Use the bissection method to approximate the root of f(x)=2x²-1 in the interval . Let ε=0.1 be the margin of error of approximation.
At first, we need to find the interval, where the roots are situated. "f(x)" is a quadratic function. "f(0)=-1" and "f(1)=f(-1)=1". Since the function is continuous, it intersects "x" -axis twice on the interval "[-1,1]". Consider the interval "[0,1]". The function intersects this interval once. I.e., the equation "f(x)=0" has one root. We denote this root by "x_1". Then, we set "x_2=-x_1". It will be the second root. It follows from the fact that "f(x)=f(-x)."
We start the bisection method:
Thus, the approximate value for one of roots is: "x_1\\approx0.7188". The approximate value for the second root is: "x_2\\approx-0.7188". We point out that exact values of roots are: "\\frac{1}{\\sqrt{2}}\\approx0.7071" and "-\\frac{1}{\\sqrt{2}}\\approx-0.7071".
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