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
votes
0answers
9 views

What function gives this inequality?

Let $i<j<k<l$ be positive integers. I want to find a "nice" function $f(x, y)$ such that $f(i, k)+f(j, l)>\max(f(i, j)+f(k, l), f(i, l)+f(j, k))$. This seems a bit tricky because the ...
0
votes
1answer
22 views

algebraic rearrangement, $C=h/[m(t_1-t_2)]$ Solve for $t_1$

$$ C=\frac{h}{m(t_1-t_2)} $$ Solve for $t_1$. The correct answer I have been given is, $t_1=t_2-h/(mC)$. I just need help in the steps taken to reach this.
0
votes
1answer
37 views

When will they meet up together? [closed]

Jabal and Michael are walking to school and agree to leave at the same time. Jabal lives 100 meters closer to school. Jabal walks 2 meters per second. Michael walks 2.5 meters per second. When ...
1
vote
1answer
46 views

Find “almost inverse” of positive definite bilinear form

Let $A$ be a positive definite $d \times d$ matrix, and define $A(x,x)=x^TAx$. Let $x$ be a point such that $\vert x^T\xi\vert^2\leq \xi^T A\xi$ for all $\xi\in\mathbb{R}^d$. Is this somehow ...
0
votes
2answers
38 views

what is the n-k derivative of $x^n$? Also, why is $n!/k! = …$

I am having troubles finding $\frac{d^{n-k}x^n}{dx^{n-k}}$ where $ k \leq n$ I believe it is equal to $n(n-1)(n-2)....k(k+1)x^k$ but htis is just from obersation, I do not know why it's that exactly. ...
0
votes
3answers
64 views

For what values ​​of a intersects $y = ax$, $y = \sin x$ just one time?

As the title says, for what values of a intersects $y = ax$, $y = \sin x$ just one time? I am not able to solve this problem, and I really want to know the answer.
-3
votes
2answers
38 views

find the value of $k$ in the term $2^{-k} = 1/n$

What is the value of $k$ if I have the following equation: $2^{-k} = \frac1n$? $$2^{-k} = \frac 1 n \implies n = 2^k \implies \log_{2} n = k$$ Is my solution correct?
2
votes
3answers
65 views

$\sin2(x) - \tan(x) = 0$ , solve for $-180\le x\le 180$

I have been unable to solve the following question, If $$\sin(2x) - \tan(x) = 0$$ Find $x$ , $-\pi\le x\le \pi$ So far my workings have been Use following identity: $$\sin(2x) = ...
3
votes
4answers
62 views

Number of integer solutions of $\frac{1}{x}+\frac{1}{y}=\frac{1}{2016}$ [closed]

How can we find number of integer solutions of $\frac{1}{x}+\frac{1}{y}=\frac{1}{2016}$ I want to ask what approach in general should be followed in such types of question?
1
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3answers
49 views

Which fraction is more

Can anyone help me prove why $$1-\left(\frac{x-10}{y}\right) \gt 1-\left(\frac{x}{y+10}\right)$$ when $y < x$. I have no idea how to show this algebraically so i'd really like some guidance.
3
votes
2answers
49 views

quadratic simultaneous equation

Solve simultaneously: $$ 12x^2-4xy+11y^2=64 $$ $$ 16x^2-9xy+11y^2=78$$ I understand that it can be solved using the quadratic formula by rearranging the equation in $ax^2+bx+c=0 $ form $$ ...
0
votes
1answer
31 views

Order of math operations

I know that this seems a childish question but i have not been able to find a proper convention that cover all math operations. Obviously the basic is that exponential takes precedence over ...
0
votes
1answer
46 views

Do we have $\frac{1}{a} - \frac{1}{b} = b - a$?

I am attempting to prove that $$\frac{1}{E'} - \frac{1}{E} = \frac{1}{m_e c^2} \cdot (1-\cos\theta)$$ can be derived from $$E + m_ec^2 - E' = c^2(p^2 - 2pp'\cos\theta + p'^2) + m_e^2c^4 $$ where ...
1
vote
1answer
25 views

Supremum of integral polynomial near origin

Let $P(x,y)$ be a polynomial with integer coefficients that is constant neither in the horizontal nor vertical direction. Prove that $\sup_{-2\leq x,y\leq 2}|P(x,y)|\geq 4$. I suspect we might be able ...
0
votes
1answer
18 views

How to use Rolle's theorem to verify the following?

How to use Rolle's theorem to verify the location of roots ? $f(x)=x^3+4/x^2+7$ has exactly one zero in ($-\infty$,$0$) I can do it without Rolle's theorem by finding the stationary point which is ...
0
votes
1answer
14 views

Difficulty simplifying nested sums with different variables

I'm trying to work out an algorithm analysis problem, and I'm having some difficulty determining how a jump is made between two steps in the answer. $$ \begin{align} ...
3
votes
4answers
41 views

$2x(1-x)$ is not onto?

How come $4x(1-x)$ is onto in $[0,1]$ but $2x(1-x)$ is not? Isn't it true that for any $y$ in the range interval, there exist two $x$ such that $f(x)=y$?
0
votes
2answers
17 views

Find real points of intersection of the equations algebraically [closed]

Can you help me to find pints of intersection of given system of equesions? Can't do it by myself. $xy + x - 2y + 3 = 0$ $x^2 + 4y^2 - 9 = 0$
18
votes
4answers
319 views

An obvious pattern to $i\uparrow\uparrow n$ that is eluding us all?

Start with $i=\sqrt{-1}$. This will be $a_1$. $a_2$ will be $i^i$. $a_3$ will be $i^{i^{i}}$. $\vdots$ etc. In Knuth up-arrow notation: $$a_n=i\uparrow\uparrow n$$ And, amazingly, you can ...
0
votes
1answer
39 views

Proving the primality of these large numbers?

In 2007, Vautier claimed that the largest known consecutive pair of prime numbers (at the time) was $2003663613\cdot2^{195000}-1$ and $2003663613\cdot2^{195000}+1$. I was wondering how Vautier found ...
0
votes
4answers
37 views

Simple Fraction needing explanation

$$\frac{x}{x^{-1/2}} = x^{3/2}$$ How? I don't see what is going on here. What rule is being used to achieve this amount?
4
votes
3answers
44 views

Find the two values of $k$ for which $2x^3-9x^2+12x-k$ has a double real root.

Find the two values of $k$ for which $2x^3-9x^2+12x-k$ has a double real root. I've found one method which is to equate $$2x^3-9x^2+12x-k=2(x-r)^2(x-c)$$ Expanding and equating coefficients I ...
1
vote
1answer
48 views

Why the identity $P_X=P_XZ(Z'P_XZ)^{-1}Z'P_X$ with $P_X=X(X'X)^{-1}X'$?

Suppose $X$ and $Z$ are matrices such that $(X,Z)$ and $P_XZ$ both have full column ranks. Here, $P_X=X(X'X)^{-1}X'$. Consider a regression model $$ P_Xy=P_XZ\zeta+v\tag{A} $$ where OLS is used ...
0
votes
3answers
34 views

Average rate of change at an exact point

So I had this question in my precalculus text book: Water is draining from a large tank. After $t$ minutes there are $160,000-8000t+t^2$ gallons of water in the tank. Estimate the rate at which ...
1
vote
3answers
20 views

simplifying complex fractions when proving inverses with function composition?

I'm working with two functions, $f(x)=\frac{x-3}{x+4}$ and $g(x)=\frac{4x+3}{1-x}$. I need to simplify $f(g(x))$ and its opposite, but i'm not sure of the procedures regarding the more complicated ...
6
votes
3answers
65 views

Finding the square root of $6-4\sqrt{2}$

I found this standupmaths video on YouTube about the A4 paper puzzle. I really liked the puzzle and managed to get the answer by using a calculator. However, the answer (which I won't spoil), led me ...
0
votes
0answers
35 views
+50

Range of inverse harmonic mean of two integers

Today I was solving an exercise and one of the things I tried (which later turned out to be useless) involved considering the following: Is there a simple way to describe in terms of $n$ the range of ...
2
votes
4answers
45 views

Show that $6^n/n! \le 6^5/5! \times 6/n$

I want to show that $$\frac{6^n}{n!} \le \frac{6^5}{5!} \cdot \frac 6n$$ without using induction, which I've done but is rather clunky. Is there a more straight forward way of doing this?
1
vote
0answers
14 views

Find the rule connecting $y$ and $x$ in the form $y=mx+c$ for $1<x≤2$ [closed]

I have attempted numerous methods but I'm not sure how to interpret the $1<x≤2$ into a rule relating $y$ and $x$.
4
votes
3answers
35 views

Prove that $a(x+y+z) = x(a+b+c)$

If $(a^2+b^2 +c^2)(x^2+y^2 +z^2) = (ax+by+cz)^2$ Then prove that $a(x+y+z) = x(a+b+c)$ I did expansion on both sides and got: $a^2y^2+a^2z^2+b^2x^2+b^2z^2+c^2x^2+c^2y^2=2(abxy+bcyz+cazx) $ but ...
0
votes
2answers
40 views

Prove $(a_1 + a_2 + a_3)^2 = a_1^2 + a_2^2 + a_3^2 + 2(a_1a_2+ a_2a_3 + a_1a_3)$

I wish to find a proof for this equality: $(a_1 + a_2 + a_3)^2 = a_1^2 + a_2^2 + a_3^2 + 2(a_1a_2+ a_2a_3 + a_1a_3)$ But then I realized there exists a more general version: $(\sum\limits_{k=1}^n ...
-1
votes
3answers
71 views

Distance between a circle and a line

Find the distance between the circle $(x-3)^2+(y+2)^2=4$ and the line $x + 2y = 9$.
4
votes
5answers
65 views

Why is the solution to $\sqrt{6-5x}=x$ only $x=1$ and not $x=-6$? [duplicate]

I solved the equation $\sqrt{6-5x}=x$ as follows: $$(\sqrt{6-5x})^2=x^2$$ $$6-5x=x^2$$ $$0=x^2+5x-6=(x+6)(x-1)$$ $$x=-6 \quad \text{or} \quad x=1$$ If I plug in $x=-6$ into the original equation, I ...
0
votes
1answer
25 views

Tangent meets curve again

If the tangent at the point $(16,64)$ on the curve $y^2=x^3$ meets the curve again at at $Q(u,v)$ then $uv$ is ? If found the tangent to the curve at $(16,64)$ but then I cannot find $uv$.Give your ...
0
votes
2answers
27 views

How to rewrite an expression

Let's say $Z=Y_1+Y_2$. I have this expression: $Y_1!Y_2!$. I want to rewrite the expression and express it by only $Z$. Is that possible?
2
votes
1answer
71 views

How does one prove that $2\uparrow\uparrow16+1$ is composite?

Just to be clear, close observation will show that this is not the Fermat numbers. I was reading some things (link) when I came across the footnote on page 21, which states the following: ...
0
votes
1answer
47 views

The uniqueness of roots of Quartic function

Define $$ f(x):=(1+ax)^3x-a(1+x)^3. $$ Would it be possible to prove that the function $f$ has only one positive real root provided that $a>0$? (There might be another root $x_0<0$, but I only ...
-2
votes
2answers
77 views

Solve for $X: X^{49} = 60$

An unknown value is raised to a known power which results in another value, also known. How do you find the unknown value?
3
votes
1answer
216 views
+200

The root of summation function

This is a calculation I need for my statistics project Big edit: simplify the function $f(x)$ a lot. Define for $f(x)$, $x\geq 0$, $$ f(x):=\sum_{k=1}^\infty ...
1
vote
0answers
31 views

Use the arithmetic-geometric inequality for this list to deduce the arithmetic-geometric inequality for $n$.

Suppose that $n$ is not a power of two. Let $2^k$ be a power of $2$ that exceeds $n$ and consider the list $$a_1,\dots,a_n,\underbrace{A,A,\dots,A}_\text{$2^k-n$ times}$$ of length $2^k$. Use the ...
4
votes
1answer
52 views

Polynomial divides set of points

Given a set of points in the plane with distinct $x$-coordinates, each point colored black or white. A polynomial $P(x)$ "divides" the set of points if no black point lies above $P(x)$ and no white ...
-3
votes
1answer
32 views

Formula $\sqrt[x]a -\sqrt[x] b$ [closed]

It is correct ? 1.) $$(\sqrt[x]{a} - \sqrt[x]{b}) \dot\ \sum_{k=0}^{\ x-1} a^{\frac{\ x-k-1}{x}} \dot\ b^{\frac{\ k}{x}}=a-b$$ 2.) $$ \lim\limits_{x \to \infty} \sum_{k=0}^{\ x-1} a^{\frac{\ ...
1
vote
1answer
19 views

Creating a four dimensional system to find a point

How can I create a four-dimensional system that has a solution of $(-2,5,-6,1)$? I know how to solve a system for its solution, but how do I work backwards?
0
votes
1answer
24 views

Line of intersection between two planes

The questions asks: Determine the line of intersection in vector, parametric, and Cartesian between the following sets of planes: $2x-y+2z+1=0$ and $-4x+2y-4z-2=0$ I realize these are parallel to ...
4
votes
2answers
72 views

Solve $\sqrt[3]{7x+19}+\sqrt[3]{7x-19}=\sqrt[3]{2}$ by algebraic methods

I was trying to solve this equation without using calculus. Is it possible to be solved by elementary algebraic methods? $$\sqrt[3]{7x+19}+\sqrt[3]{7x-19}=\sqrt[3]{2}$$
3
votes
2answers
49 views

Representation of roots of unity.

How to represent solutions of $\sqrt[26]{1}$ with solutions of $\sqrt[26]{-1}$? I know that $$w_{k}=\cos\left(\frac{0+2k\pi}{26}\right)+i\sin\left(\frac{0+2k\pi}{26}\right), \; \; ...
1
vote
2answers
42 views

Does the inverse of $f(x)=x^3$ have a non-negative domain to have a real output?

I'm not familiar with complex analysis. While playing with Mathematica (a mathematics software), I found that it keeps spitting out unexpected results, and the reason was that it considers differently ...
0
votes
3answers
46 views

How to prove geometrically the difference between line and plane?

Explain geometrically why $(x,y,z) = (1,9,17) + s(1,1,1) + t(-2,-2,-2)$ represents the equation of a line and not a plane. EDIT: the extension to this question asks "based on the above answer, does ...
1
vote
2answers
52 views

For which $a$ and $b$ the function $\frac{ax+b}{a^2x+b^2}$ is increasing?

For which $a$ and $b$ the function $$\frac{ax+b}{a^2x+b^2}$$ is increasing? I know that function is increasing if $x_1 > x_2 \implies f(x_1)>f(x_2)$ but how can I find $a$ and $b$ for ...
2
votes
0answers
9 views

Vector equation of line containing point and perpendicular to plane [duplicate]

How would one find the vector equation of the line that contains the point (x0, y0, z0) and is perpendicular to the plane Ax + By + Cz = D?