Question

For each initial approximation, determine graphically what happens if Newton's method is used for the function whose graph is shown:
x1=0,x1=4,x1=1,x1=5,x1=3

Answer 1

 The next guess for x, which will be x_2, will be the x-intercept of the line whose slope is the slope of the function at x_1. 
a) The slope is positive on a portion of the graph that has a horizontal asymptote. Newton's method will give guesses that approach negative infinity. 

b) The slope is zero. Newton's method will give a guess of infinity or "overflow" or "not a number" or "error" depending on how division by zero is treated. The tangent line is parallel to the x-axis, so there is no x-intercept. 

c) It looks like the slope is negative, but will give a guess in the same region as for answer (a). (For guesses near 2, it is likely Newton's method would converge on the solution near x=2. The function is well-behaved in a small area there, so x_1 must be "close enough" for there to be convergence.) 

d) The slope appears to be zero at x=4, so the answer is the same as for (b). 

e) The iteration method will give guesses that are likely to converge to the solution near x=6.