I want to prove the convergence of the newton method for a convex function.
Let $f: [a,b] \to \mathbb R$ be a differentiable, convex function with $f(a) \gt 0$ and $f(b) \lt 0$. Let $(x_n)_n$ be the sequence
$$x_0 = a, \quad x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$
Show that $(x_n)_...
This is a problem from a college exam which i can't figure it out.
First i add $\ln$ on both sides:
$$\ \ln y=\ln [e^{x^2}-\cot(\ln(\sqrt x+\frac 1x))]^{\sec(\frac1x)} $$
Then when i simplify it i get this: $\ \ln y=\sec(\frac1x)\ln [e^{x^2}-\cot(\ln(\sqrt x+\frac 1x))] $
I think this isn't si...
0
Suppose that φ : M → N is a smooth map between smooth manifolds; then the differential of φ at a point x is, in some sense, the best linear approximation of φ near x. It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, it is a linear map from the tangent space of M at x to the tangent space of N at φ(x). Hence it can be used to push tangent vectors on M forward to tangent vectors on N.
The differential of a map φ is also called, by various authors, the derivative or total derivative of φ, and is sometimes itself called the pushforward.
== Motivation... ==
In calculus, the differential represents the principal part of the change in a function y = f(x) with respect to changes in the independent variable. The differential dy is defined by
d
y
=
f
′
(
x
)
d
x
,
{\displaystyle dy=f'(x)\,dx,}
where
f
′
(
x
)
{\displaystyle f'(x)}
is the derivative of f with respect to x, and dx is an additional real variable...
