Questions tagged [mean-value-theorem]
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7
questions
-6
votes
0answers
47 views
What are the possible values for $f'(c)$ when $f'(c)=\frac{\sqrt{16}-\sqrt{16}}{6}$? [closed]
I am going in detail,
Question Statement -> Given $f(x)=\sqrt{25-x^2}$, find all values of $x$ in $\lbrack-3,3 \rbrack$ that satisfy Mean Value Theorem.
Solution ->
For mean value theorem,
$f'(c)...
2
votes
0answers
181 views
The collection of mean value abscissas in the Mean value theorem
The integral mean value theorem for continuous f on [0,b] and finite positive continuous measure $\mu$ we have
$$\frac{1}{\mu[a,b]}\int_{a}^{b}f(x)d\mu(x)=f(c)(*)$$
for at least one $c\in [a,b]$. We ...
1
vote
1answer
84 views
Is it possible to find an upper bound and a lower bound for $(f(\xi)-f(\tilde{\xi}))^T(\frac{\partial f}{\partial \xi}(\bar{\xi}))(\xi-\tilde{\xi})$?
For a system engineering problem I have to solve the problem below, but since I am not a mathematician, I am not sure if I have enough knowlegde to solve it.
Problem definition: Let $f(\xi) \in \...
4
votes
1answer
1k views
How to understand the integral?
In order to understander the nonlinear elliptic equation with natural boundary condition,
$$\sigma_2(D^2u)=0 \text{ in } \Omega$$
I wish to understand the following integral,
$$E(u,\Sigma)=\int_\Sigma ...
1
vote
0answers
123 views
A discrete-to-continuous approach to the Dirichlet principle?
Dirichlet principle: Let $\Omega \subset R^n$ be a compact set with $C^1$ boundary. Then, there exists a unique solution $f$ satisfying $\Delta f = 0$ in $\Omega$ and $f=g$ on $\partial \Omega$.
We ...
2
votes
1answer
193 views
Mean value theorem in terms of Wirtinger calculus?
The mean value theorem for vector-valued function in the real domain $f: \mathcal{R}^n \rightarrow \mathcal{R}^d$ can be expressed as
\begin{equation}
f(x)-f(y)=\int_{0}^{1}\nabla f(x(\tau))d \tau \...
15
votes
1answer
433 views
Bull's-eye Riemann sum
Let $f:[a,b] \to \mathbb{R}^2$ be a continuous curve on the plane.
Question: Are there numbers $a \leq x \leq c \leq y \le b$ such that $$(c-a)f(x)+(b-c)f(y) = \int_a^b f(t) \, dt \ ?$$
In other ...
