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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1
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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
35 views

please help me with this [on hold]

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
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2answers
30 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
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4answers
48 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
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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 ...
6
votes
7answers
489 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
29 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
13 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
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1answer
17 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
274 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
15 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
251 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
51 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
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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 ...
1
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3answers
129 views
+50

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
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0answers
21 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
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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
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2answers
35 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
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2answers
64 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
19 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
26 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
26 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
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5answers
43 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
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1answer
24 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, ...
2
votes
0answers
18 views

Newtonian potential for ellipsoid

Is there an explicit expression of the Newtonian potential for ellipsoid? As the expression for ball is clear by its symmetry. Definition of Newtonian potential of ellipsoid $\Omega$ at x is defined ...
2
votes
2answers
56 views

Proof of a summation of $k^4$

I am trying to prove $$\sum_{k=1}^n k^4$$ I am supposed to use the method where $$(n+1)^5 = \sum_{k=1}^n(k+1)^5 - \sum_{k=1}^nk^5$$ So I have done that and and after reindexing and a little algebra, ...
0
votes
1answer
18 views

Is $xyz=0$ a joint variation

Is $xyz=0$ a joint variation I know that a joint variation is $\dfrac{x}{yz} = k$ I just want to know if $k$ is allowed to be zero
0
votes
0answers
42 views

What is the inverse of $f(x)=x^{x^x}$?

I'm curious to find the inverse of $ f(x)=x^{x^x} $ As an added extra, I'm already familiar with the Lambert Product Log function.
1
vote
2answers
51 views

$\text{lcm} (a, b)=\text{lcm} (a+c, b+c)$

Can $\text{lcm} (a, b)=\text{lcm} (a+c, b+c)$ for natural $a, b, c$? I've tried writing out all divisors of $a, b, c$ like $p_1 p_2$ etc. And tried that maybe if $a+c> a$ and $b+c> b$ the ...
5
votes
3answers
256 views

Beautiful cyclic inequality

Prove that cyclic sum of $\displaystyle \sum_{\text{cyclic}} \dfrac{a^3}{a^2+ab+b^2} \geq \dfrac{a+b+c}{3}$ , if $a, b, c > 0$ I'm really stuck on this one. Tried some stuff involving QM> ...
6
votes
2answers
586 views

Beautiful problem on a progression

$\{x_n\}$ is a sequence defined as follows: $x_1=20,\quad x_2=14,\quad x_{n+2}=x_n - \frac{1}{x_{n+1}}$. Prove that $0$ is among the members of this sequence. Find its number. I tried some stuff ...
1
vote
2answers
30 views

Algebra Logical Pythagorean theorem help

A wire is attached to the top of a pole. The pole is 2 feet shorter than the wire, and the distance from the wire on the ground to the bottom of the pole is 9 feet less than the length of the wire. ...
1
vote
1answer
62 views

Triplets of distinct integers > 1 that return integer values.

If $(A, B, C)$ are distinct integers $> 1$, and $$f(A, B, C) = \frac{\frac{A^2-1}{A} + \frac{B^2-1}{B}}{\frac{C^2-1}{C}},$$ then for what (if any) triplets $(A, B, C)$ is $f(A, B, C)$ an integer? ...
0
votes
0answers
15 views

estimation question (I should be able to solve it but no, I failed)

Given: $ a = \frac{r+i}{r-i} $ $ b = \frac{r+j}{r-j} $ $ 1 < a < b \le 2a << r $ $ 0 < i < j << r $ How to estimate r given a and b?