Consider the following function. f(x) = ln(1 + 2x), a = 4, n = 3, 3.7 ≤ x ≤ 4.3
(a) approximate f by a taylor polynomial with degree n at the number a. t3(x) = correct: your answer is correct.
(b) use taylor's inequality to estimate the accuracy of the approximation f(x) ≈ tn(x) when x lies in the given interval. (round your answer to six decimal places.) |r3(x)| ≤ 0.3 incorrect: your answer is incorrect.
(c) check your result in part
(b) by graphing |rn(x)|.

a)

b)

Step-by-step explanation:

1) Part a

The Taylor's approximation would be given byt this formula

  (1)

Since we need the approximation T3 we just need to use this formula:

  (2)

So now we proceed to find the terms for the T3 approximation:

So then using equation (2) we have

2) Part b

We can estimate the accuracy of the approximation with the Taylor's inequality given by:

 (3)

On this case M is such that for

So we have:

Based on this we can take and

so then

So using equation (3) we got that

3) Part c

In order to plot we just need to plot:

And the graph obtained is attached below.