0
votes
3answers
58 views

Derivation of the “Combined Work Formula”

Before I get to my question, some background: Person $A$ can paint a fence at the rate $9 \frac{hour}{fence}$ (or equivalently $\frac{1}{9} \frac{fence}{hour}$) Person $B$ can paint a fence at the ...
0
votes
1answer
45 views

Finding distance using rates of change — best approach?

The question: A man drives from state $A$ to state $B$ going $60 \frac{miles}{hour}$. Then he returns from state $B$ to state $A$, driving $45 \frac{miles}{hour}$. His total driving time is $2.5 ...
1
vote
2answers
33 views

Overall difference in percent

I want to calculate the total difference in % between two investments {A,B} in the following scenario: In year t=0 revenue A is 70 % smaller than revenue B. Every year the revenue from A further ...
0
votes
2answers
26 views

Arithmetic simplification

I am asked to find $\frac{d^2y}{dx^2}$, and prove that $\frac{d^2x^2+y^2=a^2}{dx^2}$=$-\frac{a^2}{y^3}$, This is how I have proceeded: $2 y \frac{dy}{dx}+2 x=2 a \frac{da}{dx}$ => ...
14
votes
3answers
240 views

How to prove: $\left(\frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt[4]{25}-\sqrt[4]{125}}}-1\right)^{4}=5$?

Question: show that: the beautiful ${\tt sqrt}$-identity: $$ \left({2 \over \sqrt{\vphantom{\Large A}\, 4\ -\ 3\,\sqrt[4]{\,5\,}\ +\ 2\,\sqrt[4]{\,25\,}\ - \,\sqrt[4]{\,125\,}\,}\,}\ -\ ...
1
vote
1answer
17 views

Commpossed percentage over same base

Maybe this sounds a Little trivial. I have a value and Over this value apply percentage (a), then over this result apply other percentage (b) Could say : Value = 145760 a= 8% b= 40% My actual ...
1
vote
1answer
65 views

How to calculate the number of lattice points in the interior and on the boundary of these figures with vertices as lattice points?

We define a point $(x,y)$ in the plane to be a lattice point if both $x$ and $y$ are integers. Now let $$S\colon= \{ (x,y) \ | \ 0 \leq x \leq m, \ 0 \leq y \leq \frac{nx}{m} \}, $$ where $m$ and ...
1
vote
3answers
41 views

How to establish these two facts about polynomials?

Let $f(x) := \sum_{k=0}^n c_k x^k $ be a polynomial of degree $n\geq 0$ with real coefficeints such that $f(x) = 0$ for $n+1$ distinct real values of $x$. Then how to prove that each $c_k = 0$ and ...
3
votes
2answers
155 views

I want to learn math from ground-up, basic to advanced, beginner to expert

I want to learn math. I've learned math long time ago, but i hardly remember anything. I really want to relearn but have no idea where to begin. I want to learn math by reading through good books, ...
5
votes
1answer
60 views

Halmos on Definability and Luzin on Division by 0

For a successful introduction of a new symbol (e.g. '$\emptyset$') into a mathematical discourse it is necessary and sufficient that the symbol refer to something (e.g. Existence + Specification in ...
3
votes
1answer
67 views

The function $f(t)=2+\sin(t)+\sin(t\sqrt2)$

The function $f$ defined on $\mathbb{R}$ by $$f(t)=2+\sin(t)+\sin(t\sqrt2)$$ can never reach $0$. Can we find some sequence $(t_n)_{n\geq0}$ such that $$\lim_{n \to \infty}f(t_n)=0 \ \ \ ?$$ Or in ...
1
vote
1answer
25 views

Comparing powers with different bases without logarithms

I want to compare : $17^{31}$ and $31^{17}$ , this is a solution but I want another one and without using logarithms, only using the fact that $17=16+1=(2^4)+1$ and $31=(2^5)-1$ how could it ...
2
votes
2answers
136 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
54 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
784 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
51 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
43 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
205 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
117 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
6answers
286 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
109 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 ...
22
votes
6answers
801 views

If $(\sqrt{y^2-x}-x)(\sqrt{x^2+y}-y)=y$ then $x+y=0$.

Let $x,y$ be real numbers such that $$\left(\sqrt{y^{2} - x\,\,}\, - x\right)\left(\sqrt{x^{2} + y\,\,}\, - y\right)=y$$ Show that $x+y=0$. My try: Let ...
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
2answers
389 views

How prove this $x+y=0$ if $\left(\sqrt{y^2-x^3}-x\right)\left(\sqrt{x^2+y^3}-y\right)=y^3$

Question: let $x,y$ are real numbers,and such $$\left(\sqrt{y^2-x^3}-x\right)\left(\sqrt{x^2+y^3}-y\right)=y^3$$ show that $$x+y=0\tag{1}$$ before I have solve following problem: if ...
6
votes
1answer
199 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
776 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
82 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
166 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
588 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
444 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
4k 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
50 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
21 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
629 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
125 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
105 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
111 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
125 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 ...
14
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 ...