- #1
eipiplusone
- 10
- 0
Hi,
I would really appreciate it if someone can help me with the following problem, regarding a taylor polynomial:
A 2nd degree taylor polynomial to the function f around x = 1, is given by:
T_2(x) = x + x^2
Question:
What is f´(1) ?
Answer: 3
(Btw: the question is from a multiple choice test, but the answer should be evident without consulting the possible answers)
I usually solve this kind of problem simply by considering the general form of a taylorpolynomial:
f(a) + f´(a)(x-a) + (f´´(a)/2!) * (x-a)^2
and comparing it to the taylor polynomial given in the problem statement. From a little rearrangement, the answer is usually self-evident. But in this case the rearranging seems very elaborate, so I am hoping that I am missing some clever way to solve it.
In this case I really don't know how go about it. I´ve only managed to simply make sense of the answer, by the following argument:
f(1) + f´(1)(x-1) + f´´(1)(x-1)]^2 = x + x^2 ⇔
f(1) + f´(1)x - f´(1) + f´´(1)(x^2 + 1 - 2x) = x + x^2 ⇔
f(1) + f´(1)x - f´(1) + f´´(1)x^2 + f´´(1) - f´´(1)2x = x + x^2
Inserting f´(1) = 3 evaluates to x + x^2 , if f´´(1) = 2 and f(1) = 2 .
But there´s no way that I would have seen that, not knowing that f´(1) = 3 .
There must be some nice way of solving this problem?
Any help would be truly appreciated..!
I would really appreciate it if someone can help me with the following problem, regarding a taylor polynomial:
A 2nd degree taylor polynomial to the function f around x = 1, is given by:
T_2(x) = x + x^2
Question:
What is f´(1) ?
Answer: 3
(Btw: the question is from a multiple choice test, but the answer should be evident without consulting the possible answers)
Homework Equations
I usually solve this kind of problem simply by considering the general form of a taylorpolynomial:
f(a) + f´(a)(x-a) + (f´´(a)/2!) * (x-a)^2
and comparing it to the taylor polynomial given in the problem statement. From a little rearrangement, the answer is usually self-evident. But in this case the rearranging seems very elaborate, so I am hoping that I am missing some clever way to solve it.
The Attempt at a Solution
In this case I really don't know how go about it. I´ve only managed to simply make sense of the answer, by the following argument:
f(1) + f´(1)(x-1) + f´´(1)(x-1)]^2 = x + x^2 ⇔
f(1) + f´(1)x - f´(1) + f´´(1)(x^2 + 1 - 2x) = x + x^2 ⇔
f(1) + f´(1)x - f´(1) + f´´(1)x^2 + f´´(1) - f´´(1)2x = x + x^2
Inserting f´(1) = 3 evaluates to x + x^2 , if f´´(1) = 2 and f(1) = 2 .
But there´s no way that I would have seen that, not knowing that f´(1) = 3 .
There must be some nice way of solving this problem?
Any help would be truly appreciated..!