Problems in Apostol Volume 1

zonk

Hello, I am currently using Apostol for self-study. It seems to only have answers for computational problems. Some of the other problems are hard! And I see no student solution manual to guide me through these types of problems. I have no idea where to begin.

1. The problem statement, all variables and given/known data

a) (Found in Exercises 6.17, 41, part a) Let f(x) = e^x - 1 - x. Prove that f'(x) >= 0 if x >= 0 and f'(x) <= 0 if x <= 0. Use this fact to deduce the inequalities e^x > 1 + x and e^(-x) > 1 - x.

b) (Found in Exercises 7.8, 4, part b) Show that |sin(r) - r^2| < 1/(200) given that sqrt(15) - 3 < 0.9. Is the difference (sin(r) - r^2) positive or negative? Give full details of your reasoning.

2. Relevant equations

b We use the cubic taylor polynomial approximation to x^2 = sin(x), whose root is r = sqrt(15) - 3.

3. The attempt at a solution

a) f'(x) = e^x - 1. Letting x >= 0, we get e^x - 1 >= 0 by exponentiation. I have no clue where to go from here. We could do e^x >= 1, for the first, for example, but I have no clue where to go from here.

b) The book doesn't seem to cover this at all. So how do I do it?

micromass

Hi zonk!

For the first one, were you able to prove that [itex]e^x-1\geq 0[/itex] if and only if [itex]x\geq 0[/itex]?

In that case, could you tell me where the minimum of f is located?

For the second question, could you also write down question (a), because I have no idea what r is...

zonk

Yes, I was able to show it. The minima of f(x) is at x = 0. Because it's increasing in the interval I was able to get the answer. Thank you.

Part a is basically "Obtain the number r = sqrt(15) -3 as an approximation to the non-zero root of the equation x^2 = sin(x) by using the cubic Taylor polynomial approximation to sin(x)."

So what I did was expand sin(x) about 0 and got x^2 = x - (x^3)/3! and solved and got that root.

zonk

I am also having trouble with problem c.

c) Prove that [itex]\int_0^1 \frac{1 + x^{30}}{1 +x^{60}} dx = 1 + \frac{c}{31} [/itex], where 0 < c < 1.

I basically expand [itex]\frac{1 }{1 + x^{60}}[/itex] about 0 and multiply that expansion by [itex]1 + x^{30}[/itex] to get [itex]\frac{1 + x^{30}}{1 +x^{60}} = (1 + x^{30}) - (x^{60} + x^{90}) +\cdots + (-1)^n(x^{60n} + x^{60n + 30})[/itex]. Using Lagrange's form for the error, we need to find the 1st derivative of [itex]\frac{1 + x^{30}}{1 + x^{60}}[/itex], which is a long expression that couldn't possible be f'(c). Where did I go wrong?

micromass

OK, so for (b) you need to solve [itex]|sin(r)-r^2|[/itex]. So the question is how do you calculate sin(r). Well, you calculate sin(r) by taking the Taylor approximation. But this time you should add the remainder term!

micromass

For this, I would first take the Taylor expansion of

[tex]\frac{1+y}{1+y^2}[/tex]

in two terms (where the second is the remainder). And then substitute [itex]y=x^{30}[/itex].

zonk

I understood problem b but problem c is still hard to understand.

So we have [itex]\frac{1 + y}{1 + y^2} = 1 + E[/itex] where [itex]E = f'(c) x[/itex].

Also [itex]f'(y) = \frac{y'(1 + y^2) - 2yy'(1 + y)}{(1 + y^2)^2}[/itex]

So [itex]f'(x) = \frac{30x^{29}(1 + x^{60}) - 60x^{59}(1 + x^{30})}{(1 + x^{60})^2}[/itex]

As you can see f'(c) turns out to be something ghastly.

I think I'm missing something essential, so until my supplementary book arrives, I feel clueless at approximations.

Curtis1000

I am teaching myself calculus using Larson's book. It is pretty easy to understand. However, many people in the forums talk about books by Spivak and Apostol.

What benefits would I get from learning from Spivak or Apostol, that I would lack using Larson?

Skins

Apostol is very definitive and thorough in the theoretical sense. This can make it harder to understand but, whence one does understand more thorough and rewarding. I can't comment on Larson as I have never used his text.

If you are learni g Calculus on your own it is a good idea to use the material on http://ocw.mit.edu in addition to your book.

Particularly their single and multivariable calculus courses which include the complete sets of video lectures and video recitation sessions.

I particularly reccomend

This course

Single Variable Calculus

followed by this course

Multivariable Calculus

to anyone teaching themselves the calculus.

Curtis1000

Thanks for the help information and the links!

After reading the first chapters in Spivak and Apostol and completing the first 10 exercises in each book, I see how different the books are from Larson

This being said, I prefer the style of Apostol over Spivak. On the ocw.mit.edu website there is a course: Calculus with Theory, that uses Apostol volume 1. It also has notes by Professor Munkres. (Of Topology fame.)

I discovered an amazing book, "Elementary Analysis" by Ross. To get experience with proofs, terms and notation, I plan to start with Ross and then continue to Apostol, Volumes 1 and 2.

Skins

I always preferred Apostol's style myself. Yeah, I'm familiar with the "Calc w Theory" course. Too bad there are no video lectures. I would probably recommend that as a "second time around" course, after one has taken single and multivar calc.

Ross's book sounds interesting. I'll check it out. Thanks.

Oh yeah.. I have Dr. Munkres book "Topology". i need to take a refresher in Topo myself. :)