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
39 views

How $\delta_1$ and $\delta_2$ for two different limits at $a$ can be read as $\delta=\text{min}(\delta_1,\delta_2)$?

I am having trouble understanding a certain part of the proof on why a function cannot approach two different limits near $a$, so I will just list the relevant parts. If this is not enough/ambiguous ...
2
votes
1answer
469 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 ...
0
votes
1answer
68 views

A sequence with variables, find the $mn^{th}$ term given the $m^{th}$ and the $n^{th}$?

I was trying to answer this question, but I'm getting a wrong answer. The question is: If the $m$th term of an arithmetic progression is $\frac{1}{n}$ and the $n$th term is $\frac{1}{m}$ then prove ...
13
votes
2answers
471 views

Number of real positive roots of a polynomial?

Consider the polynomial $$f(x)=x((1+x^n)^n+a^n)-a(1+x^n)^n,$$ where $n\geq 2$ is a positive integer and $a$ is a positive real number. I'm interesting in deducing the number of positive real roots ...
3
votes
4answers
141 views

What is the meaning of $(2n)!$

I came across something that confused me $$(2n)!=?$$ What does this mean: $$2!n!, \quad 2(n!)$$ or $$(2n)!=(2n)(2n-1)(2n-2)...n...(n-1)(n-2)...1$$ Which one is right? The exercise is to show that ...
0
votes
2answers
645 views

Expressing a set in tabular method

Set $A=\{\,x \in\mathbb R : x^2-(a+b)x +ab=0\,\}$. We have to express it using tabular method. At first, we factorize it, getting $(x-a)(x-b)=0$ or $x =a$ and $x=b$. But how do we know that $a$ and ...
6
votes
1answer
284 views

Prove that: $\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}\leq1$

Let $a$, $b$, $c$ and $d$ are non-negative numbers such that $abc+abd+acd+bcd=4.$ Prove that: $\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}\leq1$ I simplified it and it turns out that ...
4
votes
1answer
806 views

Bill calculation (very simple math question)

I hope this question isn't too simple! If it is, let me know if I should post it elsewhere, thank you. (Also, I didn't know what to tag it with, sorry!) My wife and I live with my brother-in-law Bob, ...
5
votes
1answer
567 views

Finding the derivative of $\sin \sqrt {x^2+1}$ from the definition?

This means finding $\lim_{h \to 0} \large \large \frac{\sin \sqrt {(x+h)^2+1}-\sin \sqrt {x^2+1}}{h}$ . The only way I could think of to do this is to replace $h$ by some function $f(h)$ such that ...
2
votes
2answers
96 views

How to split the rent if two roommates live there from the beginning and a third one joins in the middle of the month?

I've been thinking over it and I can't figure it out. Consider the rent of the house is $X$. Now there are two roommates from the beginning and a third one joins in the middle of the month. Now ...
2
votes
3answers
89 views

Difference of 2 strict inequalities

Can the difference of 2 strict inequalities be a not strict one? An example : $$ a, b \in \mathbb R, n \in \mathbb N $$ $$ a< n$$ $$b < n$$ their difference is $\ a-b \leq 0$ no?, since $a$ ...
4
votes
0answers
52 views

Existence of Solutions of Two Cubic Equations in a Particular Region

If I have two cubic equations in two variables, $ax^3 + bx^2 y + cxy^2+\dots=0$ and another one with different coefficients, and I know that $(x,y)=(0,0)$ or $(1,1)$ are solutions, are there any nice, ...
2
votes
2answers
98 views

Questions on powers of a bijection $f\colon\{1,2,\dots,n\}\to\{1,2,\dots,n\}$ [closed]

Let $f$ be a one-to-one function from $X=\{1,2,\dots,n\}$ onto $X$. Let $f^k=f\circ f\circ \cdots \circ f$ denote the $k$-fold composition of $f$ with itself. Show that there are distinct positive ...
3
votes
3answers
288 views

If the $m$th term of an Arithmetic Progression is $\frac{1}{n}$ and the $n$th term is…

Problem : If the $m$th term of an A.P is $\frac{1}{n}$ and the $n$th term is $\frac{1}{m}$ then prove that the sum to $mn$ terms is $\frac{mn+1}{2}$ My working : Let $a$ be the first term of the ...
1
vote
4answers
122 views

Simplifying compound fraction: $\frac{3}{\sqrt{5}/5}$

I'm trying to simplify the following: $$\frac{3}{\ \frac{\sqrt{5}}{5} \ }.$$ I know it is a very simple question but I am stuck. I followed through some instructions on Wolfram which suggests that I ...
1
vote
1answer
65 views

Solving logarithmic equation

I'm having trouble solving this equation. I know there is a solution as my graphics calculator can solve it, but I want to see the steps on how to get the answer. The mathematical equation is: ...
1
vote
1answer
44 views

correct understanding mathematical question

suppose that we have following question,this question is not related to itself mathematics confusion,but language problem and please help me to clarify English language terms in mathematics. ...
3
votes
1answer
236 views

Factoring $a^2+b^2+c^2$?

Is it possible to factor $a^2+b^2+c^2$ ? If we make this into only two factors, I know it has to look like this: $(a+b+c+\cdot \cdot \cdot )(a+b+c+\cdot \cdot \cdot )$ . But I don't know how to get ...
3
votes
2answers
66 views

Largest square written as $p^2+pq+q^2$ where $p, q$ are primes?

I got this problem from the website Brilliant, but I have doubts about the solution presented there: $(p+q)^2-k^2=pq$ $(p+q+k)(p+q-k)=pq$ Now either $(p+q+k)=p$ and $(p+q-k)=q$ (which doesn't work), ...
6
votes
5answers
299 views

pandigital rational approximations to the golden ratio and the base of the natural logarithm

Steven Stadnicki suggested in a comment that I post the following as a question. The golden ration $\phi$ is given by $$\phi = \frac{1+\sqrt{5}}{2} \approx 1.618033988.$$ A rational approximation is ...
2
votes
2answers
291 views

How would you interpret this question focusing on problem solving?

The first step of problem solving is to understand what the problem is asking, that is where I am stuck. One of the legs of a right triangle has length 4 cm. Express the length of the altitude ...
9
votes
1answer
193 views

How to find the sum of the sequence $\frac{1}{1+1^2+1^4} +\frac{2}{1+2^2+2^4} +\frac{3}{1+3^2+3^4}+…$

Problem : How to find the sum of the sequence $\frac{1}{1+1^2+1^4} +\frac{2}{1+2^2+2^4} +\frac{3}{1+3^2+3^4}+.....$ I am unable to find out how to proceed in this problem.. this is a problem of ...
7
votes
3answers
249 views

How to understand proof of a limit of a function?

Given the following function: $$ f(x)=\left\{ \begin{array} {cc} 0, & x \text{ irrational, } 0<x<1 \\ \frac{1}{q}, & x=\frac{p}{q} \text{ in lowest terms, } 0<x<1 \end{array} ...
0
votes
0answers
58 views

Help with this challenge of the second degree equation [duplicate]

Mathematics course, and I love this science, I am very curious, and these days, studying a book of calculus found in (book from the library of the University) a sheet full of challenges, to share with ...
1
vote
1answer
100 views

how to simplify/expand $(x - 1/2)^x$

Last night, I was quite tired and asked the incorrect question for the expansion of: $$\left(x-\frac12\right)^2$$ the actual question was $$\left(x-\frac12\right)^x$$ as a multiple choice question, ...
0
votes
3answers
104 views

Find the eigenvalues of the matrices.

The characteristic equations for the two matrices are: $x^3-8x-7=0$ and $x^3-6x^2+11x-6=0$ I know that in order to find the eigenvalues, I need to factor these two equations out. I'm just having a ...
4
votes
2answers
155 views

A point $(x,y)$ moves so that its distance from the line…

A point $(x,y)$ moves so that its distance from the line $x=5$ is twice as great as its distance from the line $y=8$. Find an equation of the path of the point. I got the two equations: $$x-2y+11=0 ...
1
vote
3answers
70 views

Help on how to show two inequalities are false

Suppose I have these two inequalities: $$-\epsilon < l < \epsilon$$ $$ -\epsilon + 1 < l < \epsilon +1 $$ where $\epsilon$ and $l$ can be any number and $\epsilon \gt 0$. How can I show ...
-1
votes
2answers
216 views

A challenge on the parabola [closed]

Good Night ... Here I am editing the question and explaining the purpose of the same ... Mathematics course, and I love this science, I am very curious, and these days, studying a book of calculus ...
0
votes
5answers
204 views

What is the difference between $f(a)$ and $\lim_{x\to a} f(x)$?

I am a bit confused on this topic as I am not getting an intuition about it! For example consider slope $\frac{\mathrm dy}{\mathrm dx}$. Suppose $\frac{\mathrm dy}{\mathrm dx}$ at $x=0$ is $5$. What ...
2
votes
3answers
81 views

How to show that there is no condition that can meet two inequalities?

Here's an excerpt from Spivak's Calculus, 4th Edition, page 96: If we consider the function $$ f(x)= \left\{ \matrix{0, x \text{ irrational} \\ 1, x \text{ rational}} \right. $$ then, no ...
0
votes
2answers
52 views

Given $a > b+c$, $e>d+f$, and $i>g+h$, can the quantity $a(ei-hf) + b(-di+fg) - c(dh+eg)$ ever be zero?

Given positive reals $a > b+c$, $e>d+f$, and $i>g+h$, can the quantity $a(ei-hf) + b(-di+fg) - c(dh+eg)$ ever be zero?
-2
votes
1answer
247 views

if 30 out of 3500 drivers had an accident last year, what percent of the drivers had an accident

If 30 out of 3500 drivers had an accident last year, what percent of the drivers had an accident (round to the nearest tenth of percent)? plzz help me with this problem if you can and I will be ...
7
votes
1answer
147 views

tough algebric problem?

I wanted to know how can i prove that if $xy+yz+zx=1$, then $$ \frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2} = \frac{2}{\sqrt{(1+x^2)(1+y^2)(1+z^2)}}$$ I did let $x=\tan A$, ...
0
votes
2answers
258 views

Calculate the average sales per day in 2010.

In 2005, Company K was 730 000 units of the car market. Total sales is 50% of the total cars sold in 2010. Calculate the average sales per day in 2010. Calculation: Total sales in 2010 = $50\% ...
1
vote
1answer
62 views

Simplify $(x - 1/2)^2$

Not sure what subject this is. again the question is $$\left(x-\frac12\right)^2$$ as a multiple choice question, i chose the answer: $$x^2+\frac1x$$ Thank you.
1
vote
2answers
25 views

Calculate Danny allowances in 2010.

In 2011, Danny received an allowance of $ 260. This amount is seeing an increase of 30% compared with the previous year. Calculate Danny allowances in 2010. Calculation: = 30/100 x 260 = 7800/100 = ...
2
votes
2answers
117 views

I don't see how Cauchy's proof of AM $\ge$ GM holds for all cases?

I am reading Maxima and Minima Without Calculus by Ivan Niven and on pages $24-26$ he gives Cauchy's proof for the $AM-GM$ . The general idea of the proof is that $P_{n}$ is the proposition ...
1
vote
2answers
34 views

What is the smallest amount of the provision?

Provisions for three companies totaling $ 48 million allocated in the ratio of 8:3:1. What is the smallest amount of the provision? This is my calculation: = 8 +3 +1 = 12 = 8x12: 3x12: 1x12 = 96: ...
0
votes
1answer
50 views

number of days needed : 48-hour project with 4 employees working 6 hrs/day?

The paving of a road takes 48 hours if done by an employee. As a Project Manager, calculate the number of days required if you have a workforce of 4 people who can work for 6 hours a day?
0
votes
2answers
70 views

Root of an exponential equation

Let $0 \le a \le 1$ and $-\infty < b < \infty$. I am looking for a solution of the exponential equation. $$ a^x + abx = 0. $$ I guess closed form expression of the root in terms of $a$ and $b$ ...
0
votes
1answer
132 views

Find the value of this logarithmic expression involving fifth root of unity.

Let $\alpha$ be the fifth root of unity. We then want to evaluate the expression $$\log |1 + \alpha + \alpha^2 + \alpha^3 - 1/\alpha |$$ Thanks in anticipation for your help in solving this!
4
votes
1answer
195 views

Find the value of $\sqrt{i+\sqrt{\frac12i+\sqrt{\frac13i+\dots}}}$

Find the value of $\sqrt{i+\sqrt{\frac12i+\sqrt{\frac13i+\dots}}}$ . Is it convergent, even ?
2
votes
4answers
1k views

For what natural numbers is $n^3 < 2^n$? Prove by induction

Problem For what natural numbers is $n^3 < 2^n$? Attempt @ Solution For $n=1$, $1 < 2$ Suppose $n^3 < 2^n$ for some $n = k \ge 1$ It looks like the inequality is true for $n = 0$, $n = 1$ ...
2
votes
2answers
125 views

How to solve $x+3+\sqrt[3]{(x+1)(x^2-x+2)+\sqrt[3]{x^3+x+1}}+\sqrt[3]{x^3+x+1}=0$

solve the equalition $$x+3+\sqrt[3]{(x+1)(x^2-x+2)+\sqrt[3]{x^3+x+1}}+\sqrt[3]{x^3+x+1}=0$$ I have seen some methods, for this problem. my idea: let $\sqrt[3]{x^3+x+1}=y,$ then ...
0
votes
1answer
70 views

Algebra has me hitting a wall

Math is not my strong suit. As far as I can tell, this is what I'm looking to solve. A+B=C and B=C*D If one knows what A and D equal, can one determine the value of B and C? So far, I've tried: ...
2
votes
4answers
2k views

Calculating the number of times a digit is written when given two numbers

My homework asks me the following: If a student writes the integers from 5 to 305 inclusive by hand, how many times will she write the digit 5? I started out by writing every number that ...
-1
votes
1answer
75 views

How to find $f(xy)$ and $f(yx)$ if $f(x)$ is given [closed]

I have to find the solution of this problem.But I dont know how to start.What approach I should follow Find $f(xy)$ and $f(yx)$ if $f(x) = 3x^2-4xy+2y^2$
2
votes
2answers
101 views

How can i solve this equation $|x|\hat x= 1991x$

I found this problem in an old exam and i want to know how to do it, since i couldn't at the time, it's in spanish so i'll leave my translation and the original: Solve this equation $|x|\hat x= ...
1
vote
2answers
184 views

Word problem involving equations

Two bike riders $X$ and $Y$ both start at 2 PM riding towards each other from $40$ km apart. $X$ rides at $30 \frac{\mathrm{km}}{\mathrm{h}}$, and $Y$ at $20 \frac{\mathrm{km}}{\mathrm{h}}$. If they ...