Question

Consider the following equations. f(x)= x^3 +3x^2 -2x+1 g(x)= x^2-5x+4 Approximate the solution to the equation f(x) = g(x) using three iterations of successive approximation.

A. x =11/16

B. x=13/16

C. x=3/8

D. x= 7/8

Answer 1

A. x = 11/16

Explanation:

For the purpose here, it is convenient to rearrange the equation to f(x) -g(x) = 0. We know the root will be in the interval [0, 1] because (f-g)(0) = -3 and (f-g)(1) = +3. At each iteration, we evaluate (f-g)(x) at the midpoint of the interval to see which of the interval end points can be moved and still bracket the root.

Using the bisection method starting with the interval [0, 1] we find f(1/2)-g(1/2) < 0, so we can move the interval limits to [1/2, 1].

For the next iteration, we find f(3/4) -g(3/4) > 0, so we can move the interval limits to [1/2, 3/4].

For the third iteration, we find f(5/8) -g(5/8) < 0, so we can move the interval limits to [5/8, 3/4].

Then the root is approximately the middle of that interval:

x ≈ (5/8 +3/4)/2 = 11/16

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This value of x is 0.6875. The root is closer to 0.639802004233. The bisection method takes about 3 iterations for each decimal place of accuracy. Other methods can nearly double the number of accurate decimal places on each iteration.