Okay...
There exists a ##c_j\in [x_{j-1},x_j]## such that $$f'(c_j)=\frac{f(x_j)-f(x_{j-1})}{\Delta x_j}$$ Now, $$L(P,f')=\sum_{j=1}^n m_j\Delta x_j$$ And $$m_j=inf\{ f'(c_j):c_j\in [x_{j-1},x_j]\}$$ $$\Leftrightarrow m_j=\frac{inf\{ f(x_j)-f(x_{j-1})\} }{\Delta x_j}$$ $$\Leftrightarrow...
Okay so I think I understand this a lot better now. I drew a picture to understand all the parts of the problem and how they relate to one another and have attempted to write the proof, however I can't be sure if my explanations are adequate. Any feedback is much appreciated:
##f## is...
Following your suggestion, I have made a further attempt at solving the problem:
##f## is differentiable and continuous on ##[a,b]## so ##f## must also be differentiable and continuous on every ##\Delta x=[x_{j-1},x_j]##. Thus, there exists a ##c \in [x_{j-1},x_j]## such that...
It just occurred to me that I can't necessarily assume I can factor out ##(b-a)## unless I know that all ##\Delta x## are of equal width... which I do not. The only thing I can think to do in that case is: $$L(P,f') = \sum_{j=1}^n m_j \Delta x_j \leq f'(b) - f'(a)$$ but I am still stumped as to...
Homework Statement
Let ##f:[a,b] \rightarrow R## be a differentiable function. Show that if ##P = \{ x_0 , x_1 , ... , x_n \}## is a partition of ##[a,b]## then $$L(P,f')=\sum_{j=1}^n m_j \Delta x_j \leq f(b) - f(a)$$ where ##m_j=inf \{ f'(t) : t \in [x_{j-1} , x_j ] \}## and ##\Delta x_j =...
Is "sin(x + y) = 1" a function of x on R?
Homework Statement
Determine if the following relation is a function of x on \mathbb R:
sin(x + y)=1
The Attempt at a Solution
Rearrange to make y the subject:
y = sin^{-1}(1) - x
Then, I simply calculated some points and plotted a...
Gotcha. The fields can't cancel out unless they point in opposite directions, which happens on the left of q1 (and the right of q2, but then other conditions are not met), but not between q1 and q2 :-)
Thank you!
I have substituted the formula E = k(e) * q / r^2
So I get:
E(net) = E(1) + E(2)
Where
E(1) = k(e) * q(1) / r(1)^2
E(2) = k(e) * q(2) / r(2)^2
k(e) = 8.99 * 10^9
r(1) = x
r(2) = 1.00 - x
q(1) & q(2) as given in the OP
Homework Statement
Two point charges are placed 1.00m apart.
q(1) = -2.50 x 10^(-6) C
q(2) = +6.00 x 10^(-6) C
Task is to find where along the line, other than at infinity, the electric field will be equal to zero.
Homework Equations
E = (k * q) / r^2
The Attempt at a Solution...
Physics/Maths/CompSci student --> career in environmental science?
Hi there
I am 29 years old and after completing a tertiary foundation certificate to gain University entrance, I am about to embark on a 4.5 year long conjoint degree as follows:
BSc component: majoring in Physics
BA...
I believe I do know what ionization means.
I want to think (intuitively) that because ionization of an atom brings it to a more stable state, energy has been released. If I am misunderstanding the question, please let me know. This stuff is reasonably new to me, even if it seems obvious to you.
Homework Statement
State whether ionization is an endothermic or an exothermic process.
The Attempt at a Solution
I know what exothermic and endothermic mean, and I know that the answer is that ionization of sodium is an endothermic process, but I don't know why and I'm hoping someone can...
This may or may not be helpful:
In Adobe Acrobat Pro it is possible to create links in your document to either open a web page, or navigate instantly to another place in the document.
How I would set it up is have one front page with formulae or whatever, and set each formula up so if you...
But mostly it will be outside of your comfort zone. You CAN do this! And you will be so much better off.
The bond of family is hard to ignore, but even if people are related to you they still need to have resspect for you and earn your respect in kind. It sounds like your mother is being...
You could also consider setting up a blog to document/show future projects. Then, when applying for a position you can direct the reader to the blog in your cover letter.
This approach could also be useful in terms of networking.