1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Scilla [17]
3 years ago
14

This problem asks for Taylor polynomials forf(x) = ln(1 +x) centered at= 0. Show Your work in an organized way.(a) Find the 4th,

5th, and 6th degree Taylor polynomials forf(x) = ln(1 +x) centeredata= 0.(b) Find the nth degree Taylor polynomial forf(x) centered at= 0,written in expanded form.
Mathematics
1 answer:
stich3 [128]3 years ago
4 0

Answer:

a) The 4th degree , 5th degree and sixth degree polynomials

f^{lV} (x) = \frac{(2(-3))}{(1+x)^4} (1)= \frac{((-1)^3(3!))}{(1+x)^4}

f^{V} (x) = \frac{(2(-3)(-4))}{(1+x)^5} =\frac{(-1)^4 (4!)}{(1+x)^5}

f^{V1} (x) = \frac{(-120))}{(1+x)^6} (1) = \frac{(-1)^5 5!}{(1+x)^6}

b)The nth degree Taylor polynomial for f(x) centered at x = 0, in expanded form.

log(1+x) = x - \frac{x^2}{2} +\frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6}+\\..  (-1)^{n-1}\frac{x^n}{n} +..

Step-by-step explanation:

Given the polynomial function f(x) = log(1+x) …...(1) centered at x=0

      f(x) = log(1+x) ……(1)

using formula \frac{d}{dx} logx =\frac{1}{x}

Differentiating Equation(1) with respective to 'x' we get

f^{l} (x) = \frac{1}{1+x} (\frac{d}{dx}(1+x)

f^{l} (x) = \frac{1}{1+x} (1)  ….(2)

At x= 0

f^{l} (0) = \frac{1}{1+0} (1)= 1

using formula \frac{d}{dx} x^{n-1}  =nx^{n-1}

Again Differentiating Equation(2) with respective to 'x' we get

f^{l} (x) = \frac{-1}{(1+x)^2} (\frac{d}{dx}((1+x))

f^{ll} (x) = \frac{-1}{(1+x)^2} (1)    ….(3)

At x=0

f^{ll} (0) = \frac{-1}{(1+0)^2} (1)= -1

Again Differentiating Equation(3) with respective to 'x' we get

f^{lll} (x) = \frac{(-1)(-2)}{(1+x)^3} (\frac{d}{dx}((1+x))

f^{lll} (x) = \frac{(-1)(-2)}{(1+x)^3} (1)=  \frac{(-1)^2 (2)!}{(1+x)^3} ….(4)

At x=0

f^{lll} (0) = \frac{(-1)(-2)}{(1+0)^3} (1)

f^{lll} (0) = 2

Again Differentiating Equation(4) with respective to 'x' we get

f^{lV} (x) = \frac{(2(-3))}{(1+x)^4} (\frac{d}{dx}((1+x))

f^{lV} (x) = \frac{(2(-3))}{(1+x)^4} (1)= \frac{((-1)^3(3!))}{(1+x)^4} ....(5)

f^{lV} (0) = \frac{(2(-3))}{(1+0)^4}

f^{lV} (0) = -6

Again Differentiating Equation(5) with respective to 'x' we get

f^{V} (x) = \frac{(2(-3)(-4))}{(1+x)^5} (\frac{d}{dx}((1+x))

f^{V} (x) = \frac{(2(-3)(-4))}{(1+x)^5} =\frac{(-1)^4 (4!)}{(1+x)^5} .....(6)

At x=0

f^{V} (x) = 24

Again Differentiating Equation(6) with respective to 'x' we get

f^{V1} (x) = \frac{(2(-3)(-4)(-5))}{(1+x)^6} (\frac{d}{dx}((1+x))

f^{V1} (x) = \frac{(-120))}{(1+x)^6} (1) = \frac{(-1)^5 5!}{(1+x)^6}

and so on...

The nth term is

f^{n} (x) =  = \frac{(-1)^{n-1} (n-1)!}{(1+x)^n}

Step :-2

Taylors theorem expansion of f(x) is

f(x) = f(a) + \frac{x}{1!} f^{l}(x) +\frac{(x-a)^2}{2!}f^{ll}(x)+\frac{(x-a)^3}{3!}f^{lll}(x)+\frac{(x-a)^4}{4!}f^{lV}(x)+\frac{(x-a)^5}{5!}f^{V}(x)+\frac{(x-a)^6}{6!}f^{VI}(x)+...….. \frac{(x-a)^n}{n!}f^{n}(x)

At x=a =0

f(x) = f(0) + \frac{x}{1!} f^{l}(0) +\frac{(x)^2}{2!}f^{ll}(0)+\frac{(x)^3}{3!}f^{lll}(0)+\frac{(x)^4}{4!}f^{lV}(0)+\frac{(x)^5}{5!}f^{V}(0)+\frac{(x)^6}{6!}f^{VI}(0)+...….. \frac{(x-0)^n}{n!}f^{n}(0)

Substitute  all values , we get

f(x) = f(0) + \frac{x}{1!} (1) +\frac{(x)^2}{2!}(-1)+\frac{(x)^3}{3!}(2)+\frac{(x)^4}{4!}(-6)+\frac{(x)^5}{5!}(24)+\frac{(x)^6}{6!}(-120)+...….. \frac{(x-0)^n}{n!}f^{n}(0)

On simplification we get

log(1+x) = x - \frac{x^2}{2} +\frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6}+\\..  (-1)^{n-1}\frac{x^n}{n} +..

You might be interested in
A certain city has a 6% general sales tax. B) What will be the total cost of the purchase?
mr_godi [17]

Answer:

Final price = 1.06x

Step-by-step explanation:

If a city has a 6% general sales tax, then the total cost of any purchase will be the original price of an item plus 6%, as follows:

Final price = (x + 0.06x) = 1.06x

Therefore, the total cost of any purchase is 1.06 where 'x' represents the price of the items without taxes.

4 0
3 years ago
Camryn went to the store to buy 2.5 pounds of grapes. She paid with a $10 bill and a $5 bill and recieved $3.93 as change. She a
ankoles [38]

Answer:

3.98

Step-by-step:

$15.00 - 3.93 - 1.12 = 9.95 spent on the grapes

9.95/2.5 pounds = 3.98 per pound

6 0
2 years ago
Read 2 more answers
Ravi has 33 marbles his brother has twice as many how many marbles do they have all together​
ycow [4]

Answer

Together, they have 99 marbles.

Explanation

Ravi has 33 marbles.

His brother has twice (two times) as many marbles as him. Ravi's brother has 33×2=66 marbles.

Together, they have 33+66=99 marbles.

6 0
3 years ago
Can anyone teach plsssss........................ Given that PR= 17cm and SR= 6cm.Find the length, in cm of TS. ​
Lemur [1.5K]

Answer:

9cm

Step-by-step explanation:

  1. Since PQST is a rectangle, ∠QST=∠QSR=∠90º. So triangle QSR is a right triangle.
  2. By the Pythagorean Theorem and that SR=6 and QR=10, we can say that 6^2+b^2=10^2. So, 36+b^2=100. And b^2=64 and b=8. So QS=8.
  3. Now we know that QS=PT=8.
  4. By the Pythagorean Theorem and that PR=17 and PT=8, we can say a^2+8^2=17^2. So, a^2+64=289. And a^2=225 and a=15.
  5. We know that TR=15 and SR=6 and TR-SR=TS, so 15-6=9.
8 0
3 years ago
Does anybody know the answer <br>​
IgorLugansk [536]

Answer:C

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
Other questions:
  • Juli uses 12 ounces of cheese in her potato soup recipe. Her recipe yields 8 servings. If Juli needs enough for 20 servings, how
    14·1 answer
  • The function g(x) = –x2 + 16x – 44 written in vertex form is g(x) = –(x – 8)2 + 20. Which is one of the transformations applied
    8·2 answers
  • Solve this step by step solution please
    11·1 answer
  • 6=-w/8 please solve for w
    10·1 answer
  • HELP FAST!!! Which angles could form an obtuse triangle? A. 62°, 39°, 79° B. 46°, 92°, 42° C. 40°, 40°, 100° D. 22°, 78°, 80°
    6·1 answer
  • What's an interval. I just dont seem to get it. Its too tricky for me
    7·1 answer
  • What is 289.5 divided by 0.63
    10·1 answer
  • What is 940 is scientific notation? Thanks for helping out:))
    11·2 answers
  • Suppose that water is pouring into a swimming pool in the shape of a right circular cylinder at a constant rate of 7 cubic feet
    7·1 answer
  • I NEED HELP ASAP I’ll give Brainliest ,THANK YOU
    13·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!