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...
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.
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...
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).