Taylor's-theorem

Question

Determine the values of x for which the function can be replaced by the Taylor polynomial if the error cannot exceed 0.001.

Answer 1

Step 1:

The function is , .

Taylors theorem:

If a function is differentiable through order in an interval containing , then for each in ,there exist between and such that , ,

where error .

Here and .

Rewrite the function in polynomial form .

Apply fourth derivative with respect to on each side.

Determine by substituting corresponding values in .

Step 2:

The error cannot exceed 0.001 implies that .

Take fourth root of each side.

If , then

So,   and .

Therefore,

For , the value of is lies between

Solution:

The value of is lies between .