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
2answers
39 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
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 ...
1
vote
2answers
118 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
votes
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
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
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
vote
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
23 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
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
vote
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
17 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
582 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
29 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
60 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?
0
votes
2answers
49 views

A boy's father is 25 years older than him. The sum of their ages is 31. How old is the boy?

Q.) A boy is $y$ years old. His father is 25 years older than he is. The sum of their ages is 31. How old is he? In class we wrote the answer as $\displaystyle 2y+25=31 \rightarrow 2y=31-25=6 ...
1
vote
2answers
74 views

Finding prime solutions to $100q+80 = p^3 + q^2$

Finding prime solutions to $100q+80 = p^3 + q^2$ Does them being prime imply some patterns on division modulo 3 or some other integer? How is this done?
-3
votes
1answer
21 views

Boyles law math problem [closed]

Let $$ P_1 = 1.37 atm \\ P_2 = 0.22 atm \\ V_1 = 1 L \\ $$ Boyle's Law says $$ P_1 * V_1 = P_2 * V_2 $$ What is the missing variable answer $V_2$? I'm having trouble.
0
votes
3answers
25 views

Remainders questions help

If we divide a number by 3, 4 ,5 , 6 , we have the remainders 2, 3 , 4 , 5. Is there any way to get a pattern without guessing so many numbers and checking by 3, 4 ,5 ,6?
1
vote
1answer
29 views

How to compute $ \prod_0^n( 1- { 2 \over (2+k)(3+k)}= $?

I have spent quite some time to solve this question, before I asked Wolfram Alpha and got this: $$ \prod_0^n \left(1- {2\over(2+k)(3+k)}\right) = { n+4 \over 3(n+2)}. $$ Now that I know that this ...
3
votes
3answers
55 views

Logarithmic inequality for a>1

Is $\log_{\sqrt a}(a+1)+\log_{a+1}\sqrt a\ge \sqrt6$ always true for $a>1$? What is the approach? Do we check the first a's and then form a induction hypothesis?
1
vote
1answer
46 views

Show that if x,y,z are not divisible by 53, then $x^{26}+4y^{26} \neq\ z^{26}$

Show that if x,y,z are not divisible by 53, then $x^{26}+4y^{26} \neq\ z^{26}$ I've got that $x,y,z$ to the 52nd power are congruent to 1 modulo 53 from Fermat's. How is it continued? Help would be ...
3
votes
1answer
52 views

Arithmetic progression with common difference 2061

If there are 30 consequent members of an arithmetic progression with CD of 2061, show that among them are at most 20 squares of natural numbers. I wrote out $a_1$ through $a_{30}$ and tried to find ...
3
votes
2answers
43 views

Show that if x,y are and $ x^4y^2+x^2+2x^3y+6x^2y+8 \leq 0 $ then $x \geq -1/6 $

Show that if x,y are real and $ x^4y^2+x^2+2x^3y+6x^2y+8 \leq 0 $ then $x \geq -1/6 $ So far I've tried factoring $x^2$ and throwing the 8 on the LHS, but can't get to the needed result. Help would ...
1
vote
2answers
47 views

Some help with sin and cos

I'm having trouble to understand the following equalities in these two equations, i.e. how to apply the addition formulas. Firstly: $$ \frac {1- \frac {sin^2(\frac x2)} {cos^2(\frac x2)}} {1+ \frac ...
0
votes
1answer
17 views

Combining two liquids with different weights to achieve a desired volume and weight

I have two liquids - water and alcohol, each liquid has a different mass Water weighs 1 gram per ML Alcohol weighs 0.5 gram per ML (just for the sake of the example) I wish to combine these ...
0
votes
1answer
17 views

How many milliliters of liquid to fill [duplicate]

A right circular cone has a depth of 103 mm and a top diameter of 82.4 mm. The cone contains water to a depth of 30.0 mm. How many more millilitres of liquid need to be added in order to fill the ...