Numerical methods：練習問題

Use the Newton-Raphson method with an initial approximation \(x_0 = 1\) to find the second approximation \(x_1\) for a root of the equation \(x^3 + x - 3 = 0\). Give your answer to 3 decimal places.

Use Euler's step-by-step method with a step size of \(h = 0.1\) to find an approximate value for \(y(1.2)\) for the differential equation \(\frac{dy}{dx} = x + y^2\), given the initial condition \(y(1) = 2\). Give your answer to two decimal places.

Consider the equation \( f(x) = x^3 - 2x - 5 = 0 \).
Show that a root of this equation lies in the interval \( [2, 3] \) by calculating the values of \( f(2) \) and \( f(3) \) and observing the change of sign.

Find the root of the equation \(x^3 - 5x + 1 = 0\) in the interval \([2, 3]\) using the method of linear interpolation. Perform two iterations to provide an estimate for the root, giving your answer to two decimal places.

Use Euler's method with a step size of \(h = 0.1\) to estimate the value of \(y(0.2)\) for the differential equation \(\frac{dy}{dx} = x + y^2\), given the initial condition \(y(0) = 1\). Give your answer to three decimal places.

A curve has the equation \( \frac{dy}{dx} = x^2 + 2y \). It is given that the curve passes through the point \( (1, 0.5) \).

(a) Use Euler's step-by-step method with a step length of \( h = 0.1 \) to find an approximate value for \( y \) when \( x = 1.3 \). Give your intermediate steps to 4 decimal places.

(b) The equation \( x^3 - 5x + 1 = 0 \) has a root near \( x = 2 \). Use the Newton-Raphson method once to obtain a second approximation, giving your answer to 3 decimal places.

(c) Explain, with the aid of a sketch, why the Newton-Raphson method would fail to find a root if the initial approximation was chosen at a stationary point of the function.