Let . An engineer suspects that has a real zero between and and wishes to locate that…

MATHEMATICS · Multi-part · A Level · Mathematics · Numerical methods · Numerical methods

Problem

Let

f(x)=x^{3}-7x+1

. An engineer suspects that

f

has a real zero between

-1

and

1

and wishes to locate that zero to within

0.25

unit on the

x

–axis. (a) Evaluate

f(-1)

and

f(1)

and justify that at least one real zero of

f

lies in the interval

[-1,1]

. (b) Let

m_{1}

denote the midpoint of

[-1,1]

. Compute

m_{1}

and

f(m_{1})

. Identify which of the two half–intervals,

[-1,m_{1}]

[m_{1},1]

, must contain a zero of

f

, and explain your choice. (c) Repeat the process in part (b) once more, beginning with the interval chosen in part (b). Record the new midpoint

m_{2}

, evaluate

f(m_{2})

, and state the updated half–interval that is guaranteed to contain a zero. (d) Continue halving only as long as necessary to guarantee the interval length is at most

0.25

. State the final interval obtained, its length, and an

x

–value that serves as an approximation of the zero accurate to within

0.25

unit. Provide appropriate justification.

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