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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1answer
22 views

Steps to express $\ln \frac{8 \cdot 4^{1/3}}{\sqrt 2}$ as $\frac{19}{6}\ln 2$

Self-studying here. What are the steps to express $\ln \frac{8 \cdot 4^{1/3}}{\sqrt 2}$ as $\frac{19}{6}\ln 2$? I know the usual rules for manipulating logs and have tried several times to get the ...
0
votes
2answers
31 views

Using equation of a line to find infinitely many different solutions of $ x^2 - 2y^2 = 1$

By taking a line through $(1, 0)$ it is possible to find a point where the line crosses another point of $x^2 - 2y^2 = 1$. Then we could take a line that passes through the that point and find another ...
0
votes
1answer
20 views

How to bound this difference between two logarithmic expression

I want to bound the difference between two logarithmic expression shown below with a constant number i.e not function of $x,y,z$ where $x,y,z \in \mathbb{C}$. The difference is $$ ...
7
votes
6answers
125 views

Solutions to $\frac{1}{a} + \frac{1}{b} = \frac{1}{100}$?

I encountered this problem yesterday and successfully solved it. I'm interested in seeing other people's approach to solving this problem. Problem: How many ordered pairs $(a, b)$ are solutions to ...
0
votes
2answers
20 views

Show that there is a large value

Suppose not all 4 integers, $a, b, c, d$ are equal. Start with $(a, b, c, d)$ and repeatedly replace $(a, b, c, d)$ by $(a - b, b - c, c - d, d - a)$. Then show that at least one number of the ...
1
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5answers
66 views

Why $\frac{x}{\sqrt{x+1}-1}$ can be written as $\sqrt{x+1}+1$?

I've evaluated the first formula on $W|A$ and it says that $\sqrt{x+1}+1$ is an alternate form to the first expression. I just don't see how it's possible. The first thing I imagined was to write: ...
3
votes
3answers
41 views

Show that $\arctan x+\arctan y = \pi+\arctan\frac{x+y}{1-xy}$, if $xy\gt 1$

Show that $\arctan x+\arctan y = \pi+\arctan\frac{x+y}{1-xy}$, if $xy\gt 1$ I am stuck at understanding why the constraint $xy\gt 1$. Here is my work so far let $\arctan x =a\implies x=\tan a$ let ...
2
votes
2answers
31 views

To mark up in retail by $20$%, do I add $0.20$ times the original cost, or divide by $0.80$?

Why is it that when I take a cost of say $\$15.60$ and want to mark the item up at retail 20% that I'm being told two different ways with two different answers? The first way (my way) would be to ...
2
votes
2answers
76 views

Prove that $a^2+b^2+c^2\geq [2(a-b)^2(b-c)^2(a-c)^2]^{1/3}$

Mathematica seems to know that this statement is true, yet I am struggling to prove it. Possible useful inequalities are Minkowski and the geometric mean. Using the geometric mean inequality I can ...
2
votes
1answer
38 views

Find with proof all solutions to $2^n = a! + b!$, where $a$, $b$, $n$ are positive integers and $a \leq b$.

So far I have looked at $n=1$ with $a=1$ and $b=1$ which is $$2^1 = 1! + 1! = 2 $$ and $n=2$ with $a=2$ and $b=2$, $$2^2 = 2!+ 2! = 4$$ and finally $n=3$ with $a=3$ and $b=2$, $$2^3 = 3! + 2! = 8$$ I ...
-2
votes
0answers
11 views

volume of 3 prisms (one is one third the other and 2 are equal) [on hold]

three rectangular prisms have a combined volume of 518 cubic feet. Prism A has one third the volume of prism B. Prisms B and C have equal volume. What is the volume of each prism?
0
votes
1answer
23 views

Solving the equations .

Say , I have two equations : $$y_1=a+bx_{1}+e_1$$ $$y_2=a+bx_{2}+e_2$$ Say , $a=.5$ , $b=2.1$ , $x_1=2$ , $x_2=2.2$ . Now if $e_1=e_2$ , I have to find the relationship between $y_1$ and $y_2$ . ...
3
votes
2answers
69 views

Why do we need to rationalize fractions? [duplicate]

Teachers often take off points from students who write 1/sqrt(2) instead of sqrt(2)/2. Why do we need to write it as sqrt(2) / 2 ? Where did that convention come from? Do we need to even do it? Why do ...
2
votes
2answers
56 views

Is $ (-1)^n(x-a)^n = (a-x)^n?$ If not, why?

I came across this during an attempt at a Taylor series expansion (which I'm not very good at yet), and assumed this would be true because $(ab)^n = a^nb^n$. Plugged it into Wolfram Alpha, though, and ...
0
votes
4answers
58 views

Let $x$ and $y$ be real numbers such that $x^2 + y^2 = 1$. Find the maximum value of $2x - 5y$.

Let $x$ and $y$ be real numbers such that $x^2 + y^2 = 1$. Find the maximum value of $2x - 5y$. I do know how to solve this problem using trigonometry, however I need to solve it by using vectors. ...
3
votes
2answers
71 views

Problem in deducing the number of onto functions

Let $A, B$ have $m, n$ elements ($m > n$). Therefore, the number of onto functions from $A$ to $B$ is: $$\sum_{k = 0}^n (-1)^k \binom{n}{k} (n - k)^m.$$ How can one use the IE (Inclusion/Exclusion) ...
0
votes
0answers
25 views

Increasing/ decreasing functions

We are given a random variable x with a pdf f(x) and F(x) is its distribution function. Let $$r(x) = \frac {xf(x)} {1-F(x)} $$ Then for $x< e^{\mu} $ and $$f(x) = \frac {e^ {1/2(\log x - \mu)^2}} ...
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votes
3answers
57 views

Looking for value of equation

I have been trying to solve this MONBUKAGAKUSHO past test papers, and I am completely stuck. I have no single idea how to solve it, and I´ve tried many different things, but without any results. Here ...
0
votes
1answer
15 views

What polynomial with real coefficients generates all polynomials with real coefficients that satisfies $f(2+i)=0$?

What polynomial with real coefficients generates all polynomials with real coefficients that satisfies $f(2+i)=0$? Obviously, the polynomial $f(x)=x-2-i$ satisfies the constraint but does not have ...
1
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3answers
29 views

Divisibility of integers by integers

We are given a number $$K(n) = (n+3) (n^2 + 6n + 8)$$ defined for integers n. The options suggest that the number K(n) should either always be divisible by 4, 5 or 6. Factorizing the second bracket ...
-1
votes
0answers
16 views

distance between normal and axis? [on hold]

The job is finding the distance between the normal to the plane $x-y+4z-10=0$ and the $z$ axis...I cannot really imagine how can one do this?
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votes
0answers
17 views

angle between normal and axis? [on hold]

I was tasked with finding the angle between the normal to the plane $x-y+4z-10=0$ and the $z$ axis...I cannot really imagine how can one do this?
0
votes
3answers
62 views

What is $n+1$ factorial or $(n+1)!$? [on hold]

I have to prove by induction but first I want to know what $(n+1)!$ is? I know that $n!=n \cdot (n-1) \cdot (n-2)...$
1
vote
0answers
13 views

Find the equation of intersection of a torus and a circle on a plane without using iterative methods.

I have the equation of a circle on the plane (where $p_0$ is the centre, $\theta$ is the angle of the circle, and $w$ and $v$ are pair of orthogonal vectors from $p_0$ to the circle (having equal ...
5
votes
2answers
204 views

Solution of a Lambert W function

The question was : (find x) $6x=e^{2x}$ I knew Lambert W function and hence: $\Rightarrow 1=\dfrac{6x}{e^{2x}}$ $\Rightarrow \dfrac{1}{6}=xe^{-2x}$ $\Rightarrow \dfrac{-1}{3}=-2xe^{-2x}$ ...
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0answers
17 views
2
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3answers
34 views

Area of a parallelogram with three dimensional vectors

There is a parallelogram that has the vertices 0, a, b, and a+b, all of which are three dimensional vectors. a = \begin{pmatrix} 2 \\ -6 \\ 5 \end{pmatrix}b = \begin{pmatrix} -1 \\ -2 \\ 0 ...
1
vote
1answer
28 views

$K$ is a region in $\mathbb{R}^2$ where the area is $5$

Say that $K$ is a region in $\mathbb{R}^2$ where the area is $5$. Let B = \begin{pmatrix} 3 & 8 \\ 4 & 6 \end{pmatrix} Find the area of the region B$K$. Any starting hints? Is it possible ...
0
votes
3answers
45 views

Square Root Confusion

well we know that $$\sqrt{x^2} = \pm x$$ Then if $$x^2=y^2$$ then $$\pm x= \pm y$$ Does this mean $x = y$ or $-x = -y$ or $x = -y$ or $-x = y$ or all is true? Which is true among these?
4
votes
4answers
78 views

Proving $\binom{m}{n} + \binom{m}{n-1} = \binom{m+1}{n}$ algebraically

I am working through the exercises and have spent half a day on one problem so I decided to get some help because I can't figure it out. Show that if $n$ is a positive integer at most equal to $m$, ...
0
votes
1answer
22 views

Find the largest segment

I have seven lines with different measures. The length of each line it's a positive integer and the shortest length is equal to 1 cm. It is known that's impossible to choose three of them that makes a ...
-3
votes
1answer
60 views

Can someone help me why this equation equals zero?

I played around with some numbers and stuff and made this weird equation: $$\huge x^{- \frac{n}{n^{-x}}}$$ So the thing is, with every number I tried typing this into a calculator, I got 0. Can ...
-1
votes
1answer
30 views

Multiplying logarithms of different bases [on hold]

How do you multiply the following logs... $$\log_5(n) * \log_2(n)$$
0
votes
2answers
25 views

How to prove that eventually $(x^p/e^{x^q}) < 1/(x^2) $ for $p,q>0$

How to prove that eventually $x^p/\exp(x^q) < 1/(x^2) $ for $p,q>0$. I tried showing that $x^{p+2} > \exp(x^q)$ by using the Taylor expansion of e but this didn't really work.
3
votes
2answers
190 views

Simple 2 equations and 2 unknowns

I am reading the second partial derivative test example, but I am suck on the following step: $$f(x,y) = -x^3 + 4xy - 2y^2 + 1$$ And we have the partial derivatives as follow... $$f_x(x,y) = -3x^2 ...
4
votes
2answers
34 views

Roots of unity, where $\omega^3 = 1, \omega \neq 1$.

Say that $\omega^3 = 1$ and $\omega \neq 1$. Find the value of $(1 - \omega + \omega^2)(1 + \omega - \omega^2)$. I'm not very good at the roots of unity. May I have a couple of hints to get started? ...
4
votes
1answer
188 views

Partial fractions - different results when done in steps than not

We have: $\frac 1 {(1-x)(1+x)(1-2x)}$ If I do the partial fractions straight: $\frac 1 {(1-x)(1+x)(1-2x)}= \frac a {1-x} + \frac b {1+x} + \frac c {1-2x}$ I get: $a=-\frac 12, b = \frac 1 6, c=\frac ...
3
votes
1answer
31 views

Is there any solution to this quadratic Diophantine 3 variables equation?

Is it possible to find all positive integer triplets $(x,y,z)$ satisfying the parametric equation : $$x^2 + 2ax + y^2 + 2by = z^2 + 2cz$$ Here $a, b, c$ are fixed positive integers.
0
votes
1answer
27 views

Precalculus: Velocity addition

A boat is rowed 6.4 km up a river and back again and this takes in total 2 hours. The stream velocity is 2.4 km/h. What velocity would the boat have been moving in if the water was standstill? I ...
4
votes
2answers
43 views

If $|x|\leq 1\;,|ax^2+bx+c|\leq 1\;,$ Then $\bf{Max.}$ possible value of $|2ax+b|\;$

If $a,b,c\in \mathbb{R}$ and If $|x|\leq 1\;,|ax^2+bx+c|\leq 1\;,$ Then $\bf{Max.}$ possible value of $|2ax+b|\;$ is, Where $-1 \leq x\leq 1$ $\bf{My\; Try::}$ Put $x=1$ in ...
12
votes
7answers
1k views

How do we prove this logarithm?

Given: $$\dfrac{\log x}{b-c}=\dfrac{\log y}{c-a}=\dfrac{\log z}{a-b}$$ We have to show that : $$x^{b+c-a}\cdot y^{c+a-b}\cdot z^{a+b-c} = 1$$ I made three equations using cross multiplication : ...
1
vote
1answer
16 views

Finding orthonormal basis using orthogonal basis

I am very confused how to go about finding an orthonormal basis using a orthogonal basis. My book says to just normalize the vectors but it doesnt further explain. When i look at answers for ...
4
votes
3answers
52 views

If $f(x) = \sin^4 x+\cos^2 x\;\forall x\; \in \mathbb{R}\;,$ Then $\bf{Max.}$ and $\bf{Min.}$ value of $f(x)$

If $f(x) = \sin^4 x+\cos^2 x\;\forall x\; \in \mathbb{R}\;,$ Then $\bf{Max.}$ and $\bf{Min.}$ value of $f(x).$ My Solution:: Let $$\displaystyle y = \sin^4 x+\cos^2 x \leq \sin^2 x+\cos^2 x=1$$ ...
0
votes
0answers
12 views

Generalization of minimisation problem

First I would like indtroduce my problem ! There is an easy way to solve this one : Find the value of $$ \inf_{(a,b)\in \mathbb{R}^2} \int_0^1 (t^2-at-b)^2 dt $$ and precise for which values $a$ ...
-4
votes
0answers
50 views

Mathematics doubt. [on hold]

What is mathematics. I need a proper definition mathematics.
0
votes
3answers
31 views

When is $\theta$ obtuse or acute in sin, cosine, tan when they are positive, negative or both?

My textbook gives a non intuitive answer and tells us to memorize when the ratios are positive or negative or both based on some arbitrary rule that I don't understand. I know how to do both of ...
0
votes
1answer
26 views

$b^{\frac{m}{n}}=(b^{\frac{1}{n}})^m=(b^m)^{\frac{1}{n}}$ except $b$ is not negative when $n$ is Even.

The following property, known as Rational number property, is taken from the book (I am following now a days) College Algebra by Raymond A Barnett and Micheal R Ziegler I restate, ...
2
votes
1answer
20 views

given the following two conditions, find $f(x,y)$

Suppose that a function $f$ defined on $\mathbb R^2$ satisfies the following conditions: $f(x+t,y)=f(x,y)+ty$; $f(x,t+y)=f(x,y)+tx$; $f(0,0)=k$; then for all $x,y \in\mathbb R$, $f(x,y)=$ a) ...
-3
votes
1answer
52 views

What grade does Bob need on his final to pass his math class?

Bob's teacher has a syllabus where the grading breakdown is as follows: Homework: 10% Tests (2): 60% Final Exam: 30% Bob receives all 10% from the homework Bob receives 78/100 on Test 1 (78%) ...
0
votes
2answers
29 views

Vector Magnitude problem

I do not understand how to set up the following problem: "Forces of 20 lb and 32 lb make an angle of 52 degrees with each other. find the magnitude of the resultant force." An actually picture would ...