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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3
votes
3answers
57 views

Difficult inverse tangent identity

Prove that: $$\arctan\left(\frac{\sqrt{1 + x} - \sqrt{1-x}}{\sqrt{1 + x} + \sqrt{1-x}} \right) = \frac{\pi}{4} - \frac{1}{2}\arccos(x), -\frac{1}{\sqrt{2}} \le x \le 1$$ I'd multiply the ...
1
vote
3answers
54 views

What is the value of $a^4+b^4+c^4$?

Consider $a,b,c$ are some real numbers such that $a+b+c =1, a^2+b^2+c^2=2$ and $a^3+b^3+c^3=3$. Find the value of $a^4+b^4+c^4$, if possible. Trial: I observe that \begin{align} a^4+b^4+c^4 ...
0
votes
2answers
33 views

Complex number $\frac{z}{z+1}=2+3i$

Given that $\frac{z}{z+1}=2+3i$, find the complex number $z$, giving your answer in the form of $x+yi$. Can someone give me some hints for solving this question? Thanks
0
votes
1answer
35 views

how do put 64 squares on a cube?

I need 64 squares covering a cube. The amount of squares on each on each face must be equal to each other. The amount of squares on each must be a natural number. The root of the amount of squares on ...
2
votes
1answer
58 views

Solve for $x$: $x =\ln(x)^4$

I plotted the functions on both sides and it shows the equations has at least three solutions. Is there some non-interative (not sure if i used this term correctly - i mean the way you would solve, ...
1
vote
3answers
84 views

Why does $\frac{49}{64}\cos^2 \theta + \cos^2 \theta$ equal $\frac{113}{64}\cos^2 \theta $?

I have an example: $$ \frac{49}{64}\cos^2 \theta + \cos^2 \theta = 1 $$ then what happens next: $$ \frac{113}{64}\cos^2 \theta = 1 $$ Where has the other cosine disappeared to? What operation ...
1
vote
1answer
25 views

weird conversion of polar coordinates into rectangular coordinates.

Given $r=\frac{4}{1-\cos(\theta)}$, convert into rectangular (cartesian) coordinates. My solution: Square both sides: $r^2=\frac{16}{\sin^2(\theta)}$ Multiply the denominator by $r^2$ to get: ...
2
votes
1answer
313 views

The volume in cubic meters of water in an aquarium…

The volume in cubic meters of water in an aquarium is given by the polynomial $$V(x) = x^3 - 16x^2 + 79x - 120\;.$$ If the depth in feet (it really does say feet) can be represented by $x-3$, what ...
1
vote
1answer
20 views

Solving this finite factorial/binomial series

I am staring at the following finite series: $$ \sum_{k=1}^s A_k Y^k $$ I am trying to solve it for $Y$. $A_k$ is given by $$ A_k = {s \choose k}\nu^k (1-\nu)^{s-k}$$ I already solved that $$ ...
0
votes
0answers
8 views

Solve this equation for implicitly defined $F(x)$

For some constants/parameters $s$, $k$, $c$, $A$, $B$, I have $F(x)$ implicitly defined as $$ \sum_{k=0}^{s-1} G(s,k) (1-F(x))^k = \frac{k-c}{x-c}\frac{x}{k}A + B$$ Where $G(s,k)$ is closely ...
0
votes
1answer
26 views

Solution of polynomial equation $g(x)\cdot g(y) = g(x)+g(y)+g(xy)-2\;\forall x,y\in \mathbb{R}$

If $g(x)$ is a polynomial function satisfying $g(x)\cdot g(y) = g(x)+g(y)+g(xy)-2\;\forall x,y\in \mathbb{R}$ and $g(2)=5\;,$ Then $g(5) = $ $\bf{My\; Solution::}$ Given $$g(x)\cdot ...
-3
votes
2answers
230 views

how many numbers from 1 to 1000 are not divided by 3,4,5? [on hold]

How many numbers from $1$ to $1000$ are not divided by none of the numbers: $3,4,5$? I tried to think on a method but nothing jump my head. Thanks EDIT: it's from $1$ to $1000$. didn't noticed ...
0
votes
2answers
25 views

Factoring out an Exponent

I would like to know whether or not I can factor out an exponent on 2 variables like so $A^2 * B^2 = (A*B)^2$ ?
0
votes
2answers
34 views

Inverse of a function $xe^x$

How should I proceed about finding the inverse of the function $xe^x$? I have been wondering about it for a long time and can't think of anything to do.
0
votes
1answer
25 views

Multiply a Scalar by an Exponent

If I have the formula $A * B^2$ and I want to scale B by 2, what would I need to do to A to balance the formula. Would this be valid? $A * B^2 = (\frac{A}{2}) * (2*B)^2$
-3
votes
0answers
32 views

A simple algebra problem ($a$ fixed)

Given that $(1-a^2)(x+a)-2a(1-x^2)=0$, show that $2ax^2+(1-a^2)x-a(1+a^2)=0$. Also, show that the other solution is $x= -(1+a^2)/2a$. How does one go about solving these? Thanks.
2
votes
6answers
2k views

How do I square a logarithm?

How do I square $\log_2(3)$. Does it become $2\log_2(3)$ ? If yes,Please explain.
1
vote
1answer
22 views

Extract sum of coefficients in a binomial expression

I have two questions: (1) Given $(1-x+x^2)^{3n}=c_0 + c_1 x + \dots +c_{6n} x^{6n}$, find $c_0+c_1+ \dots +c_n$. I manage to find $c_0+c_1+ \dots +c_{6n}$ by putting $x=1$ but I do not know how to ...
0
votes
1answer
25 views

linear relations - algebra [on hold]

A baker makes a loss of $30$ when $25$ cakes are sold but makes a profit of $100$ when $90$ cakes are sold. What is the linear relation for his profit? (Please show me the steps; I can do many other ...
0
votes
0answers
23 views

Rules for multiplying exponents by scalars [on hold]

If I have the formula $A * B^2$ where I know $B= 2*C$, can I change the original formula to $A * (2*C)^2$ ?
2
votes
2answers
22 views

Solving for a three dimensional vector.

Let $a = \begin{pmatrix} 2 \\ 5 \\ -1 \end{pmatrix}$ and $b = \begin{pmatrix} -6 \\ 4 \\ -3 \end{pmatrix}$ There exists two nonzero three-dimensional vectors ${v} = \begin{pmatrix} x \\ y \\ z ...
5
votes
5answers
173 views

If $\frac{\sin^4 x}{a}+\frac{\cos^4 x}{b}=\frac{1}{a+b}$, then show that $\frac{\sin^6 x}{a^2}+\frac{\cos^6 x}{b^2}=\frac{1}{(a+b)^2}$

If $\frac{\sin^4 x}{a}+\frac{\cos^4 x}{b}=\frac{1}{a+b}$, then show that $\frac{\sin^6 x}{a^2}+\frac{\cos^6 x}{b^2}=\frac{1}{(a+b)^2}$ My work: $(\frac{\sin^4 x}{a}+\frac{\cos^4 ...
-2
votes
1answer
33 views

Write an algebraic expression in terms of $x$ (where $x > 0$) for “$\tan(2 \tan^{-1} x)$”

Write the following as an algebraic expression in terms of $x$ (where $x > 0$): $\tan(2 \tan^{-1} x)$ This is what I got so far: Let $\tan^{-1} x = \theta$. So, $\tan(\theta) = x$. So, opposite ...
0
votes
1answer
27 views

formula in Spivak calculus, ch 2-6ii f

Without going to much into details about the question itself I would like to draw attention to the fact that Spivak assumes knowledge of a formula (I got it from the solutions in the back of the book) ...
0
votes
0answers
14 views

System of Equations & Approximations

I am trying to derive Eq. (3.6) in the following thesis: http://drum.lib.umd.edu/bitstream/1903/14898/1/Khalil_umd_0117E_14726.pdf This is the equation I am trying to show: \begin{equation} ...
0
votes
1answer
26 views

Expansion for Partial Fractions for $(3-2x)/(x^2+6x+9)$

I'm trying to expand $(3-2x)/(x^2+6x+9)$ into partial fractions to integrate. I'm doing $$(3-2x)/((x+3)^2)=A/(x+3)+B(x+3)^2$$ $$(A(x+3)+B)/((x+3)^2)=3-2x$$ for x=0:$$(3A+B)/9=3$$ for x=1: ...
1
vote
3answers
46 views

How to simplify $\frac{\sqrt{x^3}}{\sqrt[3]{x^4}}$? [on hold]

Please, could someone help simplify and show the steps on how to simplify $$\frac{\sqrt{x^3}}{\sqrt[3]{x^4}}?$$ Thank you.
-1
votes
0answers
16 views
1
vote
4answers
562 views

Filling a swimming pool with two hoses

Hose A can fill a pool in 4 days. You add a second hose B, now both A and B can fill it in 3 days. How many days will it take to fill the pool with just hose B? I would like to see a formula.
1
vote
2answers
38 views

partial fraction decomposition braindead

decompose $\frac{x^2-2x+3}{(x-1)^2(x^2+4)}$ the way my teacher wants us to solve is by substitution values for x, I set it up like this: (after setting the variables to the common denominator and ...
1
vote
2answers
476 views

What annual installment will discharge a certain debt?

What annual installment will discharge a debt of $ 717.60 due in 4 years at 20% p.a. simple interest, if the installments are paid at the each end of each year? I tried the following: $ 717.60 ...
5
votes
1answer
109 views

Prove by combinatorial method that $ \frac{(2m)! \cdot (2n)!}{(m)! \cdot (n)! \cdot (m+n)!} $ is an integer [duplicate]

Prove that $$ \dfrac{(2m)! \cdot (2n)!}{(m)! \cdot (n)! \cdot (m+n)!} $$ is a positive integer, where $(m,n) \in \mathbb{Z^{+}}$ I have already solved it using Legendre's Formula ...
1
vote
0answers
26 views

Complex vector identity

Let $f=(f_1,f_2,f_3)$ be a complex vector. Can we see that $$G:=\frac{2\Im(f_2\bar{f_3})+i2\Im(f_3\bar{f_1})}{|f|^2-2\Im(f_1\bar{f_2})}=\frac{f_3}{f_1-if_2}$$ I tried using $f_j=x_{j,u}-ix_{j,v}$ ...
2
votes
1answer
480 views

Solving for radius of a combined shape of a cone and a cylinder where the cone is base is concentric with the cylinder?

I have a solid that is a combined shape of a cylinder and a concentric cone (a round sharpened pencil would be a good example) Know values are: Total Volume = 46,000 Height to Base Ratio = 2/1 ...
5
votes
3answers
4k views

Number of surjective functions from A to B

Am I on the right track? I am not sure about my reasoning... Number of surjective functions from $A$ to $B$ $$A = \{1,2,3,4\} ; B = \{a,b,c\}$$ We must count the surjective functions, meaning the ...
4
votes
3answers
70 views

Solving $7[x]+23\{x\}=191$

For every real number $x$, $[x]$ denotes the largest integer less than or equal to $x$ and $\{x\}=x-[x]$. The number of real solutions of $$7[x]+23\{x\}=191$$ is (a) 0 $\quad$ ...
6
votes
2answers
70 views

A simple way to find $\lim_{n\rightarrow\infty}{\frac{1}{n^2}\sum_{k=1}^n{\sqrt{n^2-k^2}}}$

I was reading an exam paper used to identify gifted high-school students, and I encountered the following problem: $$\lim_{n\rightarrow\infty}{\frac{1}{n^2}\sum_{k=1}^n{\sqrt{n^2-k^2}}}$$ Using ...
1
vote
2answers
58 views

Graph of the function $y = 2 + (x + 1)^3$

I know that this function will have the behavior of $Y = X^3$ but as I will translate for this function $(Y = X^3)$? I do this: $$(x + 1)^3 = x^3 + 3x^2 + 3x + 3 \quad y = x^3 + 3x^2 + 3x + 5$$ But ...
1
vote
1answer
13 views

To find, wether '1' lies in the range of f, where $f(x)=[ln(\frac{7x-x^2}{12})]^\frac{3}{2}$?

$f(x)=[ln(\frac{7x-x^2}{12})]^\frac{3}{2}$, For the given function, the question is whether, f(x) can equal 1 for some real value of x?
1
vote
1answer
41 views

Is this a correct way to use triangle inequality

If I have: $$|g_1(x) - g_2(x) - (g_1(a) - g_2(a))| \leq f(x^*)$$ Can I proceed to say: $$|g_1(x) - g_2(x) - (g_1(a) - g_2(a))| \leq |g_1(x) - g_2(x)| - |(g_1(a) - g_2(a))|$$ $$ \implies |g_1(x) - ...
0
votes
2answers
23 views

Rewriting a particular sequence in respect to inverses

I'm having a large amount of difficulty on piecing together the intermediate algebra between the following formulas. $$ \frac{n^2 + 1}{2n^2 - 3} = \cdots = \frac {1 + \frac{1}{n ^ 2}}{2 - ...
4
votes
0answers
80 views

Find $\Big\{ (a,b)\ \Big|\ \big|a\big|+\big|b\big|\ge 2/\sqrt{3}\ \text{ and }\forall x \in\mathbb{R}\ \big|a\sin x + b\sin 2x\big|\le 1\Big\}$

Find all (real) numbers $a $ and $b$ such that $|a| + |b| \ge 2/\sqrt{3} $ and for any $x$ the inequality $|a\sin x + b \sin 2x | \le 1$ holds. In other words, find the set $Q$ defined as ...
2
votes
3answers
122 views

Is there a way to expand the cube of a trinomial easily?

Is there a way to expand $(x+y+z)^3$ easily/fast? It takes some time for me to expand this type of quantities, I was just thinking if there's like a shortcut. Motivation: whenever I do ...
0
votes
0answers
40 views

Show that the expansion of $(1+x)^n$ by Binomial Theorem is convergent when $x<1$

To show that the expansion of $(1+x)^n$ by Binomial Theorem is convergent when $x<1$ Let $u_r, u_{r+1}$ represent the $r^{th}$ and $(r + 1)^{th}$ terms of the expansion; then ...
2
votes
1answer
38 views

If $ x=\frac{\sin^3 t}{\sqrt{\cos 2t}}$ and $y = \frac{\cos^3 t}{\sqrt{\cos 2t}}\;,$ Then $\displaystyle \frac{dy}{dx}$ in terms of $t$

If $\displaystyle x=\frac{\sin^3 t}{\sqrt{\cos 2t}}$ and $\displaystyle y = \frac{\cos^3 t}{\sqrt{\cos 2t}}\;,$ Then $\displaystyle \frac{dy}{dx}$ in terms of $t$ $\bf{My\; Try::}$ Using The Formula ...
13
votes
2answers
173 views

Finding all solutions to the equation $|||||x|-1|-1|-1|-1|=0$

I was presented this question by a student I was tutoring: Suppose $x \in \mathbb{R}$. Find all solutions of the equation $$|||||x|-1|-1|-1|-1|=0.$$ What I explained to the student: Given ...
1
vote
1answer
64 views

Number theory / decimal representation

Prove that for any $n\in\mathbb{N}$ there exists a number $m\in\mathbb{N}$ such that the decimal representation of $m^2$ has $n$ ones at the beginning and some combination of $n$ ones and twos at ...
1
vote
1answer
362 views

Algorithm to find the maximum/minimum of a polynomial without graphing.

For a quadratic equation of the form $y=ax^2+bx+c$ the max/min occurs at $x=-\frac{b}{2a}$. Is there any hard and fast equation like this for polynomials of degree $\geq 4$?. For such polynomials the ...
-2
votes
1answer
46 views

Polynomial and squares

Let f be the polynomial $f\in\mathbb{Z[x]}$ defined by $f(x)=x^4-22x^3+135x^2-154x-34$. How many times f(n) is a perfect square when $n\in\mathbb{Z}$ ? This problem I solved another way than the ...