Mathematics, 12.08.2020 05:01, marivi3cazares

Assume that f(x)=ln(1+x) is the given function and that Pn represents the nth Taylor Polynomial centered at x=0. Find the least integer n for which Pn(0.2) approximates ln(1.2) to within 0.01.

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In this exercise we have to use the knowledge of Taylor Polynomial to calculate the requested function, this way we will have;

the least integer for n is 2

The function given in this exercise corresponds to:

f(x) = ln(1+x)

knowing that the x point will be centered on:

$x=0\\Pn(0,2)\\Error < 0.01$

By rewriting the equation we have to:

$[[Max(f^{(n+1)} (c))]/(n + 1)!] *0.2^{(n+1)} < 0.01$

So doing the derivatives related to the first function given in the exercise we have to:

f(x) = ln(1+x)

First derivative: f'(x) = 1/(x + 1) < 0! = 1

2nd derivative: f"(x) = -1/(x + 1)^2 < 1! = 1

3rd derivative: f"'(x) = 2/(x + 1)^3 < 2! = 2

4th derivative: f""(x) = -6/(x + 1)^4 < 3! = 6

Following this we have to:

$Max|f^{(n+1)} (c)| < n!$

Thus, error is;

$(n!/(n + 1)!) * 0.2^{(n + 1)} < 0.01$

$(1/(n + 1))* 0.2^{(n + 1)} < 0.01$

Let's try n = 1

$(1/(1 + 1)) *0.2^{(1 + 1)} = 0.02$

This is greater than 0.01 and so it will not work. Let's try n = 2

$(1/(2 + 1)) * 0.2^{(2 + 1)} = 0.00267$

This is less than 0.01. So,the least integer for n is 2.

See more about Taylor polynomial at link

answered: yasarhan2

the least integer for n is 2

Step-by-step explanation:

We are given;

f(x) = ln(1+x)

centered at x=0

Pn(0.2)

Error < 0.01

We will use the format;

[[Max(f^(n+1) (c))]/(n + 1)!] × 0.2^(n+1) < 0.01

So;

f(x) = ln(1+x)

First derivative: f'(x) = 1/(x + 1) < 0! = 1

2nd derivative: f"(x) = -1/(x + 1)² < 1! = 1

3rd derivative: f"'(x) = 2/(x + 1)³ < 2! = 2

4th derivative: f""(x) = -6/(x + 1)⁴ < 3! = 6

This follows that;

Max|f^(n+1) (c)| < n!

Thus, error is;

(n!/(n + 1)!) × 0.2^(n + 1) < 0.01

This gives;

(1/(n + 1)) × 0.2^(n + 1) < 0.01

Let's try n = 1

(1/(1 + 1)) × 0.2^(1 + 1) = 0.02

This is greater than 0.01 and so it will not work.

Let's try n = 2

(1/(2 + 1)) × 0.2^(2 + 1) = 0.00267

This is less than 0.01.

So,the least integer for n is 2

answered: Guest

answer: 1,380

step-by-step explanation:

1: you turn %15 into a decimal by moving the decimal point two places to the left.

2: and then you multiply .15 by 1,200 , which will get you 180.00

3: from there you will add 180 to 1,200

4: which will get you 1,380

hope it *im kind of slow so idk if my answer is right*

answered: Guest

reduce the friction between its moving parts

step-by-step explanation:

in fact, by reducing the friction between the moving parts of the machine, it is possible to reduce the energy wasted due to this friction; therefore, more input energy is converted into useful work, and this will improve the efficiency of the machine.