2
votes
2answers
85 views

number of terms

The following problem maybe tedious if done by hand and requires patience. After factorizing the following variables find the number of terms and the sum of the number of terms. ...
1
vote
1answer
41 views

I need to do math from ground up, so what is a good workbook?

Can you guys recommend me a workbook that begins with arithmetic and ends with calculus. Or from pre-algebra to calculus. Like all "Master Math Series" books but in one complete book. It would really ...
0
votes
1answer
45 views

Nice proof for $\lim_{h\to 0}\frac{f(x+nh)-f(x)}{h}=nf'(x)$ besides LHR

Why is $$\lim_{h\to 0}\frac{f(x+nh)-f(x)}{h}=nf'(x)?$$ A cheap answer would be L'Hospital's rule, but I think there should be a more direct way to prove it, appealing to the first principles of the ...
2
votes
4answers
447 views

Limit of $\sqrt{x^2-6x+7}-x$ as x approaches negative infinity

What is $\lim\limits_{x\to-\infty}(\sqrt{x^2-6x+7}-x)$ ? Don't understand how to approach this question
1
vote
3answers
50 views

I need some basic introduction to limits

So, I know you can obviously cut out a value if it is multiplying and dividing something at the same time, right? Like: $$\frac{4h-2xh-h^2}{h} = \frac{h(4-2x-h)}{h} = 4-2x-h$$ But then I saw this ...
3
votes
2answers
41 views

Question about arc length.

I am having some trouble finishing an arc length problem. Specifically, what is $\int_{0}^{1}|x'(t)| dt=?$ Is it just $\int_{0}^{1} |x(t)| dt=|x(1)-x(0)|$? If so why?
7
votes
2answers
133 views

Come up with some fun “equation Limericks”

We were discussing "Limericks" in my Calculus class. Specifically, "equation Limericks". A Limerick is a poem with five lines. The first, second, and fifth lines should have nine syllables each and ...
0
votes
2answers
66 views

Help with limit of radical expression

$$\lim_{x \to \infty} (\sqrt{x^2-49}-\sqrt{x^2-16} ) $$ I multiplied by the conjugate radical expression: $$=(\sqrt{x^2-49}-\sqrt{x^2-16}) \times (\sqrt{x^2-49}+\sqrt{x^2-16}) $$ $$= ...
0
votes
7answers
256 views

Why Not Define $0/0$ To Be $0$?

For every number $x$, $x\times 0=0$, hence $\dfrac{0}{0}$ can be any number! So $\dfrac{0}{0}$ "is knows as indeterminate" [1]. But what if we define it to be $0$? I already have an answer, but ...
2
votes
3answers
89 views

Is multiplication by zero in an equation allowed?

If we have equal quantities, we cannot divide with zero. But, we can multiply both sides with zero. But, my friend said, even multiplication with zero also wrong it seems. Unfortunately, he is not ...
1
vote
0answers
30 views

Is $x_1^{\alpha_1} + \dotsb + x_n^{\alpha_n}\geq x_1^{h/n}\dotsb x_n^{h/n}$ an example of power means?

I learned here that there is a relation between weighted means of the form $x_1^{\lambda_1}\dotsb x_n^{\lambda_n}$ and $(\lambda_1 x_1^r + \dotsb + \lambda_nx_n^r)^{1/r}$, namely that the former is ...
5
votes
1answer
144 views

Why is the $0$th power mean defined to be the geometric mean?

Mentioned in the wikipedia article, the $0$th power mean is defined to be the geometric mean. Why is this? I understand that a convenient consequence is that the means are ordered by their exponent. ...
6
votes
3answers
573 views

Explanation for $\lim_{x\to\infty}\sqrt{x^2-4x}-x=-2$ and not $0$

I am trying to intuitively understand why the solution to the following problem is $-2$. $$\lim_{x\to\infty}\sqrt{x^2-4x}-x$$ ...
1
vote
2answers
80 views

How do I improve my approach to solving integrals to get this and similar ones in the future correct?

$$\int \sqrt{ 8 (\cos t \sin t)^2 } dt = \sqrt{2} \int 2\sin t\cos t dt = \sqrt{2} (\sin t)^2 + C$$ Which seems correct to me, but if I take the definite integral from $0$ to $\pi$, then: $$\sqrt{2} ...
3
votes
1answer
165 views

What is the maximum of the self root $f(x) = x^{1/x}$

This is a knowledge sharing question as I have answered it below. I am demonstrating how one would differentiate an expression such as $x^{1/x}$ and proving the following statement. What is the ...
1
vote
3answers
445 views

The limit of $\lim\limits_{x \to \infty}\sqrt{x^2+3x-4}-x$

I tried all I know and I always get to $\infty$, Wolfram Alpha says $\frac{3}{2}$. How should I simplify it? $$\lim\limits_{x \to \infty}\sqrt{(x^2+3x+4)}-x$$ I tried multiplying by its conjugate, ...
6
votes
3answers
338 views

Does half-life mean something can never completely decay?

Caffeine has a half-life of approximately six hours. I understand this to mean that every six hours, the amount of caffeine in the body is half of what it was six hours prior. Does that mean that ...
0
votes
1answer
3k views

How does the difference quotient with a square root in the numerator end up with square roots in the denominator?

I don't understand when I apply the difference quotient to: $f(x) = \sqrt{x} $ , to get: $$\frac{\sqrt{x+h} - \sqrt{x}}{h}$$ To simplify it.. How does it end up like this?: $$\frac{x + h - x}{h ...
0
votes
2answers
49 views

How to find bounds for $x$ and $y$ for this triple integral?

I want to find the volume of the region enclosed by $z=x^2+y^2$ and $z=x+y$. How can I find the bounds for $x$ and $y$?
0
votes
0answers
18 views

Get a representative vector from a large set, and compare it with samples.

We're a small team of programmers and we're triying to solve a little problem, but we think we need some advices from professional mathematicians. We want to know if a picture of a card is an ...
2
votes
1answer
502 views

How to work faster on the GRE.

I'm fixin' to go to grad school next year, so lately I've been studying for the mathematics subject GRE. What I have heard, from MSE and other places, is that the most important thing is to learn to ...
3
votes
2answers
118 views

Different Representations of Numbers in Subsets of $\mathbb R$

I think I've mentioned sometime before about logarithmic number system. In this system, a real number $r$ is represented by $(\text{sgn}(r), \log|r|) \in \{-1, 0, 1\} \times \mathbb R$. If $\mathbb R$ ...
2
votes
1answer
102 views

Why does this equation converge to 1?

The following simple equation takes in an N-length (real) vector, and spits out a (real) number between 0 and 1. (I believe this means that it is a transformation mapping $\mathfrak{R}^N \rightarrow ...
1
vote
2answers
103 views

bound on recursive series

Let $a_1 = 0$ and $a_i \le \alpha + \beta a_{i-1}$. I am looking for an upper bound on $a_i$ that depends only on $\alpha$ and $\beta$ and $i$. If it helps, $\alpha, \beta \ge 0$.
3
votes
3answers
121 views

Implicit differentiation misunderstanding

I'm trying to see why my textbook's solution is correct and mine isn't. "Find an expression in terms of $x$ and $y$ for $\displaystyle \frac{dy}{dx}$, given that $x^2+6x-8y+5y^2=13$ First, the ...
13
votes
4answers
3k views

Aunt and Uncle's fuel oil tank dip stick problem

This problem first came to me in high school, and a couple times since, and I even assigned it for extra credit in one of my calculus classes after I became a teacher. So I know the solution. What I ...