This problem asks for Taylor polynomials for f(x) = ln(1 + x) centered at a = 0. Show your work in an organized way. (a) Find the 4th, 5th, and 6th degree Taylor polynomials for f(x) = ln(1 + x) centered at a = 0.
(b) Find the nth degree Taylor polynomial for f(x) centered at a = 0, written in expanded form.
(c) Find the nth degree Taylor polynomial for f(x) centered at a = 0, written in sigma (summation) notation.
(d) Use the 7th degree Taylor polynomial to estimate ln(2).
(e) Compare your answer to the estimate for ln(2) given by your calculator. How accurate were you?
(f) Looking at the Taylor polynomials, explain why this estimate is less accurate than the estimate you found for sin(3°) in Problem 6 of Homework 10. (In that problem, you were asked to use a 7th degree Taylor polynomial centered at a = 0 for sin(x) to approximate sin(3°)).


Answer 1

Taylor's series expansion about x=0 was written by using the expansion formula.

What is Taylor's series expansion of f(x) about c?

Taylor's series expansion of f(x) about c is given by

So, Taylor's series expansion about x=0 will be given by

Hence, Taylor's series expansion about x=0 was written by using the expansion formula.

To get more about Taylor's series visit:

Answer 2

Compute up to the 6th order derivative:

You might start to see a pattern here: the -th order derivative is

For , we have

The -th degree Taylor series for centered at is

a. Plug in above:

b. This would just be

c. See above part (a).

d. The 7th degree Taylor polynomial is


e. The actual value is closer to , giving an error of about 0.0664.

f. The short answer is that this series converges much slower than the Taylor series for .

Related Questions


What is the product of -2x^3 + x - 5 and x^3 - 3x - 4? Are the products equal to those of x^3 - 3x - 4 and -2x^3 + x - 5? Explain I am soooo lost on working this out.



Step-by-step explanation:

(  -2x³ + x - 5) × ( x3 - 3x - 4)

= ( -2x³)×( x3 - 3x - 4) +(x)×( x3 - 3x - 4) - (5)( x3 - 3x - 4)

= - 2x^6  +6x^4 +8x³ + x^4 - 3x² -4x - 5x³ +15x + 20

= - 2x^6  +7x^4 +3x³-3x²+11x +20


One angle of a triangle measures 72∘ more than the smallest, while a third angle is twice the smallest. Find the measure of each angle.


Answer: 27 degrees, 99 degrees, and 54 degrees

Step-by-step explanation:

Let x = the smallest angle of the triangle

Let x + 72 = the second angle

Let 2x = the third angle

They have to add up to 180

Degrees, so:

x + (x +72) + 2x = 180; now add like terms:

4x + 72 = 180; now subtract 72 from both sides;

4x = 108; now divide both sides by 4 and you get x = 27.

So the smallest angle is 27, another angle is 27+72=99, and the last angle is 27x2=54. You can check, so they add to 180? 27 + 99 + 54 = 180


What is 3² in xpoient




Step-by-step explanation:

It saying 3 times to the power of 2


3 squared

so basically it's saying



Xander goes to the movies with his family. Each family member buys a ticket and two boxes of popcorn. If these are five members of his family,let t represent the cost of a ticket and p represent tye cost of a box of popcorn. Write two different expressions that represents the total amount his family spent.


Members in his family = 5
Boxes of popcorn for each member = 2
Boxes of popcorn for 5 members = 2 x 5 = 10

Total Spent = (5t + 10p)___(1st Expression)

Now, For 2nd Expression,

Money spent on each member = (t+2p)
Money spent on 5 members = 5×(t+2p)

2nd Expression will be 5(t+2p).

!! Hope It Helps !!

Subtract. What is the difference (9 x10^10) - (4 x 10^10) in scientific


Answer and explanation:

The answer would be 5 × . Both numbers are being multiplied by 10 to the tenth power, so all you need to do is sabtract the numbers in the front and keep the 10 to the tenth power the same.


The point (2, 3) is on the terminal side of angle θ, in standard position. What are the values of sine, cosine, and tangent of θ?


(2,3) so x=2, y=3 and h=(2^2+3^2)^(1/2)=√13

sina=3/√13, cosa=2/√13, tana=3/2

Two lines interesting at a right angle​



Perpendicular lines are lines that intersect at right angles.

Step-by-step explanation:

I'm assuming that your question is asking what the term is for two lines that intersect at a right angle. If I'm correct then your answer is perpendicular lines.

Hope this helps


One hundred times the quantity nine plus four


If you need an equation, than it would be 100 x 9+4
100 x (9+4)= 100 x 13= 1300

if you were a water molecule in a glass of water how would you best describe your relationship with the water molecules around you


I would describe how theres space. Because there is no shape of water do molecules are more spaced out. I think that's how it is.

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