Proving Inequalities and Solving 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.

Homework Statement

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.

Homework Equations

b We use the cubic taylor polynomial approximation to x^2 = sin(x), whose root is r = sqrt(15) - 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 $e^x-1\geq 0$ if and only if $x\geq 0$?

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 $\int_0^1 \frac{1 + x^{30}}{1 +x^{60}} dx = 1 + \frac{c}{31}$, where 0 < c < 1.

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

 

[micromass]

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.

OK, so for (b) you need to solve $|sin(r)-r^2|$. 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]

I am also having trouble with problem c.

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

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

For this, I would first take the Taylor expansion of

$$\frac{1+y}{1+y^2}$$

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

 

[zonk]

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

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

Also $f'(y) = \frac{y'(1 + y^2) - 2yy'(1 + y)}{(1 + y^2)^2}$

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

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

http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/"

followed by this course

http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/"

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]

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.

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. :)