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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2
votes
1answer
53 views

Wonder how to evaluate this factorial $\left(-\frac{1}{2}\right)!$

I've learned factorial. But today I saw a question which I don't know how to start with: $$\left(-\frac{1}{2}\right)!$$ Can anyone explain how to solve it? Thanks
1
vote
5answers
80 views

In the sequence $1,3,7,15,31\ldots$ each term is $2\cdot\text{immediately preceding term}+1$. What is the $n$-th term?

I readily see that it is $2^n-1$, but how can I deduce the $n$-th term from the given pattern i.e. $n$-th term $= 2\cdot(n-1)\text{th term} + 1$ without computation.
0
votes
1answer
17 views

Is there a relation between arithmetic and geometric series for the same a and d=q.

Multiplication is repeated addition, so is there an explicit relation between arithmetic and geometric series if the first term is a and common difference d is equal to the common ratio q. Is it even ...
1
vote
0answers
28 views

Clarify the definition of highest coefficient of a polynomial.

The highest coefficient of a polynomial is given to be 1. Case 1) $ ax^2+bx+c $ Case 2) $ c(x-a)(x-b) $ I am specifically interested in the case 2 polynomial. Edit: I am actually working on a ...
1
vote
1answer
30 views

Finding the domain of a difficult inverse

$f(x)=\frac{3x+5}{-6x+2}$ , largest possible domain Find $f^{-1}(x)$ of this 1-1 function and the domain. So I wrote the equation as $$y=\frac{3x+5}{-6x+2}$$ Interchanged x and y, and made y ...
0
votes
0answers
31 views

Biological modelling math question?

I am trying to write a biological model that models protein interaction. I am having an issue with one aspect. Lets say protein A and protein B interact with eachother to form complex AB. Now every A ...
0
votes
1answer
58 views

Operation with Sigma

How demonstrate that operation: $$1)\sum _{k=1}^{2n}\frac{\left(-1\right)^{k-1}}{k}\:+2\sum _{k=1}^n\left(\frac{1}{2k}\right)\:=\:\sum _{k=1}^{2n}\left(\frac{1}{k}\right)$$ $$2)\sum ...
2
votes
2answers
41 views

Solving for Explicit Form (Differential Equations step)

I have a differential equation $(\alpha)$: $$I'=r\cdot I(S-I) \text{ , }\ I(0)=I_0.$$ Where $r$ is a positive constant. Also, we are given the fact that: $$ \lim_{t\to\infty} ...
0
votes
1answer
23 views

Is it possible to rearrange this term in the form I need?

If I have a term in the following form: $$ 2\frac{(ak + bk - ab)}{(a^2+b^2+k^2)} $$ is it possible to rearrange it into a term like this? $$ 2*f(a, k, b) + f(a, k, b)^2 $$ f can by any type of ...
1
vote
0answers
18 views

$k \in \mathbb{N}$ such that given prime set $X= \{p_1,p_2,,..,p_a\}$, $p_1p_2…p_a - nk$ cannot be decomposed into $a$ prime factors in $X$

How does one find minimum $k \in \mathbb{N}$ such that given prime set $X= \{p_1,p_2,,..,p_a\}$ and $X \subset \mathbb{N}$, $\prod\limits_{i=1}^{a}p_i - nk$ for any $n \in \mathbb{Z}$ cannot be ...
21
votes
13answers
817 views

How to explain to a 14-year-old that $\sqrt{(-3)^2}$ isn't $-3$?

$\require{cancel}$ I had this problem yesterday. I tried to explain to the kid this: $$\sqrt{(-3)^2} = 3,$$ and he immediately said: "My teacher told us that we can cancel the square with the square ...
0
votes
2answers
50 views

Prove, that x, y, z fulfill this equation

I've got the following problem I can't solve myself. Prove, that if x, y, z fulfill this equation: $$ x+y=z+2015 $$ $$ x^2+y^2=z^2+2015^2 $$ then: $$ x^3+y^3=z^3+2015^3 $$
0
votes
1answer
22 views

Pre-Algebra Fractional Exponent Question

Why does $t^{\frac{3}{2}} \cdot t^{\frac{1}{2}} = t^2$? What I tried to do was multiply the exponents together $\frac{3}{2} \cdot \frac{1}{2} = \frac{3}{4}$ so my final answer was $t^{\frac{3}{4}}$ ...
0
votes
1answer
36 views

Predicting increasing income within a given time

I have this game where player earns "gold" and "reputation". With higher reputation, the player nets more income (gold). Each reputation points yields 1% bonus in income (gold) For every 20 gold ...
1
vote
2answers
38 views

Assistance with polynomial factorization

Is it possible to factorize $x^6 + 4x + 8$? If so what is the fully simplified form, and what would be the steps to get there?
0
votes
3answers
39 views

Solve the Equation.

$$ \begin{bmatrix} 1 & 1 \\ 1 & 1 \\ \end{bmatrix} \begin{Bmatrix} v_1 \\ v_2 \\ \end{Bmatrix}= \begin{Bmatrix} 0 \\ 0 ...
0
votes
1answer
19 views

Determine if a function is even or odd

Let $f:\mathbb{R}\to\mathbb{R}$. Define $h:\mathbb{R}\to\mathbb{R}$ by $$h(x)=f(x)\{f(x)+f(-x)\}$$ Then, which of the following option(s) is/are correct ? (A) h is even for all f (B) h is odd for ...
3
votes
2answers
48 views

How does $y=|x+3|+4$ become $y=\frac{1}{2}|2x+3|+4$ (compositions and translations)

Today, I had a test question that was bothering me because my friend and I had different answers to it. It's a grade 12 math question. It's telling us to explain the changes that were made to the ...
1
vote
4answers
81 views

Roots to the quartic equation, $(x+1)^2+(x+2)^3+(x+3)^4=2$

Solving with Mathematica gives me the four roots, $$x=-4,-2,\dfrac{-7\pm\sqrt5}{2}$$ Is there some trick to solving this that doesn't involve expanding and/or factoring by grouping?
1
vote
1answer
19 views

Equation for calculating percentile of data point on normal curve

I'm taking pre-calculus right now, and I'm trying to make a statistics program, so I want to calculate the percentile of a score on a normal curve without using a "standard normal table." I can ...
0
votes
1answer
32 views

limt of the function as $\mu\rightarrow\infty$ or $\mu\rightarrow-\infty$ .

$\lim_{\mu\rightarrow\infty}\frac{\exp(\bar x-\mu)^2}{(\bar x-\mu)}=? $ Also, $\lim_{\mu\rightarrow-\infty}\frac{\exp(\bar x-\mu)^2}{(\bar x-\mu)}=? $ I know, ...
4
votes
1answer
48 views

Is $x(x!)^{1/x}$ an increasing function of $x$, for $x > 0$?

Is $x(x!)^{1/x}$ an increasing function of $x$, for $x > 0$? Here $x!$ is the factorial of $x$. Sure, I do know differential calculus, but my problem is that I do not know how to compute for the ...
0
votes
0answers
27 views

When the company will stop production?

Given total costs function $C(q)=100+10q-6q^2+3q^3$. For which price the company will stop production given that all of the fixed costs are sunk? Do not know how to approach those type of questions, ...
0
votes
2answers
32 views

Proposition from Liber Quadratorum

Prove the following proposition: For any odd square number $x$, there is an even square number $y$, such that $x + y$ is a square number
1
vote
2answers
68 views

How to solve for $x$ in $2(x-5) + 4 (x-3) = -30$

In $2(x-5) + 4 (x-3) = -30$... I'm very confused as to how to solve for $x$, the correct response is $-2$, but I keep getting $4/3$.
1
vote
2answers
47 views

Can I make an assumption about arbitrary numbers in a proof?

Assume $ x_1, x_2... x_{10} $ are different numbers and $ y_1, y_2... y_{10} $ are some arbitrary numbers. Prove that there exists some unique polynomial of degree not exceeding 9 such that: ...
0
votes
2answers
40 views

What is the area of $[r = \frac{4}{2 - \cos \theta}]$?

It makes an ellipse, but I'm unsure where to go from here.
7
votes
2answers
71 views

Can $y=\frac{e^\frac{t^4}{12}}{e^{\frac{t^3}{3}}}$ be simplified?

Can $y=\frac{e^\frac{t^4}{12}}{e^{\frac{t^3}{3}}}$ be simplified? So, I'm working on a problem and I encounter this problem instead. For some reason, the way it looks is intimidating. So this is how ...
4
votes
2answers
80 views

How is $x^\frac{9}{4} \cdot x^\frac{1}{2} = x^\frac{11}{4}$

How is $x^\frac{9}{4} \cdot x^\frac{1}{2} = x^\frac{11}{4}$? I don't really understand it.
2
votes
2answers
52 views

help me understand this given solution from gelfand algebra text.

Find the polynomial of $2$nd degree such that a. $p(1)=0,p(2)=0,p(3)=4$ b.$p(1)=0,p(2)=2,p(3)=0$ c. $p(1)=6,p(2)=0,p(3)=0$ and d. $p(1)=6,p(2)=2,p(3)=4$. The solutions for $a,b,c$ are ...
17
votes
3answers
1k views

What skill do I lack to factor multivariate polynomials?

Ok so I can factor easily regular quadratic polynomials, i.e. $5x^2+7x+9$ (I'm not sure whether that's prime, just made it up), and I was working on solving $y^2+(x^2+2x−2)y+(x^3−x^2−2x)$ by ...
8
votes
2answers
95 views

How to evaluate $ \sum\limits_{n=1}^{\infty} \left( \frac{H_{n}}{(n+1)^2.2^n} \right)$

Evaluate $$ \sum_{n=1}^{\infty} \left( \dfrac{H_{n}}{(n+1)^2.2^n} \right)$$ Where $H_{n}$ is the $n^{th}$ Harmonic Number, i.e., $H_{n} = \displaystyle \sum _{k=1}^n \frac{1}{k}$ I ...
8
votes
4answers
169 views

Evaluating $ \sum_{n=1}^\infty \frac{1}{n^2 2^n} $

Evaluate $$ \sum_{n=1}^\infty \dfrac{1}{n^2 2^n}. $$ I have tried using the Maclaurin series of $2^{-n}$ but it further complicated the question. Moreover, I have also tried taking help ...
0
votes
1answer
16 views

Combine these two equations?

I have two equations: $$f(x,y,z)=\frac {z} {\cos\left(x-\frac {180\lfloor\frac{xy} {180}\rfloor}{y}\right)}$$ which is correct when $\left\lfloor\frac {xy} {180}\right\rfloor$ is odd and... ...
2
votes
4answers
63 views

Subtraction of a number

I have a number $x$. If I remove the last digit, I get $y$. Given $x-y$, how can I find $x$? For example x=34 then y=3 given 34-3=31, I have to find 34. if x=4298 then y=429 , given 4298-429 = 3869 . ...
1
vote
3answers
88 views

Mid '|' in math?

What does this equation mean? What does the $|$ mean? $446617991732222310 | mn(m^k - n^k)$ Here is the complete question for reference - What is the smallest positive integer $k$, such that for ...
0
votes
2answers
15 views

Find if lines intersect, only knowing gradient.

Is it possible to find out if and where two lines intersect with only knowing the gradient of said lines?
3
votes
1answer
44 views

Simplifying algebra fraction expression

The below equation is after taking the derivative for a calculus problem, but when I try to simplify I don't see how. My calculator shows that $${5(\sqrt{1+x^2}-x) \over ...
1
vote
1answer
37 views

Integrating the log-normal function

Compute $$F(t)=\int_0^t \frac{1}{\sqrt{2\pi}\sigma t} \exp\left[-\frac{1}{2}\left(\frac{\log t-\mu}{\sigma}\right)^2\right]\,dt; t>0$$ My Attempt: $u=\frac{1}{t}\Rightarrow du=-\frac{1}{t^2}dt$ ...
1
vote
2answers
29 views

Calculating terms in an arithmetic series

I don't know how I'd go about actually solving this problem. Using brute force I was able to determine there are between $5000$ and $5300$ terms added to get the result, but I want to know how I'm ...
0
votes
1answer
21 views

Find minimum surface area of half cylinder given volume

I want to build a half-cylinder structure, with the volume $1200$ cubic feet. But I want to find the minimum surface area needed to achieve this volume. I'm struggling. I know that the equation for ...
1
vote
2answers
46 views

Show that $\frac{z}{z+1} = 1 - \frac{1}{z+1}$

I'm doing some complex analysis work and I came upon this equivalency: $$f(z) = \frac{z}{z+1} = 1 - \frac{1}{z+1}$$ I was trying to find out how to go from $z/ (z+1)$ to the equivalent expression ...
0
votes
0answers
11 views

Assistance with exponential inflation equations

We are trying to model an exponentially inflationary currency exchange, let's say converting dollars into clams, based on a compounding rate. Our cost equation looks like this: c = b(b*r)^e Where: c ...
4
votes
5answers
1k views

Why/How does this sqrt term work? The inverse of a fraction in a sqrt

Why does this term work? $$ \frac{1}{\sqrt{\frac{g}{l}}} = \sqrt{\frac{l}{g}} $$
0
votes
1answer
23 views

System of equations with 3 variables - Does order matter?

I'm studying Algebra and I'm now at topic 'System of equations with 3 variables'. I'm having a hard time with the following example: $$ \begin{cases} 2x + 2y + 3z = 10\\ 3x + y-z = 0\\ x + ...
4
votes
2answers
94 views

How to compute the integral $\int_{-\infty}^\infty e^{-x^2/2}\,dx$?

Yes, I know that this is very similar to $\int_{-\infty}^\infty e^{-x^2}\,dx$, which has been answered a million times, but I still don't know how to apply the technique from that integration to ...
0
votes
1answer
18 views

Understanding the conditional in a specific case

The problem given was as follows: Let $f: A \rightarrow B$ be a surjective function. Let us define a relation on A by setting $a_0$ ~ $a_1$ if $f(a_0) = f(a_1)$ Does this definition mean that ...
6
votes
3answers
44 views

If $x^2 +px +1$ is a factor of $ ax^3 +bx+c$ then relate $a,b,c$

Suppose If $x^2 +px +1$ is a factor of $ax^3 +bx+c$ then relate $a,b,c$ such that $a,b,c \in R$ I can write $$ax^3 +bx+c=(x^2 +px +1)(\lambda x +D)$$ $$\implies ax^3 +bx+c =\lambda x^3 + x^2.p\lambda ...
1
vote
3answers
41 views

How to calculate $\lim_{k \to \infty} \frac{(2k)!}{(2k+2)!}$

I need to calculate limit number 1, and I don't understand how to get out the factors. $$ (1) \lim_{k \to \infty} \frac{(2k)!}{(2k+2)!}$$ $$ (2) \lim_{k \to \infty} \frac{(k)!}{(k+1)!}$$ When I ...
0
votes
4answers
51 views

Please, I'm using the comparison test for a series?

can i use the comparison test to decide if the folowing series is divergent? $$\sum_{n=1}^{\infty} \frac{{n3^n}}{7^n+1}$$ my working is as follows ...