Numerical solution of equations: Practice Questions

The continuous function \(f(x) = x^3 - 2x - 5\) is known to have a root in the interval \([n, n+1]\), where \(n\) is an integer. Find the value of \(n\).

The equation \(x^3 + x - 5 = 0\) has a root near x = 2. Using the Newton-Raphson process once with an initial value \(x_0 = 2\), find the value of the next approximation \(x_1\).

The continuous function \(f(x) = e^x + x - 4\) has a root \(\alpha\) in the interval \([1, 1.1]\). Use linear interpolation on this interval to find an estimate for \(\alpha\), giving your answer to 3 decimal places.

The continuous function f is defined by f(x) = x³ - x - 1. It is known that a root \(\alpha\) of the equation f(x) = 0 lies in the interval \([1, 2]\). Using the interval bisection method once, which of the following intervals can be shown to contain \(\alpha\)?

The continuous function \(f(x) = x^3 + 2x - 5\) has a root \(\alpha\). Show that \(1 < \alpha < 2\) by evaluating the function at the boundaries of the interval.

Using the Newton-Raphson process once with an initial value \(x_0 = 1\), find a first approximation \(x_1\) to the root of the equation \(x^3 + x - 1 = 0\).

The function \(f(x) = x^3 - 3x^2 + 5\) has a root \( {\alpha}\) in the interval \([-2, -1]\).
(a) Use linear interpolation once on the interval \([-2, -1]\) to find an approximation for \( {\alpha}\).
(b) Taking \(x_0 = -1.1\) as a first approximation, use the Newton-Raphson process once to find a second approximation for \( {\alpha}\), giving your answer to 3 decimal places.