Hi~ Thank you for your help!
I was trying to work on a problem about Taylor series, but i don't think im approaching the problem the right way.
I have to find the fifth order Taylor polynomial and Taylor series for the function f(x) at x = 0.
f(x) = 1/(x+2)
I tried to just write out the first and second and third derivatives, and this was fine, but then i got to the fourth and fifth derivatives and the quotient rule made the derivative look really confusing....is there a better way to approach this problem?
I might be doing this totally wrong since we just started learning Taylor series today in class, so any advice is much appreciated.
It turns out that if you had done your derivatives correctly, you would have found the rule:
for f(x)=1/(x+2)
f(n)(x) = (-1)(n)n!/(x+2)n+1
I got f(5)(x)=-120/(x+2)6.
Hope that helps.
Hi there! I'd be happy to help you with your problem involving Taylor series.
To find the Taylor series for a function f(x) centered at x = 0, we need to find its derivatives at x = 0 and use them to construct the polynomial. In your case, the function f(x) = 1/(x + 2).
Instead of directly finding the higher order derivatives using the quotient rule, let's try a different approach to simplify the process.
1. Rewrite f(x) as a power series centered at x = 0:
f(x) = 1/(x + 2) = (1/2) * 1/(1 + (x/2)).
2. Recall the formula for the geometric series: 1/(1 - t) = 1 + t + t^2 + t^3 + ..., valid when |t| < 1.
3. Notice that by substituting t = -(x/2), we get:
f(x) = (1/2) * 1/(1 - (x/2)) = (1/2) * (1 - (x/2) + (x/2)^2 - (x/2)^3 + ...).
4. Now we can expand the geometric series:
f(x) = (1/2) * (1 - (x/2) + (x/2)^2 - (x/2)^3 + ...)
= (1/2) * (1 - x/2 + x^2/4 - x^3/8 + ...).
This is the Taylor series representation of f(x) centered at x = 0. However, since we want the fifth-order Taylor polynomial, we need to truncate the series after the fifth term.
5. The fifth-order Taylor polynomial for f(x) is obtained by keeping only the terms up to x^5:
T5(x) = (1/2) * (1 - x/2 + x^2/4 - x^3/8 + x^4/16 - x^5/32).
Now you have the fifth-order Taylor polynomial for f(x) at x = 0, which approximates the original function f(x) within a certain range around x = 0.
I hope this helps! Let me know if you have any further questions or if there's anything else I can assist you with.