Recent content by IntroAnalysis

  1. I

    Final exam - help- Inverse function Theorem

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

    Final exam - help- Inverse function Theorem

    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...
  3. I

    Show [itex]\phi[/itex][itex]\circ[/itex]f is Riemann integrable

    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?
  4. I

    Show [itex]\phi[/itex][itex]\circ[/itex]f is Riemann integrable

    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]
  5. I

    Show [itex]\phi[/itex][itex]\circ[/itex]f is Riemann integrable

    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...
  6. I

    Find derivative of Square root (x + square root(x + x^(1/2))) Help

    Thank you. Your approach makes sense and is clear.
  7. I

    Find derivative of Square root (x + square root(x + x^(1/2))) Help

    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?
  8. I

    Find derivative of Square root (x + square root(x + x^(1/2))) Help

    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...
  9. I

    Intersection of Rationals and (0 to Infinity)?

    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?
  10. I

    Intersection of Rationals and (0 to Infinity)?

    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...
  11. I

    Is Function f Continuous at x=0?

    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.
  12. I

    Is Function f Continuous at x=0?

    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!
  13. I

    Is Function f Continuous at x=0?

    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...
  14. I

    Proving the Recursive Formula for an Using Induction

    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 -...
  15. I

    Proving the Recursive Formula for an Using Induction

    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).
Back
Top