Recent content by IntroAnalysis

Thanks so much, you save me from messing with exponents! Since I know cosh (x) = √(1 + sinh(x)^2 = √1 + y^2 !

Homework Statement Let f(x) = sinh(x) and let g be the inverse function of f. Using inverse function theorem, obtain g'(y) explicitly, a formula in y. Okay the Inverse function Theorem says (f^-1)'(y) = 1/(f'(x)) If f is continuous on [a, b} and differentiable with f'(x)\neq0 for all x\in[a...

Can't we just say that since \phi is monotone increasing, that we know that \phi(f(ci)) (resp. \phi(f(di))) is where \phi reaches its maximum (resp. minimum)? Thus, 0\leql \phi(f(ci) - \phi(f(di)) l \leq2K?

The upper Riemann sum = (i=1\rightarrown) \sumMiΔxi where Δxi=[xi-xi-1] The lower Riemann sum = i=(1\rightarrown) \summiΔxi where Δxi=[xi-xi-1] Mi=sup[f(xi): x\in[xi, xi-1] mi=inf[f(xi): x\in[xi, xi-1]

Homework Statement Let f:[a, b]\rightarrow[m, M] be a Riemann integrable function and let \phi:[m, M]\rightarrowR be a continuously differentable function such that \phi'(t) \geq0 \forallt (i.e. \phi is monotone increasing). Using only Reimann lemma, show that the composition \phi\circf...

Thank you. Your approach makes sense and is clear.

I think the limit as x -> xo, if you multiply numerator and denominator by it conjugate you get, f'(xo) = 1/2(xo+(xo + (xo)^(1/2))^(1/2))^(1/2)) Is that it?

Homework Statement Define f(x)=\sqrt{}(x + (\sqrt{}(x + \sqrt{}x) Determine where f is differentiable and compute the derivative Homework Equations f'(xo)= lim as x approaches xo (f(x) - f(xo))/(x - xo) The Attempt at a Solution By the definition, f(x) = \sqrt{}x does not have a...

Then back to my original question why is the intersection of rationals and (0,∞) = (0,∞) in other words, why don't irrationals come out of this intersection?

Homework Statement Let A = [Q\bigcap(0,\infty)] \bigcup {-1} \bigcup(-3, -2] Homework Equations So A = (0,\infty) \bigcup{-1} \bigcup(-3,-2] The Attempt at a Solution I understand that the Rational numbers are cardinally equivalent to (0,\infty), but why isn't...

It is 1/2 which is f(0). So this approach I show: 1) the point c is in the domain 2) the limit of f(c) exists and 3) lim x->c f(x)=f(c) I should have thought of this, it is a lot easier to show. Thank you.

It just says to show, it doesn't specify δ-ε proof. I've spent hours working on this one problem, any suggestions greatly appreciated!

Homework Statement Define f:[-1,∞]→ℝ as follows: f(0) = 1/2 and f(x) =[(1 + x)^(1/2) - 1]/x , if x ≠ 0 Show that f is continuous at 0. Homework Equations Definition. f is continuous at xo if xoan element of domain and lf(x) - f(xo)l < ε whenever lx - xol < δ The Attempt at...

Got the induction part: Assume ak+1 - ak = (-1/2)k(a1-a0), then ak+2 - ak+1 = (ak+1 + ak)/2 -[(-1/2)k(a1 - a0) + ak] = (ak+1 + ak - 2ak)/2 - [(-1/2)k(a1 - a0)] = (ak+1 - ak)/2 - [(-1/2)k(a1 - a0)] = (1/2)(-1/2)k(a1 - a0) - [(-1/2)k(a1 -...

No, I did not have the power wrong. This is from Introduction to Analysis, Gaughan, Prob. 23. It says you may want to use induction to show that: an+1 - an = (-1/2)^n(a1 - a0).