Tagged Questions

Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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0
votes
3answers
49 views

Simple real life problem

Let say my girlfriend makes $2000$ Euros per month and I make $3400$ Euros, let say all our living costs sum up to $1540$ Euros. How can i calculate how much each of us must pay (off the $1540$ Euros) ...
2
votes
3answers
23 views

Substitution with trigs (not integrals)

I am currently trying to find out the values of $x$ and $\theta$, given the two following statements. I have tried using substitution, but I am not quite getting it. I would be extremely glad if I ...
1
vote
1answer
35 views

Annuity/Finance

I'm trying to determine the question below: Mr. Learnwell wants to setup a scholarship of $4500 paid at the end of every six months. If the interest rate is 6.4% compounded semi-anually, how much ...
0
votes
2answers
19 views

Partial Fractions (3 Factors)

This document outlines a shortcut for partial fractions involving 2 factors in the denominator (P(x) + a) and (P(x) + b). At the end of the document it gives a challenge to find a similar shortcut ...
1
vote
0answers
23 views

Is shunting-yard algorithm needed if there is no parenthesis?

I am trying to find a permutation (with replacement) of operators(such as addition and multiplication) that makes numbers 1 2 3 , ..., 9 result in to some numbers. My guess is to find all possible ...
0
votes
2answers
38 views

current value of a forward contract

Suppose we hold a forward contract on a stock with expiration 6 months from now. We entered into this contract 6 months ago so that when we entered into the contract, the expiration was T=1 year. The ...
0
votes
2answers
48 views

Difficulty understanding addition of exponents

If you rewrite $(3^{12})(3^{-12})$ in the form $3^n$ what does it equal? What is the intuition behind it? Do exponents cancel each other out so it is just $3$? or do the negatives cancel out $((-x) ...
0
votes
2answers
33 views

Locus given by a pair of scissors sliding along the ground.

I came up with this problem when dragging a pair of scissors along the ground. The question is, more mathematically: Suppose there is a point (a,0) and a point (0,b) with a fixed distance m between ...
1
vote
3answers
32 views

Use a function to represent positive real numbers?

Is it correct to define the positive real numbers as $\{f(x) = x^2\mid x \in \mathbb R\}$?
0
votes
0answers
36 views

When Is This Equation an Integer?

I'm working on a math problem, but I've gotten stuck. When does $$-\frac{4(2^Q-3^Q)}{2^{Q+S}-2*3^Q}$$ produce an integer?
1
vote
1answer
14 views

Determine values of the constants a & b so the limit exists and is equal to f(2) in a piece-wise function

I am trying to determine the values of the constants a & b in a piece-wise function that has to satisfy these parameters: The limit f(x) as x approaches 2 does exist and is equal to f(2) The ...
-4
votes
1answer
36 views

please help me with this [closed]

the cooling system of a car has a capacity of 15 liters if the system is currently filled with a mixture that is 10% antifreez how much of this mixture should be drained and replaced with pure ...
0
votes
1answer
15 views

Maximal value of $\vert r^2-n\vert$ with a special condition

let $M,n\in \mathbb{N}$ and $R=\lbrace r\in \mathbb{N} \mid \vert r- \sqrt{n}\vert <M<2\sqrt{n}\rbrace $. I have to show that the maximal value of $\vert r^2-n\vert $ for $r\in R$ is at most ...
1
vote
2answers
32 views

How is it sometimes helpful to use cross multiplication in order to complete proportions with a variable?

How can it be helpful to do cross multiplication with proportions with variables such as ${2\over 4}={3\over x}$? In this one, the value of x has to be found. It can be found this way: 1. Do the ...
1
vote
4answers
51 views

systems of linear equations intuition

I want to know why in a system of linear equations I'm allowed to sum or subtract the equations. I can't get the intuition of why I can do that to solve for the equations.
1
vote
1answer
21 views

Simplification of $|a+b|^2$ for $a,b \in \mathbb{C}$

How do I simplify $|a+b|^2$, where $a,b \in \mathbb{C}$ and $|a|=|b|=1$? I know that the result is $4-|a-b|^2$, but I would like to be be explained how to do the simplification in the most elegant ...
7
votes
7answers
515 views

Is the number 0.2343434343434.. rational? [duplicate]

Consider the following number: $$x=0.23434343434\dots$$ My question is whether this number is rational or irrational, and how can I make sure that a specific number is rational if it was written in ...
0
votes
4answers
26 views

Help in proving an algebraic identity involving powers of binomials.

For some reason I found this equation: $(1 + x)^n - 1 = x \sum\limits_{k=0}^{n-1} (1+x)^k$ I think that this is an identity. If for instance one expands the powers and the sum for n = 4, the ...
2
votes
1answer
31 views

How fast does the water level decrease in a cylindrical tank?

Is this solution correct? What I know is that the volume of the tank is $V = \pi r^2 h$, where r and h are in meter. Water is drained by a rate of $2,7\frac{m^3}{min}$. How fast does the water level ...
0
votes
1answer
14 views

Contradiction - Equivalence of polynomials

I think I'm having a brain fart. Please tell me if my reasoning is correct. Suppose you have a polynomial-function $f(x)$ of degree $N$ that has coefficients $a_{0 \leq j \leq N}$ and roots $r_{0 ...
0
votes
1answer
18 views

Deal with non standard form of conic

I want to know how can I calculate latus rectum, tangent at vertex, vertex and axes of a parabola whose equation is not standard. For example, the parabola: $$ 4x^2 - 4xy + y^2 - 10 y - 19 = 0 $$
3
votes
3answers
280 views

Show that inequality holds

How would you show that the following inequality holds? Could you please write your reasoning by solving this problem too? $a^2 + b^2 + c^2 \ge ab + bc + ca$ for all positive integers a, b, c I ...
0
votes
2answers
41 views

Asymptotic approximation of the arctangent?

That is, I am looking for an algebraic function $f(x)$ that approximates $\arctan x$ for large values of $x$. The approximation could be reasonably modest -- perhaps something like $$\tan (f(x)) = ...
0
votes
2answers
100 views

How to solve $100x^{99} + \cos x = 0$?

I want to know how to solve such equations without a graphing calculator, so please show the steps. $$100 x^{99} + \cos x = 0$$
0
votes
1answer
10 views

Compare sales growth

So I’m trying to measure the sales growth of specific salesman. Any salesman has the highest sales growth (min $4\%$) will receive a giftcard. However I found it not fair to compare when a saleman ...
0
votes
0answers
16 views

Polynomial Long Division with Divisor<Dividend

So here's the problem... 20x^3-4/5x^2-3 When I divide this I get 20x^3 -4 -20x^3 +12x 12x-4/5x^2-3 So 5x^2 goes into 12x how many times? It doesn't seem to. So how do I solve this?
2
votes
3answers
254 views

What do these extra solutions mean?

I'm trying to find a constant $a$ such that $n(n+1)(n+2)(n+3)$ is equivalent to $(n^2+an)(n^2+an+2)$. Clearly by inspection, we have $a=3$. However, say I wish to substitute $n=-1$. We get our ...
2
votes
2answers
53 views

Cool little system of equations.

Solving the system of equations for integers: $2^a+3^b=5^b$ $3^a+6^b=9^b$ How is it done? I tried substituting the $2^a$ from the first equation into the second, and dividing the two equations by ...
-3
votes
1answer
53 views

Ladder against a wall.

Having a bit of a problem with a question. There is a 4m ladder leaving against a wall. There is a box in between The ladder and wall. The box is a cubic metre. I have found a quartic to find the ...
1
vote
0answers
30 views

Question about series and how the pattern idea works

Two Questions: When you are given: $1, 2, 3, .... , n$ How do you know that in the $...$ that it continues the $x_{n-1} + 1$ pattern? Is it the definition of series? Secondly: Do partial sums ...
5
votes
3answers
189 views

Find the number of roots of the equation in $\mathbb{R}$

How many roots does the equation $$\\x^{x^x}=(x^x)^x\\$$ have in $\\\mathbb{R}$? My observations:I observed that $x=-1,1,2$ are its roots. Are there other roots of this equation?And how we can find ...
0
votes
1answer
44 views

Is it possible to find $n-1$ consecutive composite integers

Given an integer $n\geq 2$ ,can we always find an integer $m$ such that each of the $n-1$ consecutive integers $m+2,m+3,.....,m+n$ are composite?
0
votes
0answers
22 views

Prove $\log_ab+\log_bc+\log_ca\geq1+\log_{ab}bc+\log_{bc}ab$

Prove inequality $$\log_ab+\log_bc+\log_ca\geq1+\log_{ab}bc+\log_{bc}ab$$ for $a>1,b>1,c>1.$ We noted $x=\lg a,y=\lg b, z=\lg c $ and wrote inequality in the form ...
0
votes
2answers
33 views

Solve for x, y and z

$$x+y-2z=5 (1)\\ x +z=4 (2)\\ -z=6 (3)$$ $$2\cdot(2): 2x+2z=8 (4)\\ (1)+(4): 3x+y =13 (A)$$ $$(2)+(3): x=10 (B)$$ $$(A)-(B)= 3+y=3\\ y=3-3\\ y=0$$ Substitute $y$ into $(A)$ ...
1
vote
2answers
36 views

Expressing $\frac{1}{4n^2-1}$ as a partial fraction

I was asked to express $$\frac{1}{4n^2-1}$$ as a partial fraction. I have no clue as to what I should break this into. For example I know : $$\frac{1}{n(n-1)}= \frac {A}{n} + \frac {B}{n-1}$$ ...
0
votes
1answer
30 views

Expand $(\frac{x}{3}+\frac{x^2}{4})^2$

$(\frac{x}{3}+\frac{x^2}{4})^2$ I know that the special product of $(a+b)^2$ is $a^2+2ab+b^2$ they said the answer is $\frac{x^4}{16}+\frac{x^3}{6}+\frac{x^2}{9}$ I don't understand how they got ...
1
vote
2answers
66 views

$ay^3 + xy = ab^3$, can I isolate $y$?

I was wondering how much force it would take to compress a sphere of air (assuming Boyle's Law instead of the Real Gas laws, ignoring the engineering method of applying said force), so I started with ...
3
votes
1answer
30 views

Prelude to Cauchy-Schwarz, Quadratic proof.

I have a problem in trying to prove the following observation: "Show that if $ a,b,c \in \mathbb{R} $ are such that for all $ \lambda \in \mathbb{R} $, $a\lambda^2 + b\lambda +c \geq 0 $ then $ b^2 - ...
-1
votes
2answers
20 views

Prove that the image of $f: (0, \infty) \to R$ is contained in $[2, \infty)$. [closed]

where $f(x) = x + 1/x$ Any help is appreciated, what I did was completely wrong haha..
-1
votes
1answer
27 views

How to compute inequality that involves logarithm

So I was reading a math book and I faced with expression I could not solve. Well, I even do not know how to begin, really. I understand that in order to compute power we need to find a logarithm. ...
0
votes
1answer
27 views

simplify using the difference quotient when $f(x)=2/x^2$

I am trying to simplify a difference quotient with the form $$\frac{f(x+h)-f(h)}/{h}$$ if $f(x)=2/x^2$ I have attempted to cancel out the denominator of the numerator by the least common denominator ...
0
votes
2answers
36 views

Factorial simplification rules

I want to know if the following simplification that i did holds true $$(2(n+1))! = 2(n+1)! = 2(n+1)(2n)!$$ and if not true what other simplification can work for it , it's a question about series ...
0
votes
1answer
48 views

Abstract algebra

Assuming there is a real number $x$ with $ x^3 =7$, prove that $x$ is irrational. I started the proof by contradiction, and I got to the point that $7q^3 = p^3$, but I don't know what should I do ...
2
votes
1answer
29 views

I need help solving $3e^{2x}-1=\frac{1}{2}$

I am trying to solve $3e^{2x}-1=\frac{1}{2}$. Here is my work: $3e^{2x}-1=\frac{1}{2}$ $3e^{2x} =\frac{1}{2}+1$ $e^{2x} =\frac{1.5}{3}$ $\ln{e^{2x}} =\ln{(\frac{1}{6})}$ $2x ...
2
votes
2answers
18 views

Simplify a limit problem with the difference quotient $(g(x)-g(a))/(x-a)$ given $g(x) = -3x^2+8x+12$

I am trying to simplify this limit problem using the difference quotient, but I am not sure how to cancel out the a,s since it is g(x) and not g(x+a). Here is my work so far: $$ ...
3
votes
3answers
25 views

Calculus 1: Find the limit as x approaches 4 of $\frac{3-\sqrt{x+5}}{x-4}$

I understand how to find limits, but for some reason I cannot figure out the algebra of this problem. I tried multiplying by the conjugate and end up with 0/0. When I check on my calculator, or apply ...
1
vote
5answers
45 views

rational function limit involving factorials

I posted something similar but someone edited the question and added the wrong equation, which gave irrelevant responses. Lim (2n-1)!/(2n)^n as n approach infinity. Any method, I would just like a ...
4
votes
4answers
50 views

How to solve a convoluted absolute value inequality?

$$ \lvert \lvert x-2\rvert -3\rvert \lt 5 $$ How can I attack this the best way? I see that both sides are positive. Squaring yields: $$ \lvert x-2\rvert ^2 -6 \lvert x-2\rvert +9\lt 25 $$ $$ ...
0
votes
0answers
53 views

Help in writing a nasty expression in nice closed form

This question is abouting re-writing a product in nice closed form. I have the following $$f(v_1) = \left(\sum_{i=1}^K \pi \lambda_i \delta_1 v_1^{\delta_1-1} P_i^{\delta_1} e^{-\beta_i ...
1
vote
1answer
26 views

Beautiful problem of a set of a,b,c.

A set of a,b,c was changed to this set: $a^4-2b^2, b^4-2c^2, c^4-2a^2$. It happened that these two sets are identical. Find a,b,c, if a+b+c=-3. $a^2(a^2-2)+b^2(b^2-2)+c^2(c^2-2)=a+b+c=-3$ I guess, ...