Question

Hello! I would like some guidance here. Can somebody give me steps? (General steps, not literally giving the answer)

If f(1) = 5 and f'(x) = 6/(x^2+2), what is the best approximation for f(1.03)?

Answer 1

By "best approximation", and the other information they provided,
I'm assuming they want a good "linear approximation".

So ummm

Answer 2

If you need an explanation as to where this formula comes from,
we can do that,
but for now, here is the formula we generally use for linear approximation.\[\large\rm f(x)\approx f(a)+f'(a)(x-a)\]

Answer 3

We can approximate a "weird" value of x by using a convenient point that is close by.

Answer 4

So for this problem, we would like to approximate x=1.03,
with the nearby value a=1,\[\large\rm f(1.03)\approx f(1)+f'(1)(1.03-1)\]

Answer 5

And then just uhhh, ya,
plug in the values.

Whatchu think broski?
Good stuff? Got it?

Answer 6

@Jerobrien : OpenStudy says you are "just looking around." If you post a question and then decide to explore other problem solving sessions, at least please check back now and then to see if anyone has tried to help you. I'm moving on.

Answer 7

Ah wow, thanks!

Answer 8

And I'm sorry to offend you, @mathmale, I got distracted.

Answer 9

Did you learn trig substitution?
Through a lot of pain you can get an exact answer, but for that you will as well need to find a taylor series for arctangent, as well as taylor series for √x at some a (not sure which).

Answer 10

I have not. I'm struggling to understand my online calc class and I find solace and better explanations here.

Answer 11

Solomon, forgive me, but I see no relevance to this problem of the arctangent.

Answer 12

Arctan would be the outcome of the integral

Answer 13

I think of (√x/2) and so coeff...

Answer 14

Ah no, we haven't reached integrals yet! I'll come back to you when I have a better understanding.

Answer 15

yes, I was thinking so.... ): (:

Answer 16

Then do as zepdrix wrote... linearization

Answer 17

If f(1) = 5 and f'(x) = 6/(x^2+2), what is the best approximation for f(1.03)?

The function here is not given, but at least we know that it (the function) has the value 5 when x has the value 1, and that the derivative of the function is 6 over (x^2+2).

What are we supposed to do here? Note how 1.03 is "close" to 1. We use a linear approximation to estimate the value of f(1.03).

I'm stopping here for a moment, to ask you, jerobrien, whether any of this sounds familiar to you and / or whether you have any questions.

Answer 18

But, you can draw any taylor polynomila there. (not just degree 1 := linearization, but a quadratic, a cubic or any nth, and the larger the better)... but I would still assume they mean linearization (tho).

Answer 19

Solomon, you're right, as usual, but why complicate this problem any more than necessary by talking about advanced possibilities when we haven't finished discussing the basics?

Answer 20

that's why I said linearization is assumed still.

Answer 21

Jer: still waiting to hear from you in regard to my questions.