Homework Statement
Spivak's "Calculus" Chapter 1, Problem 4v
Find all numbers x for which
x^2-2x+2>0
Homework Equations
The Attempt at a Solution
x^2-2x+2>0
x^2-2x>-2
x^2-2x+1>-2+1
(x-1)^2>-1
x\in R
Would that be an adequate proof? Anything...
Homework Statement
f is a function that satisfies
f(x+y)=f(x)+f(y) and f is continuous at 0.
prove f is continuous everywhere
Homework Equations
The Attempt at a Solution
its easy to see that f(0)=0
My hunch is that the only soln f= cx, and f=0;
but otherwise can't make...
I am looking for a calculus book that gives the reader deep understanding of how calculus works, not just rote memorization. I have heard that the books by Micahel Spivak and Tom Apostol are good. Which of the books provides the best understanding of calculus: Spivak's or Apostol's? My parents...
I have the 4th edition of Spivak's Calculus. Problem 13(b) in Chapter 7 says:
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Suppose that f satisfies the conclusion of the Intermediate Value Theorem, and that f takes on each value only once. Prove that f is continuous.
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Well, what about this function:
f(1) = 2
f(2)...
Next year I'm going to begin studying physics as freshman and this summer I was thinking about studying Calculus on my own. Would Spivak be feasible as an introduction to calculus? I'm in Pre-Calculus right now and find it pretty easy (though I know that is not necessarily any indicator of...
Please forgive any stupid mistakes I've made.
On p.85, 4-5:
If c: [0,1] \rightarrow (R^n)^n is continuous and each (c^1(t),c^2(t),...,c^n(t)) is a basis for R^n , prove that
|c^1(0),...,c^n(0)| = |c^1(1),...,c^n(1)| .
Maybe I'm missing something obvious, but doesn't c(t) =...
Prove the following:
x^n - y^n = (x-y)(x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1})
Hmm ... I factored x then -y and came out with:
x^{n}+...x^{2}y^{n-2}-x^{n-2}y^{2}...-y^{n}
Argh. What's up with the middle part? I'm not sure where to go from here.
I just got Spivak's book a couple days ago, I've flipped through it and was looking through the Index. Does Spivak cover Related Rates or Optimization? I did not find it at all, so I am curious and wondering if it's under some other name. Thanks!
Must be something missing from my repertoire-- Spivak got me in Chapter 1! :-)
Trying to find all x that satisfy:
x + 3^x < 4
I've tried everything I can think of. Here are a few lines I've run down, to no avail:
x + 3^x < 4 \Rightarrow e^{x+3^x} < e^4 \Rightarrow e^x \cdot...
I honestly enjoyed using James Stewart (yes heathen) when I did calc I, II, III and now these forums are buzzing with Spivak - as if he's the new Giancoli/Feynman of math textbooks
whats the big deal?
Oh - and i did check out the Amazon.com reviews - surprisingly the one bad review I read...
Hi.
I have a question on proof of proposition 2 in chater 7 Spivak volume2.
In the proof, he says that the n-dimensional distribution
\Delta_{p}=\bigcap^{n}_{i,j=1}ker\Lambda^{i}_{j}(p)
in R^(n+n^2) is integrable.
Could anyone explain why it is an n dimensional distribution?
Thanks.
I am reading a Vol2 of geometry book by Spivak. On page 220-221 he says that:
"Notice that the possibility of defining covariant derivatives depends only on the equation..."
The equation is some equality involving Christoffel Symbol.
If anyone has this book, could you explain why what he...
I own both volumes of Apostol, but I must confess that when I bought them I had never heard of Spivak's book. I think it's pretty much a toss up for me now. What about you guys?
Suppose
f(a) = g(a) = h(a)
and f(x) <= g(x) <= (x) for all x
Prove g(x) is differentible and that
f'(a) = g'(a) = h'(a).
So.. I need to prove that the following limit exists:
lim h -->0 (g(x+h) - g(x)) / h
but how can i use the fact that f(x) <= g(x) <= (x) for all x?
Thanks
Does anyone know if there's worked out solution to the problems in spivak's calculus on manifolds? It's awfully easy to get stuck in the problems and for some of them I don't even know where to start...
Also, if there isn't any, any good problem and 'SOLUTION' source for analysis on manifolds...
1) Find a function, f(x) which is discontinuous at 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4} ..., but continuous at any other points.
Solution (I have come across, probably wrong and a half):
f(x) = { 1 for all real x; 0 for 1/x where x is natural numbers.
Can anyone tell me the answer to...