Numerical Methods
The equation \(\displaystyle e^x = \frac{1}{x}\) has a solution somewhere between \(0\) and \(1.\) Equivalently, we seek a root of the continuous function \( x e^{x}-1 \) in the interval \( (0,1).\) If we apply the bisecton method 4 times, which of the following intervals will we end up with?
One of the roots of the equation \(\displaystyle f(x)=3x+\sin x-e^x = 0\) lies between \(0\) and \(0.5.\) If we apply the bisecton method 5 times, which of the following intervals will we end up with?
Determine the value of \(\displaystyle \sqrt[3]{5}\) by using the bisecton method. Let the width of the final interval less than \(0.02,\) and start with the interval \([1,2].\) Then, guess the upper boundary of the final interval as the value of \(\displaystyle \sqrt[3]{5}.\)
The equation \(\displaystyle (x-2)^3+(x-2)^2-1=0\) has a root between \(2\) and \(3.\) If we apply the bisecton method 6 times, which of the following intervals will we end up with?
The equation \(\displaystyle 2^x+2^{-x} = 3\) has a root between \(1\) and \(2.\) If we apply the bisecton method 5 times, which of the following intervals will we end up with?