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
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1answer
16 views

Differents between $lnx^2$ and $ln(x^2)$ Find derivative

I have this problem Find derivative for $lnx^2$. It seems that $lnx^2 \neq ln(x^2)$ since the derivative are differents using Wolfram Alpha. I don't understand how to calculate the derivative for ...
2
votes
5answers
28 views

Working Out Easy Equations

does anyone know how to do this equation? I know it's easy but I can't work out what the question means. When I expanded the first equation: $(y+4)-(y-3)$ $y^2 -3y -4y - 12$ $y^2-7y-12$ Not ...
0
votes
1answer
27 views

Getting ready for Calculus?

So I wanted to start a Masters program but they require that I have Calculus III. I want to take that course at the university, but I need to be ready for it. As I look at Khan Academy and do some ...
2
votes
1answer
19 views

Calculating $\sum_{k=0}^{n-1}\frac{1}{a+bk^2}$.

I want to calculate the following summation: $$\sum_{k=0}^{n-1}\frac{1}{a+bk^2}$$ Any hint how I can calculate this? Is there any kind of closed form for this summation?
2
votes
0answers
20 views

Algebraic values of the sine function

First question: For which angles $x$ is $\sin(x)$ a real number that can be expressed using only integers, addition, subtraction, multiplication, division and the extraction of $n$th roots? (With ...
2
votes
1answer
28 views

Inequality $(a+b)^2 + (a+b+4c)^2\ge \frac{kabc}{a+b+c}$ for $a,b,c \in\mathbb{R}$

Find biggest constans k such that $(a+b)^2 + (a+b+4c)^2\ge \frac{kabc}{a+b+c}$ is true for any $a,b,c \in\mathbb{R}$ Could you check up my solution? I'm not sure it's ok - $(a+b)^2 + (a+b+4c)^2 \ge ...
2
votes
1answer
44 views

Inequality $x^4+y^4+(x^2+1)(y^2+1)\ge x^3(1+y) +y^3(1+x)+x+y$ for $x,y \in\mathbb{R}$

Prove for $x,y \in\mathbb{R}$ that such inequality exists ; $x^4+y^4+(x^2+1)(y^2+1)\ge x^3(1+y) +y^3(1+x)+x+y$ And here is what I realised ; because $(x^2+1)(y^2+1) >=1$ and $x^4+y^4 \ge 0$ ...
-2
votes
4answers
60 views

How to determine the nth term of a sequence, given only the first four terms? [on hold]

I need to calculate the tenth term of the following sequence: $1 \quad 8 \quad 27 \quad 64\quad \ldots$
2
votes
0answers
20 views

Solving A Certain Diophantine Equation

I am stack on finding the solution of the diophantine equation: $d(2^{k+1}-1)-b^2(2^{k+1}-2)=1$. where $k\geq 1$ and $b^2>d$ for $b$ an odd composite integer. Is there a solution to this ...
-4
votes
1answer
34 views

Linear-algebra problem [on hold]

How do you solve this equation? $$ -7x-28=7+6(1-8x)$$
0
votes
1answer
45 views

How can the modulus of something be less than zero?

I've been asked to prove that for $\epsilon>0$, $$| a-x| < \epsilon \iff a-\epsilon<x<a+\epsilon,$$ and as a hint to consider both $| x-a|>0$ and $| a-x|<0$. I used the fact that ...
2
votes
2answers
50 views

How many positive integers less than $2013$ are divisible by none of $2, 3, 4 ,5$?

How many positive integers less than $2013$ are divisible by none of $2, 3, 4 ,5$? This was an olympiad question. I thought of writing a number $x \le 2012$ in the form: $x = 2^{a}3^{b}4^{c}5^{d} = ...
0
votes
2answers
13 views

FV (Future Value) of annual payments

My client will receive $\$881$ now and twice that amount in a year. He will get 3 times $\$881$ in 2 years . 4 x $\$881$ in 3 years etc. for as long as he lives. Assuming he lives 20 years and each ...
0
votes
1answer
39 views

sum of exponentials to non-integer power

I have the expression \begin{equation} (e^{at}+e^{bt}+e^{ct})^{v} \end{equation} for some a,b,c which isn't important. I'd like to take a limit $t\rightarrow \infty,v\rightarrow 0,vt=\text{constant}$. ...
1
vote
2answers
42 views

how old are Inee and Imee now? [on hold]

Inee and Imee are twins. Their mother is 28 years older than they are and 4 times as old as the sum of their ages. How old are Inee and Imee now?
1
vote
1answer
88 views

Number of real roots of $2 \cos\left(\frac{x^2+x}{6}\right)=2^x+2^{-x}$

Find the number of real roots of $ \cos \,\left(\dfrac{x^2+x}{6}\right)= \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 ...
0
votes
1answer
18 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
26 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
3answers
19 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
4 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 ...
-1
votes
2answers
25 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
votes
0answers
28 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
68 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 ...
-5
votes
4answers
52 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
41 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
62 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
26 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
11 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
66 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
47 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
30 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
17 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
28 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
vote
2answers
68 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
64 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
38 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
99 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
24 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
27 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
39 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
19 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
12 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
12 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
38 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
44 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
78 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
35 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