Questions tagged [algebra-precalculus]

For questions about algebra and precalculus topics, which include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomial, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

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27 views

Why not the easier format for the derivatives of inverse functions

The derivatives of the inverse trig functions in almost all the textbooks or online documents are not presented in format A, but format B. Are there reasons for why we do this or is it just out of ...
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1answer
31 views

Concise proof that $\lvert a-b\rvert < \frac{\lvert b \rvert}{2} \implies \lvert a\rvert > \frac{\lvert b \rvert}{2}$

I am only considering $\mathbb{R}$ in this context. I understand why the statement holds and can prove it geometrically, but I can’t think of a concise way to derive the antecedent from the precedent ...
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38 views

Maxima of combined cosine function

I'm new to these kinds of functions but I'm interested in mathematical proofs. Take this function: $f(x) = 1 - \cos(a x) + \cos(b x)$ I assume this can often have no period, so how do you find the ...
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40 views

Need help with this induction proof

I’d appreciate some help regarding this problem. $x_1, x_2,...,x_n$ are variables $y_1, y_2,...y_{2^n-1}$ are sums of non-empty sunsets of $x_i$ $p_k(x_1, x_2,...,x_n)$ is k-th elementary symmetric ...
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0answers
33 views

$f(x-a)+f(x-b)$ vs $f(x-a)f(x-b)$

Consider the condition $f(x)=0$ if $x<c$ and now we wish to determine the values of $x$ for which : $f(x-a)f(x-b)$ is equal to $0$, for $a,b\in\mathbb{R}$ $f(x-a)+f(x-b)$ is equal to $0$, for $a,...
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0answers
25 views

Problem with a derivation using the continuity equation

Consider the continuity equation for an electron gas: $$\tag{1} \nabla \cdot\left[n(\boldsymbol{r}, t) \frac{\partial}{\partial t} \tilde{\boldsymbol{r}}(\boldsymbol{r}, t)\right]=-\frac{\partial}{\...
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2answers
77 views

SMO problem: Sequence and series.

Problem on Series and Sequences (SMO Test): For each positive integer $n \ge 1$ , we define the recursive relation given by $a_{n+1}=\cfrac{1}{1+a_n}$. Suppose that $a_1=a_{2012}$.Find the sum of ...
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1answer
25 views

Fraction manipulation and binomial coefficients

I am working through a proof and I do not understand why the following equation holds: $$\frac{m(m-1)(m+1)}{6} = \binom{m + 1}{3}$$ relative to the following definition of the binomial: $$\binom{n}{k} ...
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2answers
43 views

Sigma notation but each result is stored in a new sequence of equal length (no adding)

My background is Computer Science, so I recognise $\Sigma$ as a for loop that exclusively sums, returning one element/ answer. What is the notation used to loop over a finite sequence, to perform any ...
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10 views

How to find undefined singularity point of a polynomial equation

I was given two equations and I had to find if they are equal or are they undefined and the condition given was: (https://latex.codecogs.com/gif.latex?x\neq&space;0) The equation : (https://latex....
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1answer
63 views

Test to determine if a polynomial has only real roots?

Given a polynomial $p(x)=x^n + c_{n-1} x^{n-1} + \cdots + c_0$ with real coefficients $c_{n-1}, \ldots, c_0$, is there an efficient method to determine whether all roots to the polynomial are real and ...
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1answer
59 views

Could Someone Please Verify My Proof? (fun with two primes)

Am I wrong? Is my proof clear? Is this already a known result? If so, are there any simple implications or corollaries of this proof? (if that makes sense) I know some about groups, rings and fields, ...
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28 views

$1+e^{\sqrt{-1}5t}$ form change

I have a question. I am concerning if $1+e^{\sqrt{-1}5t}$ is exponential form or not. If it is not exponential form, how to change it into exponential form? I mean, exponential form is like $$x(t) = |...
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2answers
35 views

Require help to solve an Inequality

$|0.022\cdot e^{-2t}\cdot sin(11t)|<0.01$ Values of $t$ for which the inequality holds. Thanks in advance
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1answer
57 views

Summing Concentric Circles.

This should be a simple problem. I want to sum lengths of concentric circles with the outer circle at radius L. Then each smaller circle is at radius $2\pi R$ less than the previous until the inner ...
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1answer
67 views

How to prove $1-\frac12+\frac13-\frac14+\ldots+\frac1{199}-\frac1{200}=\frac1{101}+\frac1{102}+\ldots+\frac1{200}$?

I want to prove $$1-\frac12+\frac13-\frac14+\ldots+\frac1{199}-\frac1{200}=\frac1{101}+\frac1{102}+\ldots+\frac1{200}$$ First I added $\frac12+\frac14+\ldots+\frac1{200}$ to both sides of the equation ...
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0answers
19 views

Rule for Multiplying The LB and The UB of The Sum

Here i have: $$\sum_{i=2}^{13} i \tag{1}$$ I need to deform $(1)$ like the following expression $$\sum_{i=2(c)}^{13(c)} x,\quad c>0 \tag{2}$$ How to find $x$ in terms of $i$ such that $(1)=(2)$ ??? ...
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0answers
18 views

Is any information lost when I add some equations in the given system?

I have a system of equations with four unknowns $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ $$ a\;\alpha _3 +b\;\alpha _4 +c\;\alpha _1 +d\;\alpha _2=0,\qquad\qquad\qquad\qquad(A)\\ e\;\alpha _3 +f\;\alpha ...
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3answers
80 views

Polynomial with real coefficient which has root $\sqrt{5}+\sqrt{7}$ [closed]

Is it possible to find a polynomial with real coefficients with a root of $\sqrt{5}+\sqrt{7}$ ?
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1answer
17 views

Need help to understand different formula for Trapezoidal's rule

My book (Numerical Analysis by Douglas) said the formula for Trapezoidal's rule is $$\int_a^b f(x)\, \Bbb dx \approx \frac h2 \left(f(a) + f(b) + 2\sum_{k=1}^{n-1} f(x_k)\right)$$ Here is the ...
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1answer
36 views

How to sove this inequality: $2 \cdot \sqrt{x^2-x-2} -2x +1 > 0$. I know it looks obvious, but there something into it

I get $-8>1$, which mean that $2\sqrt{x^2-x-2} -2x +1 > 0$ is always false , but actually it isn't. Edit: here is how I got it. $2\sqrt{x^2-x-2} -2x +1 > 0 $ $2\sqrt{x^2-x-2} > 2x -1 $ $(...
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1answer
51 views

When to use exponent rule $\sqrt x^2$ and when to square both sides?

I am solving equations to see if the equation is a function or not. I would like to know when to use exponent rules and when to square both sides? The equation is $$x= \sqrt {1 - y^2}$$ I want to ...
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1answer
27 views

For $0<b \leq a$ prove that $\frac{1}{8}\left(\frac{(a-b)^2}{a} \right) \leq \frac{a+b}{2}-\sqrt{ab} \leq \frac{1}{8}\left(\frac{(a-b)^2}{b} \right) $

For $0<b \leq a$ prove that $$\frac{1}{8}\left(\frac{(a-b)^2}{a} \right) \leq \frac{a+b}{2}-\sqrt{ab} \leq \frac{1}{8}\left(\frac{(a-b)^2}{b} \right) $$ I was trying compare the following ...
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0answers
34 views

$\nabla\left(\nabla \boldsymbol{r}_{0} e^{i \boldsymbol{k} \boldsymbol{r}} e^{-i \omega t}\right) $- meaning?

My physics professor wrote the following equation: $$ \nabla\left(\nabla \boldsymbol{r}_{0} e^{i \boldsymbol{k r}} e^{-i \omega t}\right)=-e^{-i \omega t} e^{i \boldsymbol{k r}} \boldsymbol{k}\left(\...
2
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3answers
67 views

Prove: There exists two real numbers $a$ and b such that the cube root of $a^3 +b^3 = a+b$

I am getting tripped up by some proof concepts. Can someone breakdown how to prove: $$\sqrt[3]{a^3+b^3} = a+b$$
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2answers
76 views

If $a,b\in \mathbb R^+$, $ |a-2b|\leq\frac {1}{\sqrt{a}}$, $|b-2a|\leq\frac {1}{\sqrt{b}}$ Prove $a+b\leq 2$

Question: If $a,b\in \mathbb R^+$, $|a-2b|\leq\frac {1}{\sqrt{a}}$, $|b-2a|\leq\frac {1}{\sqrt{b}},$ prove $a+b≤2$. I figured out that $a+b\leq \frac {1}{\sqrt{a}}+\frac {1}{\sqrt{b}}$, but I am ...
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1answer
19 views

Time and Work : Alternating Workers

Two taps $A$ and $B$ can fill a tank in $20$ minutes and $30$ minutes respectively. An outlet pipe $C$ can empty the full tank in $15$ minutes. $A$,$B$ and $C$ are opened alternatively each for $1$ ...
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1answer
19 views

Finding the amplitude of the difference between two sinusoids

I have the following function: $$y(t)=a\sin(wt+p)-b\sin(wt+p+\frac{2\pi}{3})\tag1$$ Question: what is the amplitude of the function $y(t)$ in terms of $a,b,w$ and $p$? I thought I can find: $$\frac{\...
2
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1answer
29 views

Absolute error formula

Trying to figure out a problem in my textbook. In my exercise in a textbook, a problem says that Determine the error in the approximation given that the actual length is $3.7437137$. And in the ...
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1answer
64 views

When is the sine of radians and degrees the same?

I need to find the two smallest positive values of $x$ such that $$sin(x °) = sin(x)$$. This is what I did. First, I converted degrees to radians by doing $x ° = {x\pi \over 180}$. So our starting ...
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3answers
58 views

how can $\sqrt{2}/4=1/2^{3/2}$

$\sqrt{2}/4=1/2^{3/2}$ this is a subpart of a work example question I have but I don't understand how I can convert the first part into the second.
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19 views

Can $N$ take infinite values?

What is the value of $N$, and can it have infinite solutions? $Nx+Ox=Ny+Py=Nz+Dz$ with $N$, $O$, $P$, and $D$ positive integers; and $D > P > O$
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34 views

Arithmetic in Lambert-W number system

The principal branch of Lambert-W function is defined like so: $$W(x) = f^{-1}(x)\\ \text{where} f(x)=xe^x$$ So $W(e) = 1$, $W(2e^2)$ is 2, $W(3e^3)$ is 3 etc. We define Lambert-W number system to ...
2
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1answer
47 views

Major doubt in Mathematical Induction [duplicate]

I am doing problems in Induction. What i know is: First we need to test Base case, that is $P(n_0)$ should be verified. Second we need to assume that $P(k)$ is True which is the inductive Hypothesis ...
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0answers
21 views

How can I prove the following inequality. [duplicate]

If n is a positive integer, prove that $\dfrac{1}{2\sqrt{n+1}} < \dfrac{1\cdot3\cdot5\ldots(2n-1)}{2\cdot4\cdot6\ldots2n}$
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1answer
60 views

How to remember the derivatives of inverse trigonometry function?

I remember the derivatives of trig functions by naming 3x basic right triangles in a specific way and using ONE simple multiplication. Just wondering if there are similar approaches to remember ...
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3answers
79 views

A very small disc winds around a larger circular disc of radius $R$ connected to it by a string. How long is the spiral it travels?

A very small disc winds around a larger circular disc of radius $R$. It is connected to it by a string of length l that remains tight. What distance does it travel before it hits the larger disc of ...
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2answers
30 views

Expectation of minimum of uniformly random subset of $\{2^1,2^2,⋯,2^{10}\}$?

Choose a random subset of $\{2^1,2^2,⋯,2^{10}\}$ by selecting each of the 10 elements independently with probability $\frac{1}{2}$. Find the expected value of the smallest element in the subset (e....
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1answer
42 views

Finding all monic complex polynomials $P(x)$ such that $P(x)|P(x^2)$ [duplicate]

Find all monic complex polynomials $P(x)$ such that $P(x)|P(x^2)$. My progress so far is that I have find that for degree 1, $P(x)=x, x^2$ are the only ones. For degree 2, they are $P(x)=x^2+x+1, x^2,...
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4answers
41 views

Determining the value of $k$ if $x+2$ is a factor of $3x^3-kx^2+kx-4$ [closed]

Determine the value of $k$ if $x+2$ is a factor of $3x^3-kx^2+kx-4$. I have absolutely no idea how to tackle this problem, I have tried trial and error but I think I am doing something wrong. Could ...
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3answers
28 views

Finding a polynomial $f(x)$ that when divided by $x+3$ yields quotient $2x^2-x+7$ and remainder $10$

I'm struggling to grasp this particular question: When a polynomial $f(x)$ is divided by $x+3$, the quotient is $2x^2-x+7$ and the remainder is $10$. What is $f(x)$? This is what I did: $$\begin{...
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0answers
24 views

Does fractal decomposition work with irreducible quartics and higher powers?

I am curious to know if one can write for example $$\frac{5x}{(1-x)(x^4+1)^2}=\frac{A}{1-x} + \frac{Bx^3+Cx^2+Dx+1}{x^4+1} + \frac{Ex^3+Fx^2+Gx+H}{(x^4+1)^2}$$ and more generally can one use this ...
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3answers
50 views

Solve for $x,y$ : $x^4+y^4+2 = 4xy$

Solve for $x,y$ : $x^4+y^4+2 = 4xy$ Obviously, the very obvious solutions are $(1,1)$ and $(-1,-1)$. But I don't know how to reach the answer. I got the answer through hit and trial. I tried adding ...
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3answers
62 views

What is the value of $a+b$ Satisfies these two equations.

Let $a,b\in \mathbb R$ Such that : $$\cases{a^3-3a^2+5a-17=0 \\ b^3-3b^2+5b+11=0}$$ Find the value of $a+b$ I’ve tried to add the two equation and factoring $a+b$ but it didn’t help me. I think that ...
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0answers
31 views

If $\frac{1}{a^2-bc}+\frac{1}{b^2-ac}+\frac{1}{c^2-ab}=0$ then show that $\frac{a}{(a^2-bc)^2}+\frac{b}{(b^2-ac)^2}+\frac{c}{(c^2-ab)^2}=0$. [duplicate]

I am trying to use this lemma : If $ab+bc+ca =0$ then $\frac{1}{a^2-bc}+\frac{1}{b^2-ac}+\frac{1}{c^2-ab}=0$. The given condition implies that $xy+yz+xz=0$ where $x=a^2-bc$, $y= b^2-ac$ and $z=c^2-ab$....
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0answers
23 views

Fourier transform with factor $(-1)$ rather than $\frac{1}{\sqrt{2\pi}}$

My physics professor defined the following expression for the Fourier transforms of the current density in space and time: $$ \Delta j_{i}(\mathbf{k}, \omega) \equiv \frac{1}{(2 \pi)^{2}} \iiiint e^{-...
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1answer
49 views

Necessary and sufficient condition for a degree 4 polynomial to be a sum of 4th powers

This is a problem I encountered in Hall and Knight's Higher Algebra. Suppose we have a polynomial $p(x,y)=a_0x^4+4a_1x^3y+6a_2x^2y^2+4a_3xy^3+a_4y^4.$ I want to find a necessary and sufficient ...
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0answers
32 views

System of equations wrong solution $(x, y, z) = (3, 3, 2)$

Find the real solutions for the following system of 2 equations: $z^2 - 4z = -9 - 2xy$ $x(x - 6 - y) + y(y - 6 - x) = -13$ (A book of problems for 8th graders) The book says that $(x, y, z) = (3, 3,...
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1answer
49 views

Finding $r$ in quadratic integer equation

$$M=f(n)=\frac{n(n+1)}{2}$$ $$M=g(k)=40k+30+r$$ I have found that $r=2,6$ but $r$ can only be one. Since $n, k \in \mathbb{N}$, I wanted to see which vaue for r would lead to integer solutions for k ...
2
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1answer
75 views

For real $a$, $b$, $c$ all greater than $1$, show $\frac{a^a}{b^b}+\frac{b^b}{c^c}+\frac{c^c}{a^a} \;\ge\; \frac{a}{b}+\frac{b}{c}+\frac{c}{a}$

An inequality question from an olympiad book: If $a,b,c$ are real numbers satisfying: $a>1$, $b>1$ and $c>1$, then prove that: $$\frac{a^a}{b^b}+\frac{b^b}{c^c}+\frac{c^c}{a^a} \;\ge\; \...

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