Quantitative Finance

Numerical Methods

Root Approximation - Bisection

         

The equation ex=1x\displaystyle e^x = \frac{1}{x} has a solution somewhere between 00 and 1.1. Equivalently, we seek a root of the continuous function xex1 x e^{x}-1 in the interval (0,1). (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 f(x)=3x+sinxex=0\displaystyle f(x)=3x+\sin x-e^x = 0 lies between 00 and 0.5.0.5. If we apply the bisecton method 5 times, which of the following intervals will we end up with?

Determine the value of 53\displaystyle \sqrt[3]{5} by using the bisecton method. Let the width of the final interval less than 0.02,0.02, and start with the interval [1,2].[1,2]. Then, guess the upper boundary of the final interval as the value of 53.\displaystyle \sqrt[3]{5}.

The equation (x2)3+(x2)21=0\displaystyle (x-2)^3+(x-2)^2-1=0 has a root between 22 and 3.3. If we apply the bisecton method 6 times, which of the following intervals will we end up with?

The equation 2x+2x=3\displaystyle 2^x+2^{-x} = 3 has a root between 11 and 2.2. If we apply the bisecton method 5 times, which of the following intervals will we end up with?

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