The question is:
Let f be a function that has derivatives of all orders for all real numbers. Assume f(0)=5, f'(0)= -3, f"(0)=1, f"'(0) = 4.
a) Write a third-degree Taylor polynomial for f about x=0 and use it to approximate f(0.2)
b) Write the fourth-degree Taylor polynomial for g, where g(x) = f(x^2), about x=0.
c) Write the third-degree Taylor polynomial for h, where h(x) = integral(from 0 to x) f(t) dt, about x=0.
d) Let h be defined as in part c). Given that f(1) = 3, either find the exact value of h(1) or explain why it cannot be determined.
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I got through a - c, but am stuck on d.
Here are my solutions:
a) T_3 (x) = 5 - 3x + (x^2)/2! + (4x^3)/3!
f(0.2) = 4.42533
b) M_3 (x) = 5 - 3x^2 + x^4/2!
c) N_3 (x) = c + 5x - 3x^2/2! + x^3/3!
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First a general question on notation: for part a), b), and c), I renamed the polynomials M or N or some other random letter with a subscript showing the value of highest exponent on x, only because I figured they were polynomials, and thus approximations of the actual function g(x) or f(x^2). Am I correct in doing so?
d) I was stuck on this. I had a few thoughts...
One was that it had something to do with the interval of convergence for the original function, f(x). We know that if a function is differentiated or integrated, the new function's interval of convergence will be the exact same, except for the endpoints, which may or may not converge. So, in the problem, if f(x) had an interval of convergence of which 1 was an endpoint, then h(x) might not converge there...However, I have no idea how to find the interval of convergence of f(x). I don't think I have enough information for that (I'm not given enough derivatives...and I can't seem to find a pattern between the terms to construct my own general term).
So, then I started to think that you can't find the exact value because h(x) is the integral of f(x) and not all derivatives of f(x) are given, f(x) being a function that has "derivatives of all orders for all real numbers." You should be able to find an approximation of h(1), though, by plugging into the polynomial you found in part c), but otherwise, nothing else...and that's assuming convergence at 1.
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Thanks for reading this far...
Let f be a function that has derivatives of all orders for all real numbers. Assume f(0)=5, f'(0)= -3, f"(0)=1, f"'(0) = 4.
a) Write a third-degree Taylor polynomial for f about x=0 and use it to approximate f(0.2)
b) Write the fourth-degree Taylor polynomial for g, where g(x) = f(x^2), about x=0.
c) Write the third-degree Taylor polynomial for h, where h(x) = integral(from 0 to x) f(t) dt, about x=0.
d) Let h be defined as in part c). Given that f(1) = 3, either find the exact value of h(1) or explain why it cannot be determined.
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I got through a - c, but am stuck on d.
Here are my solutions:
a) T_3 (x) = 5 - 3x + (x^2)/2! + (4x^3)/3!
f(0.2) = 4.42533
b) M_3 (x) = 5 - 3x^2 + x^4/2!
c) N_3 (x) = c + 5x - 3x^2/2! + x^3/3!
--
First a general question on notation: for part a), b), and c), I renamed the polynomials M or N or some other random letter with a subscript showing the value of highest exponent on x, only because I figured they were polynomials, and thus approximations of the actual function g(x) or f(x^2). Am I correct in doing so?
d) I was stuck on this. I had a few thoughts...
One was that it had something to do with the interval of convergence for the original function, f(x). We know that if a function is differentiated or integrated, the new function's interval of convergence will be the exact same, except for the endpoints, which may or may not converge. So, in the problem, if f(x) had an interval of convergence of which 1 was an endpoint, then h(x) might not converge there...However, I have no idea how to find the interval of convergence of f(x). I don't think I have enough information for that (I'm not given enough derivatives...and I can't seem to find a pattern between the terms to construct my own general term).
So, then I started to think that you can't find the exact value because h(x) is the integral of f(x) and not all derivatives of f(x) are given, f(x) being a function that has "derivatives of all orders for all real numbers." You should be able to find an approximation of h(1), though, by plugging into the polynomial you found in part c), but otherwise, nothing else...and that's assuming convergence at 1.
--
Thanks for reading this far...