Answer:
-61/16
Step-by-step explanation:
The version of successive approximation you are using here is one that starts with an interval containing the solution, then cuts that interval in half with each iteration. The end result of each round of iteration is a new approximation to the solution: the midpoint of the smaller interval.
Setup
The graph shows the curves cross at a point between x = -4 and x = -3, so the initial interval is [-4, -3]. We have defined the function h(x) = g(x) -f(x) so that h(-4) < 0 and h(-3) > 0. The sign of h(x) will tell which interval limit gets replaced:
h(x) > 0 ⇒ x replaces the upper limit
h(x) < 0 ⇒ x replaces the lower limit
The initial estimate of the solution is (-4 +(-3))/2 = -7/2. (iteration 0)
Iterations
h(-7/2) > 0, so the interval becomes (-4, -7/2), and the approximate solution after iteration 1 is x ≈ -15/4.
h(-15/4) > 0, so the interval becomes (-4, -15/4), and the approximate solution after iteration 2 is x ≈ -31/8.
h(-31/8) < 0, so the interval becomes (-31/8, -15/4) and the approximate solution after iteration 3 is x ≈ -61/16
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Additional comment
Note that we don't really care about the function value when we do iteration this way. We only care about the sign of the function value. Other iteration methods use the function values to try to approximate the root. The secant method uses a linear approximation between the function values at the end points of the interval to choose a new value for x.
