Consider the error in using the approximation ex1x on the in

Consider the error in using the approximation ex1x on the interval [1,1].

(a) Reasoning informally, on what interval is this approximation an overestimate? ______
An underestimate? ___________
(For each, give your answer as an interval or list of intervals, e.g., to specify the intervals 0.25x<0.5 and 0.75<x1, enter [-0.25, 0.75), (0.75,1] Enter none if there are no such intervals.)

(b) Use the Error Bound for Taylor Polynomials to find a good smallest bound for the error in approximating ex with 1x on this interval:
error bound = ____________

Now, consider the error in using the approximation ex1x+x^2/2!x^3/3! on the same interval.

(c) Reasoning informally, on what interval is this approximation an overestimate? ___________
An underestimate? ________________
(For each, give your answer as an interval or list of intervals, e.g., to specify the intervals 0.25x<0.5 and 0.75<x1, enter [-0.25, 0.75), (0.75,1] Enter none if there are no such intervals.)

(d) Use the Error Bound for Taylor Polynomials to find a good smallest bound for the error in approximating ex with 1x+x^2/2!x^3/3! on this interval:
error bound =____________

Solution

To find the error in linear approximation. we must require a function f

if it is knows error is given by e = f - l

As here the open interval is -1.

for the given function substitute given intervals in f(x) say -0.25,0, 025,0.5,0.75 whatever the values that are

greater than 0 for all vales if f(t)>0 than the approximation is a under-estimate. If f(t) <0 for all t then the

approxiation is an over-estimate.