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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2answers
53 views

Roots of $\cos (\frac{x^2+x}{6})=2^x+2^{-x}$

Find the number of real roots of $ \cos \,(\dfrac{x^2+x}{6})= \dfrac{2^x+2^{-x}}{2}$ 1) 0 2) 1 3) 2 4) None of these My guess is to approach it in graphical way. But equation seems little ...
0
votes
1answer
16 views

Simplify $\frac{\sum_{i = 1}^{n}x_{i}}{n} - \frac{n - \sum_{i = 1}^{n}x_{i}}{1 - \theta} = 0$ to show that $\theta = \bar{x}$

Simplify $\frac{\sum_{i = 1}^{n}x_{i}}{n} - \frac{n - \sum_{i = 1}^{n}x_{i}}{1 - \theta} = 0$ to show that $\theta = \bar{x}$ $\frac{\sum_{i = 1}^{n}x_{i}}{n} = \frac{n - \sum_{i = 1}^{n}x_{i}}{1 - ...
-1
votes
1answer
21 views

Simplify the function

I am having problems solving this, any help would be appreciated. Find $f(x+h)-f(x)$ and simplify if $f(x)=2+3-x^3$ Thanks in advance.
0
votes
2answers
16 views

Need help in clarifying relation of square root and logarithm to do a correct substitution

This might be so basic and obvious, but I am stuck on how to do substitution that involves logarithm and square root. If we have $$\lfloor\sqrt{n}\rfloor$$ and we do the following substitution ...
0
votes
0answers
3 views

Order of Dilated horizontally and translated horizontally

I have a parent function $f(x) = x^2$, and $g(x) = (6[x-2]))^2$ is a transformation from $f(x)$. The question is: $g(x)$ is from $f(x)$ by Dilated horizontally by a factor of 1/6, then translated ...
-2
votes
2answers
20 views

how many jelly beans did each girl have at first?

Martha and Mary had $375$ jelly beans in all. After Mary ate $24$ jelly beans and Martha ate $\frac 17$ of her jelly beans, they each had the same number of jelly beans left. How many jelly beans did ...
0
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0answers
22 views

Quention about the historical definition of determinant

$$ax+by = k_1\\cx + dy = k_2$$ If I want to solve for $y$ in the first equation: $$by = k_1 - ax\implies y = \frac{k_1-ax}{b}$$ Then substitute $y$ in the second equation: $$cx + d\frac{k_1-ax}{b} ...
0
votes
2answers
65 views

Let $p^3+q^3=4$ and $pq=2/3$ . Find $p+q$.

Let $p^3+q^3=4$ and $pq=\frac{2}{3}$ . Find $p+q$. A graphing calculator can find values of $p$ and $q$ numerically. As one can see from the graph below, the two solutions are approximately ...
-6
votes
4answers
51 views

24 hours before Wednesday [on hold]

I have a procedure scheduled for 11 a.m. on Wednesday. I can't take certain medications for 24 hours, so what time should I be able to take my last dose?
0
votes
4answers
30 views

Finding maximum of a function with unknown constants

I have a function in the form: $$y = \frac{ax}{b + \frac{x^2}{c} + x}$$ Supposedly, the maximum of this function is equal to $\sqrt{bc}$. I've tried substituting in $\sqrt{bc}$ for $x$, but I don't ...
0
votes
4answers
57 views

How many solutions has this third degree equation?

how many solutions has this equation: $$ {x}^{3}+4\,{x}^{2}-1=0 $$ i tried ruffini so far and it is not working, now i'm stuck and no idea of how to aproach this.
0
votes
0answers
23 views

Integer solutions to an equation

Let $x,y,z$ be positive integers and $S$ be the set of all the solutions to the equation $x^y+y^z=z^x$. Is $S$ finite or infinite? Lots of thanks for any help in advance.
0
votes
0answers
9 views

Find a cyclic rational function such that…

I'm looking for a function of the form $\frac{f(a,b,c)}{f(b,c,a)}$ (or close to this form, e.g. $\frac{(a+b)^2}{b^2+bc+c^2}$) which is roughly equal to $\frac{b^3-a^2-b^2-a^3-ab^2}{b^2c+a^2b+b^3}$ (I ...
3
votes
2answers
60 views

Prove, inequality ,positive numbers

$$\frac{a}{e+a+b}+\frac{b}{a+b+c}+\frac{c}{b+c+d}+\frac{d}{c+d+e}+\frac{e}{d+e+a}<2$$ Prove that for positive numbers $a,b,c,d,e$ there is such inequality
-1
votes
3answers
23 views

How can we make this expression small? [on hold]

How can we make the following expression small: $$(bx-ay)^2+(cx-az)^2+(cy-bz)^2+(ay-bx)^2+(az-cx)^2+(bz-cy)^2$$, where $a,b,c,x,y,z$ are nonnegative reals? Note: I'm not looking for an exact answer, ...
2
votes
2answers
42 views

divide 6 people in group of 2 in same size

Exercise: divide 6 people in group of 2 in same size. My solution: The exercise tells us to calculate the combination without repetition. If I start by calculating the number of ways to select how ...
0
votes
1answer
29 views

Algebraic manipulation and logarithms

How can i manipulate $3\left(\dfrac{n}{2}\right)^{\log_2 3}$ to equal $n^{\frac{\log 3}{\log 2}}$? I understand that $$\log_b a = \dfrac{\log_2 a}{log_2 b}$$ but i'm not sure how the $3/2$ went away.
-5
votes
1answer
16 views

is there constant $k$ such that nth fibonnaci number $F_n$ satisfies $F_n > k2^n$ and vice versa? [on hold]

Is there constant $k$ such that nth fibonnaci number $F_n$ satisfies $F_n > k2^n$? Also is there constant $k$ that $k2^n>F_n$?
2
votes
0answers
27 views

Extremal points relative to origin for an ellipsoid

Suppose I have an ellipsoid of the form $ax^2 + by^2 + az^2 - cxy -cyz = d$ How would I find the points nearest to, and furthest from, the origin?
-1
votes
2answers
53 views

Can anyone help me find an $x$ for which $\sin x=-1/2$ and $\sin x=\sqrt{2}/2$?

I know that $\sin x=0$ when $x$ is of the form $x=n\pi$ for $n\in\mathbb{Z}$. But, I can't figure out an $x$ for which $\sin x=-1/2$ and $\sin x=\sqrt{2}/2$ are both true. Can anyone help me?
1
vote
2answers
62 views

Sum of $1/n+1/(n-2) + 1/(n-4) + \cdots $

How does one calculate $$\frac{1}{n} + \frac{1}{n-2} + \frac{1}{n-4} \cdots $$ where this series continues until denominator is no longer positive? $n$ is some fixed constant positive integer.
1
vote
2answers
37 views

polynomial of $4^\text{th}$ degree, prove

There is a polynomial $f$ of integer coefficients such that $\text{deg(f)} \geq 4$. Let's assume that there are four integers $a,b,c,d$ for which $f(a)=f(b)=f(c)=f(d)=5$. Prove that there is no ...
1
vote
2answers
98 views

Proving a function is onto?

Let $f: \mathbb{R}\setminus \{3\} \to \mathbb{R}\setminus \{1\}$ be defined by $f(x)=\dfrac{x+3}{x-3}$ Prove that $f$ is onto: Okay, here is the deal. I just started my first abstract algebra ...
0
votes
2answers
23 views

Manipulating an expression into another equivalent form

I have an expression (shown below) and I want to show that $$(n+1)(n)(3n^2+11n+10) = (n)(n+1)(n-1)(3n+2) + \text{some other stuff}$$ How can I do this?
-3
votes
1answer
26 views

…is the closed form for sequence A_n. Find c using the Fibonacci and Lucas number sequences. [on hold]

Let $$\begin{align*} A_0 &= 6 \\ A_1 &= 5 \\ A_n &= A_{n - 1} + A_{n - 2} \; \textrm{for} \; n \geq 2. \end{align*}$$ There is a unique ordered pair $(c,d)$ such that $c\phi^n + ...
0
votes
3answers
38 views

Why $|x-y|<1\implies|y|\leq |x|+1$?

I have the following passage in one of the proofs in my workbook: $$|x-y|<1\implies|y|\leq |x|+1$$ Why is this valid?
0
votes
1answer
18 views

Multiplying brackets in $n(n+1)/2+n+1$

Why does: $$n(n+1)/2+n+1 = (n^2+3n+2)/2 $$ and not $$ (n^2+2n+1)/2 $$ ? Additionally, why is: $$(n^2+3n+2)/2 = ((n+1)(n+1)+1)/2$$ rather than: $$((n+1)(n+1)+1n)/2$$
0
votes
0answers
9 views

How to calculate recurrence $F(n) = F(n/u) + \Theta(n^k)$ where $u,k \in \mathbb{N}$

$\Theta$ is used as in Bachmann-Landau notation (often called as Big-O notation convention). How does one in general the recurrence relation of the following from: $$F(n) = F(n/u) + \Theta(n^k) ...
1
vote
1answer
11 views

How to calculate direct proportionality with logarithms and constant terms added

For the equation: $$y=a-b-c\log(x)$$ How do I calculate how $y$ scales with $x$? This is simple without the logarithms. For example: $$y=a+bx$$ $$y=b(\frac{a}{b}+x)$$ $$y\propto(\frac{a}{b}+x)$$ ...
1
vote
1answer
37 views

Cubic curve with a point of inflection

Not quite what I wanted to ask. What I really wanted to know is why you can't have a cubic curve that starts from top left and ends top right.
5
votes
0answers
42 views

Evaluate $S=\left|\sum_{n=1}^{\infty} \frac{\sin n}{i^n \cdot n}\right|$

Evaluate $$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$ where $i=\sqrt{-1}$ For this question, I did the following, Let $$ \begin{align*} S &= \sum_{n=1}^{\infty} ...
-4
votes
1answer
17 views

If two people temporarily covered the cost of \$20 for the 3rd person by paying \$10 each, how much would the 3rd person owe person 1 and 2? [on hold]

If two people temporarily covered the cost of 20 for the 3rd person by paying 10 each, how much would the 3rd person owe person 1 and 2? ( so that everyone is paying the same amount in the end.)
0
votes
2answers
77 views

A closed form for the sum $S = \frac {2}{3+1} + \frac {2^2}{3^2+1} + \cdots + \frac {2^{n+1}}{3^{2^n}+1}$ is …

A closed form for the sum $S = \frac {2}{3+1} + \frac {2^2}{3^2+1} + \cdots + \frac {2^{n+1}}{3^{2^n}+1}$ is $1 - \frac{a^{n+b}}{3^{2^{n+c}}-1}$, where $a$, $b$, and $c$ are integers. Find $a+b+c$.
1
vote
1answer
34 views

How can I solve a system of two equations, like $A + B = 13$ and $2D + B = 13$?

I am currently studying for my SSAT and this question appeared in my practice book: When $A + B = 13$ and $2D + B = 13$, what is the value of $D$? (A) 13 (B) 5 (C) -5 (D) -7 ...
1
vote
2answers
32 views

Factorial formula problem [duplicate]

Prove that $(n-r)!(r!)$ divides $ n! $ i know its a factorial formula and it might be easy but i stuck .I tried induction to $n$ or analyzing the factorials but im missing something
0
votes
1answer
14 views

Values of $w$ while $y$ changes

I know this is very simple, but I just can't manage to find it. I have $w, y \in \mathbb{N}^*$. Assume that $0 < y < 255$ and $500 \ge w \ge 138$. This is for an animation controlled by the ...
0
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2answers
26 views

Spivak's Calculus, chapter 1 problem 19 (inequalities)

I'm having trouble with problem 1-19 in Spivak's Calculus. I have to prove that if $|x-x_0| < \frac{\epsilon}{2} $ and $ |y-y_0| < \frac{\epsilon}{2} $ then $ |(x-y)-(x_0-y_0)| < \epsilon $. ...
0
votes
2answers
28 views

Show using inequality of means that $a\cdot n \cdot \frac{1}{n} \le a^2n^2+\frac{1}{n^2}$

Show using inequality of means that for $a>0$ and $n\in\mathbb{N}$: $$a\cdot n \cdot \frac{1}{n} \le a^2n^2+\frac{1}{n^2}$$ I'm sure it's not that complicated, but I'm probably missing ...
3
votes
1answer
54 views

Relationship between increasing integer sequences

Suppose that $\mathcal X\cap \mathcal Y=\emptyset$, that $\mathcal X\cup \mathcal Y=\Bbb N$ and that $X(n),\;Y(n)$ are increasing surjections $\Bbb N\to \mathcal X$ respectively $\Bbb N\to \mathcal ...
1
vote
1answer
21 views

Transformation of an equation

How do you get from the left side to the right side in this equation? $$\frac{1+\sqrt{5}}{2} + 1 =\left(\frac{1+\sqrt{5}}{2}\right)^2$$
3
votes
0answers
42 views

How does one solve $y^y-x^x=x$ for $x$ as a function of $y$?

In order to find the answer to this question I started thinking that as a first step to obtain the first and second column, one would have to solve the equation: $$y^y-x^x=x$$ for $x$ as a function ...
12
votes
1answer
94 views

$P(z)=0$ iff $Q(z)=0$, $P(z)=1$ iff $Q(z)=1$. Prove that $P(x)=Q(x)$ for all $x$

Assume $P(x)$ and $Q(x)$ are polynomials with complex coefficients with degree greater than or equal to $1$ such that $P(z)=0$ if and only if $Q(z)=0$, $P(z)=1$ if and only if $Q(z)=1$. Prove that ...
0
votes
0answers
31 views

quadratic formula for polynomials with variable coefficients

I have trouble calculating equations like the one in last comment in the first answer; Solve system of 3 equations there are variable coefficients which I can calculate using quadratic formula - if ...
0
votes
2answers
21 views

Computing an academic grade when relative weights are changed

My grade is 88.6% (High B) and we get 80%(Assessment Grade) and 10%(Homework). My teacher is now making this 70%(Assessment Grade) and 30%(Homework). I have done all my homework 100% and I've been ...
2
votes
2answers
57 views

If 2 people pay 10 each, how much would a 3rd person have to pay to have an equal share?

If person 1 and 2 pay $\$10$ to equal $\$20$, how much would person 3 have to pay person 1 and 2 to become even? My solution: 20 divided by 3 is 6.66 so wouldn't the 3rd person just have to pay ...
2
votes
2answers
36 views

Find the number of children, given that the estate was divided evenly between them [on hold]

Problem of the Week at University of Waterloo: A man died leaving some money in his estate. All of this money was to be divided among his children in the following manner: $x$ to the first ...
-3
votes
3answers
34 views

The closed form sum of $12 \left[ 1^2 \cdot 2 + 2^2 \cdot 3 + \ldots + n^2 (n+1) \right]$… [on hold]

The closed form sum of $12 \left(1^2 \cdot 2 + 2^2 \cdot 3 + \ldots + n^2 (n+1) \right),n \geq 1$ is $n(n+1)(n+2)(an+b)$. Find $an + b$.
-6
votes
1answer
66 views

you know root square of -1, what is the larger of the square? [on hold]

there is a square ABDC, $BD = \sqrt{-1}$ what is the value of AB=BC=DC=AD?
1
vote
1answer
40 views

Find polynomial f(n) such that for all integers $n$ $\geq 1$, we have

Find polynomial f(n) such that for all integers $n \geq 1$, we have $3\left( 1\cdot2 + 2\cdot3 + \ldots + n(n+1) \right) = f(n)$. Write f(n) as a polynomial with terms in descending order of $n$.
0
votes
1answer
31 views

How to solve $D=\sqrt{X^2+MX^2}$ for $X$?

How I to solve $D=\sqrt{X^2+MX^2}$ for $X$? With my rudimentary experience, I find myself incapable. I apologize for asking a question after asking a similar one previously (several days ...