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What does -(-... mean in this equation? - Answer 2

I'm trying to understand what is meant by the -(-w etc. in the equation below. Can someone enlighten me? Source: Section 2.2.3 Case Amplification

Finding a basis for all polynomials $f(t)$ of degree 3 such that $f(1)=0$ and $\int_{-1}^{1} f(t) dt$=0 - Answer 1

this is a problem from my textbook I am working on. I am not sure how to approach it -- I would like to solve it using a 'matrix' way if possible -- that is, to set up a matrix with coefficients etc. and solve for the basis of the kernel, which would be the basis of what I am looking for (not sure if that's possible). How to find a basis for all polynomials $f(t)$ of degree 3 such that $f(1)=0$ and $\int_{-1}^{1} f(t) dt$=0 ...

The maximum of a function of a single variable where all derivatives $f^{(n)}=0$ for $n\le 2$? - Answer 1

I was recently reading about classifying the extrema of a continuous function of a single variable. I came across the information that if the second derivative is zero then we can examine the higher derivatives such as $f^{\prime\prime\prime}$,$f^{(4)}$ and so on. Suppose $f^{(n)}$ is the first non-zero derivative. If n is odd, then the point is an inflection point and if n is even then a positive nth derivative means a minimum and a...

Evaluating the definite integral $\int_0^\infty \mathrm{e}^{\sum_ia_ix^{n_i}}\mathrm{d}x $ - Question

As a follow-up to a question on evaluating the definite integral $\int_0^\infty \mathrm{e}^{-x^n}\,\mathrm{d}x$, I wish to know if there is a general analytic solution to the related integral where $-x^n$ is replaced by a polynomial of arbitrary degree, namely $$ \int_0^\infty \mathrm{e}^{\sum_ia_ix^{n_i}}\mathrm{d}x $$ for $n_i\in\mathbb{Z}$ and where the individual coefficients $a_i\in\mathbb{R}$ can be positive or negative, but in a ...

Uniform Continuity Proof - Answer 1

$$f(x) = \frac{\sin x^3}{x+1}$$ If $f(x)$ is defined for $x \in [0,\infty)$, I can see that its derivative is bounded in the interval so it is just the matter of proving it. Im going for an $e-s$ proof, but I'm stumbling in terms of finding an appropriate $s.$ Also, would be nice if someone could give an explanation if possible what it means for a function to be uniformly continuous graphically - such that I can tell almost in...

Minimising an expression - involving polynomial - Answer 1

I found this one on a forum but it has been unanswered from long there. I am curious to know if there is a solution to this problem. Here it is: Let n be a positive integer. Determine the smallest possible value of $$|p(1)|^2+|p(2)|^2 + .........+ |p(n+3)|^2$$ over all monic polynomials p with degree n. I don't have much idea about the problem. I can go about by assuming $p$ to be some monic polynomia...

What is the accepted syntax for a negative number with an exponent? - Answer 7

A friend is taking a college algebra class and they are teaching him that $$-3^2 = -9$$ Their explanation is: $$-3^2 = -(3^2) = -9.$$ It has been a long time for me but I thought that in the absence of any parenthesis that: $$-3^2 = (-3) \times (-3) = 9.$$ They are even contradicting themselves because they teach the odd/even shortcut for exponents in another part of the book. i.e.:...

Prove the Radical of an Ideal is an Ideal - Answer 1

I am given that $R$ is a commutative ring, $A$ is an ideal of $R$, and $N(A)=\{x\in R\,|\,x^n\in A$ for some $n\}$. I am studying with a group for our comprehensive exam and this problem has us stuck for two reasons. FIRST - We decided to assume $n\in\mathbb{Z}^+$ even though this restriction was not given. We decided $n\ne 0$ because then $x^0=1$ and we are not guaranteed unity. We also decided $n\notin\mathbb{Z}^-$ b...

How to show that a funtion is Continuous? - Question

So I have to show that this function is continuous for $a\in\mathbb{R}$, where $a>0$ and $n\in\mathbb{N}$. For some reason this seems so obvious but I don't know how to proof it. My idea was to show that $nx^n$, $-x^n$, $a$ and $nx^{n-1}$ are all continuous as that would proof that the whole function is continuous. I would really appreciate if someone could help me with this. I also need to find a fixed point of the functio...
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