a polynomial approximation of degree 2, we assume the form $P_2(x) = b_0 + b_1x + b_2x^2$ .

STEP 9

The error function $E(b_0, b_1, b_2)$ for degree 2 is given by:
$E(b_0, b_1, b_2) = \int_{0}^{1} \left( \sqrt[3]{x} - (b_0 + b_1x + b_2x^2) \right)^2 \, dx$

STEP 10

To minimize the error, we need to find the partial derivatives of $E(b_0, b_1, b_2)$ with respect to $b_0$ , $b_1$ , and $b_2$ , and set them to zero:
$\frac{\partial E}{\partial b_0} = 0$ $\frac{\partial E}{\partial b_1} = 0$ $\frac{\partial E}{\partial b_2} = 0$

STEP 11

Calculate the partial derivative with respect to $b_0$ :
$\frac{\partial E}{\partial b_0} = -2 \int_{0}^{1} \left( \sqrt[3]{x} - (b_0 + b_1x + b_2x^2) \right) \, dx = 0$

STEP 12

Calculate the partial derivative with respect to $b_1$ :
$\frac{\partial E}{\partial b_1} = -2 \int_{0}^{1} x \left( \sqrt[3]{x} - (b_0 + b_1x + b_2x^2) \right) \, dx = 0$

STEP 13

Calculate the partial derivative with respect to $b_2$ :
$\frac{\partial E}{\partial b_2} = -2 \int_{0}^{1} x^2 \left( \sqrt[3]{x} - (b_0 + b_1x + b_2x^2) \right) \, dx = 0$

STEP 14

Solve the system of equations obtained from the partial derivatives to find $b_0$ , $b_1$ , and $b_2$ .

STEP 15

The solutions for $a_0$ , $a_1$ (degree 1) and $b_0$ , $b_1$ , $b_2$ (degree 2) will give us the least squares approximations.

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