Maclaurin remainder interval estimate

[chief10]

Homework Statement

The question asks to estimate the remainder on the interval |x|≤ 1.
f(x) is given as sinh(x).

I solved the polynomial P₃(x) = x + (1/6)(x³)

I then went ahead and solved R₃(x) up to the point shown below.

R₃(x) = (sinh(c)*x⁴)(1/24)I then don't know how to go about getting the estimate. It clearly lies between 0 and x but it escapes me.. Any help would be great. Thanks a lot.

 

[pasmith]

Re-read whatever is in your notes or textbook on estimating the remainder, or just look it up on wikipedia. That should give you an idea of what to do next.

 

[chief10]

I did do that.. however I can't seem to procure the answer given in the text of

|R₃(x)| < (1/12) as sinh(1) < 2

I can't make sense of that.

 

[pasmith]

The question is basically asking you for an upper bound (not necessarily a least upper bound) for $|x^4\sinh(c)|/24$ if $|x| \leq 1$ and $|c| \leq |x|$.

So how you maximize $|x^4\sinh(c)|$ subject to those constraints?

 

[chief10]

sub in 1 i guess since it's inclusive?

i still don't get however how you ascertain that R3(x) is less than 1/12th

 

[chief10]

No ideas anyone?

 

[chief10]

i'm going to beg lol

 

[pasmith]

sub in 1 i guess since it's inclusive?

Correct. That gives you $|R_3(x)| \leq \sinh(1)/24$.

Now you need to show, from the definition of sinh, that $\sinh(1) < 2$. Then you can conclude that
$$|R_3(x)| \leq \sinh(1)/24 < 2/24 = 1/12.$$

 

[chief10]

why would you need to show that it's less than 2 though? i mean of all the numbers to pick, if you didn't know the answer, why would you pick 2?

 

[pasmith]

why would you need to show that it's less than 2 though? i mean of all the numbers to pick, if you didn't know the answer, why would you pick 2?

The basic answer is that simply leaving the estimate as sinh(1)/24 feels incomplete. For many reasons it is preferable to give a rational estimate, which had better be bigger than sinh(1)/24 (although not so much bigger as to be useless).

The crudest way is to note that $\sinh(1) = (e - e^{-1})/2 < e/2 < 2$, since $e < 4$.

But that's really the largest useful estimate: an immediate improvement is to note that actually $e < 3$, so $e/2 < 3/2$, which gives an estimate of $|R_3(x)| < 3/(2 \times 24) = 1/16.$

 

[chief10]

okay i'll make note of that

thanks a lot for your help pasmith, it's been much appreciated.. do you mind if i shoot a few more maclaurin's your way in the future if i have an issue? possibly via PM?