18. f(x) = ln(1 + 2x), a = 1, n = 3, 0.5 ≤ x ≤ 1.5

Name: 18. f(x) = ln(1 + 2x), a = 1, n = 3, 0.5 ≤ x ≤ 1.5
Uploaded: 2020-01-03T22:14:57-08:00
Duration: 6 min 17 s
Channel: Doruk Isik
Description: 18. f(x) = ln(1 + 2x), a = 1, n = 3, 0.5 ≤ x ≤ 1.5

Transcript

00:01 In this problem, we're as to approximate initial log of 1 plus 2x using tailors polynomials of order 3 around 1a is equal to 1.

00:11 We know that tailors of polynomials of order 3 will be of the given form right here.

00:17 So we need to find, evaluate the function at the given point, first derivative, second derivative and third derivative.

00:22 So let's first start by finding the derivative of the function.

00:25 So f prime of x, f double prime of x, and f triple prime of x.

00:30 Now knowing that f of x is natural lock of 1 plus 2x, we find the first derivative as 2 over 1 plus 2x.

00:37 Second derivative is then equal to negative 4 over 1 plus 2x squared and the third derivative is then 16 over 1 plus 2x cubed.

00:50 All right now we have the functionality derivatives.

00:53 All we need to do is to plug those into this given equation and we're going to use we're going to take a to be equal to 1.

01:01 From here we find 3 of x as natural log of 3 plus 2 over 3 times x minus 1 minus 4 over 9 times x minus 1 squared over 2 factorial.

01:17 We have plus 16 over 27 x minus 1 cubed divided by 3 factorial.

01:27 Now from this we find t3 of x as an actual log of 3 plus 2 over 3 times x minus 1 minus 2 over 9 times x minus 1 squared plus 8 over 81 times x minus 1 cubed that is part a in part b whereas to find the accuracy using taylor's inequality since we're interested in accuracy we're interested in n plus 1 term so since ns3, that will be the fourth term.

01:59 First of all, why are we interested in a plus one term? because don't forget taylor's approximation, taylorless polynomial is a series expression of a function.

02:10 And since we truncated this series at the third term, the next term will determine the accuracy of this approximation.

02:17 So that's why we're interested in the next term that is in plus one term.

02:20 In this case, since ns3, we're interested in fourth term.

02:23 So actually we want to know what's happening to the fourth derivative of this function, f4 of x.

02:29 Knowing third derivative, we find f4 of x as negative 96 over 2x plus 1 to the fourth derivative.

02:38 And better to know if this function is increasing or decreasing, we need to look at derivative of that.

02:44 So we want to know what's happening to the fifth derivative of this function, x...

Question

18. $f(x) = \ln(1 + 2x)$, $a = 1$, $n = 3$, $0.5 \le x \le 1.5$

Question

18. $f(x) = \ln(1 + 2x)$, $a = 1$, $n = 3$, $0.5 \le x \le 1.5$

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