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.

learn more… | top users | synonyms (2)

0
votes
0answers
35 views
0
votes
0answers
11 views

What is the angle that an Archimedean conical spiral makes with the floor?

I have a spiral in the form $$r = r_0(1-{\theta\over2\pi k }) \{r >= 0\}$$ Where $r_0$ is an initial radius, and k is the number of turns. (it is a spiral that decays from $r_0$ to 0 as $\theta$ ...
12
votes
2answers
387 views

Interesting Question on Ants

A horizontal stick is one metre long. Fifty ants are placed in random positions on the stick, pointing in random directions. The ants crawl head first along the stick, moving at one metre per minute. ...
0
votes
1answer
14 views

Unclear Application of Cauchy's Inequality

I was looking for a solution to a problem (both found here), where I came across the following ($a, b, c > 0$): Applying Cauchy's inequality, we get $(\frac{c}{a+2b} + \frac{a}{b+2c} + ...
0
votes
1answer
24 views

How can I solve this expression for x?

I would like to solve for $x$ given that \begin{equation} e^{-x}-\gamma-\eta e^{-\lambda(z-x)} = 0 \end{equation} where $\gamma, \eta, \lambda$ are positive constants and $z$ is a real number.
0
votes
1answer
24 views

questions related to progression [on hold]

Along a road lie an odd number of stones and distance between consecutive stones is 10m. A person can carry only one stone at a time and his job is to assemble all the stones around the middle stone. ...
0
votes
1answer
21 views

Giving a geometric representation of Cartesian products

What is being asked of me? Question 4 of Zorich(page 11) is exactly the following Give geometric representations of the following Cartesian products a) The Product of two line segments (a ...
0
votes
0answers
14 views

Is there a way to arrive at a funtion or a formula based on the outcome

The following table shows the input and the output. I'm trying to create a function that would relate the input and the output. SNU C020 C100 C300 C600 0 0 0 0 0 ...
3
votes
3answers
62 views

Solve: $\sin x - y\cos x = z$ for $x$.

I am working on programming a series of algorithms into a project, however I have run into trouble trying to solve this equation for $x$: $$ \sin x - y\cos x = z $$ It should be noted that $y$ and ...
0
votes
3answers
41 views

Best argument to prove $|x|\le a \iff -a\le x \le a$

$$|x|\le a \iff -a\le x \le a$$ I can only verify the integrity of this by talking about distances on the number line. But is there a algebraic argument that proves this?
0
votes
0answers
10 views

Locus of intersection between $y= 8\lambda/(\lambda ^2 + 4)$ and $y =2 \lambda x/(4-\lambda^2)$

I have the equations $$y=\frac{4\lambda}{\frac{1}{2}\lambda^2+2}\quad \text{and}\quad y=\frac{\lambda x}{-\frac{1}{2}\lambda ^2 + 2}$$ each representing a line. I'm asked to find the locus of the ...
0
votes
3answers
41 views

simplifying $-\pi i/8 (e^{i\pi/8} + e^{i3\pi/8} + e^{i5\pi/8} + e^{i7\pi/8})$

simplifying $-\pi i/8 (e^{i\pi/8} + e^{i3\pi/8} + e^{i5\pi/8} + e^{i7\pi/8})$ in my lecture notes somehow my lecture got from$-\pi i/8 (e^{i\pi/8} + e^{i3\pi/8} + e^{i5\pi/8} + e^{i7\pi/8})$ to ...
0
votes
0answers
35 views

Neutron-Density cross-plot interpretation

I have a question about solving a particular graphical problem. This is a picture of a Neutron-Density cross-plot: It's a little bit confusing as plots go, so allow me to try to explain the salient ...
-1
votes
0answers
19 views

Problem about a focal chord

Given parabola $y^2=4ax$ with length of the focal chord equal to $l$ and the length of the perpendicular from vertex to the chord is $p$. Which one of these statements is true? 1) $l⋅p$ is constant ...
4
votes
2answers
59 views

Solving $\sin(2v) = \sin(v)$

$$\sin(2v) = \sin(v)$$ Why can't this equation be solved by setting: $$2v = v + 2\pi n \quad \leftrightarrow \quad v = 2\pi n\\2v = \pi - v + 2\pi n \quad\leftrightarrow \quad 3v = \pi + 2\pi n ...
1
vote
2answers
39 views

Inequality using only algebraic ''moves''

How can I verify the following inequality using only algebraic passages? $$ 5^\frac{1}{3} + 6^\frac{1}{2} > or < 4 $$
1
vote
1answer
41 views

Beautiful sines equation

If $θ_1,θ_2,θ_3,θ_4$ are four real numbers, then any root of the equation $\sinθ_1z^3+\sinθ_2z^2+\sinθ_3z+\sinθ_4$=3, lying inside the unit circle $\vert z\vert$=1, satisfies which inequality? ...
0
votes
0answers
9 views

no. of solution of the equation $\arccos(1-x)+m\cdot \arccos(x) = \frac{n\pi}{2}\;$

(1) The no. of solution of the equation $\displaystyle \arccos\left(\frac{1-2x-x^2}{(x+1)^2}\right) = \pi\left(1-\{x\}\right)\;,$ Where $x\in \left[\;0,76\;\right]$ Where $\{x\}$ denote fractional ...
2
votes
1answer
59 views

Beautifully looking little geometry/trigonometry problem

Given triangle ABC, a,b,c as its sides, p is a half perimeter, such that $\dfrac{p-a}{11}=\dfrac{p-b}{12}=\dfrac{p-c}{13}$. We need to find $(\tan\dfrac{A}{2})^2$ (A)$\dfrac{143}{432}$ ...
1
vote
5answers
48 views

solve this equation for $x$ : $y=x-6\sqrt{x}$

solve for $x$ this equation : $$y=x-6\sqrt{x}$$ I've tried raising everything to the power of two but it doesn't work $x$ shouldn't have two values.
0
votes
1answer
24 views

How to find $f^{−1}([9,0])$ and $f([1,4])$ for $f(x)=x-6\sqrt{x}$?

$f$ is a the function defined by $$\eqalign{ f\colon& \Bbb R &\rightarrow \Bbb R_+\\ & x&\mapsto x-6\sqrt{x} }$$ Find $f^{−1}([-9,0])$ and $f([1,4])$.
0
votes
1answer
26 views

simple problem of calculus.

A company wishes to manufacture a box with a volume of $36ft^3$ that is open on top and twice as long as it is wide.Find the dimensions of the box produced from the minimum amount of material. My ...
3
votes
4answers
55 views

Finding inverse of a function $h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$

I have a function: $$h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$$ With just pen and paper, how can I determine if there exists an inverse function? Am I supposed to sketch it on paper to see if it can ...
4
votes
3answers
187 views

Why do non-real solutions of a polynom occur pairwise complex-conjugated?

So if I have a polynom with real coefficients and the solution $x+iy$, why is $x-iy$ always a solution too? Let $z$ and $w$ be complex numbers, with $w^{\ast}$ = complex-conjugated of $w$, then ...
3
votes
1answer
22 views

Specific piecewise-function SAT2 question

Taken from Barron's SAT Math Level 2 prep book: If f(x) = i, where i is an integer such that i ≤ x < i + 1, the range of f(x) is ...
0
votes
1answer
15 views

Solving for joint angles in 2-segment robot leg

I am trying to program a robot leg with 2 segments and two joints, such that for a given location of the foot, I can calculate the angles of both joints. From here on out, the positive Y direction is ...
3
votes
1answer
33 views

$\phi(v)/\Phi(v)$ is decreasing for $\phi$ and $\Phi$ being the PDF and CDF of $N(0,1)$

Let $\phi(v)$ and $\Phi(v)$ denote, respectively, the PDF and CDF of the standard normal distribution. How would one show that $$ \frac{\phi(v)}{\Phi(v)} $$ is decreasing? I tried the quotient rule ...
-4
votes
3answers
60 views

Multiplying a fraction [on hold]

Example: $\frac{x-1}{2}\cdot \frac{1}{3} =$ I'm really starting to hate math even though I've aced couple of times still can't calculate simple math. Please show work thanks! Okay basically what I ...
-2
votes
1answer
36 views

Can somebody simplify this radical? [on hold]

how do you simplify the radical shown above, please show as many steps as you can. I am very confused!!!
0
votes
2answers
22 views

Formal power series question

$$(1-t)^d \sum_{k = 0}^{\infty} \binom{d+k-1}{d-1} t^k = 1$$ How can this be proven? Thanks in advance.
0
votes
0answers
35 views

What is the proper term to describe algebraic techniques of equation manipulation?

Is there a term to describe the category of algebraic "tricks" that include: polynomial division completing the square quadratic formula partial fraction expansion etc. These are related since ...
0
votes
4answers
49 views

Solve system of equations

$$\sin(x+y)+1.6x=0$$ $$x^2+y^2=-1$$ Can this system be solved? Please help me with it. I managed to make graphs of it but can't get it solved without graph. Graph:
2
votes
4answers
49 views

Determining whether $\sum_{k=1}^\infty \frac{x^k}k$ converges [on hold]

$$\sum_{k=1}^\infty \frac{x^k}k$$ Does this series converge, if yes, then for what values of $x$?
0
votes
0answers
9 views

Calculate ratio of volumes in mixture with given ratio of masses

I have two components of a mixture: $a$ and $b$. I know that using $3.5 \text{ kg}$ of component $a$ and $0.5 \text{ kg}$ of component $b$ will give a proper mixture, and its density will equal $1.45 ...
1
vote
2answers
49 views

Find the domain of $\frac{x}{\sqrt{6x^2+3x+3/4}+x}$.

My attempt: Let's assume that $\sqrt{6x^2+3x+\frac{3}{4}}+x$ $>$ 0$$ \rightarrow\sqrt{6x^2+3x+\frac{3}{4}} > -x \rightarrow {6x^2+3x+\frac{3}{4}} > x^2\\ \rightarrow ...
0
votes
0answers
17 views

On Equivalence Relations and monoids

Let $R$ be an equivalence relation on a monoid $G$ such that $$ a_1 R a_2 \; \; and \; \; b_1 R b_2 \implies a_1b_1 R a_2b_2 $$ for all $a_i,b_i \in G$. Then the set $G / R = \{ [g] : g \in G \}$ ...
1
vote
3answers
56 views

Using mathematical induction to show that for any $n\ge$ 2 then $\prod_{i=2}^n\left(1-\frac{1}{i^2}\right)=\binom{n+1}{2 \cdot n}$

I'm trying to work through some practice problems but I've been stuck on this for god knows how long now and I've no idea where to even start. Just wondering if it would be possible for someone to ...
2
votes
2answers
41 views

If $(1-i)^n = 2^n$ , then find $n$.

If $$(1-i)^n = 2^n$$ then find $n$. If anything raised to $0$ is $1$, but according to my book $ n \ne 0$. Is the print wrong?
0
votes
1answer
18 views

Algebra - fraction problem

"The cooler in a car contains $8$ litres. The coolant fluid contains $\dfrac3{10}$ of glycol and rest is water. To increase the glycol content to $\dfrac35$ you drop some of the coolant fluid and fill ...
9
votes
4answers
618 views

Show that $\frac{2a_1^2}{a_1+a_2}+\frac{2a_2^2}{a_2+a_3}+…+\frac{2a_n^2}{a_n+a_1}\geq a_1+a_2+…+a_n$

Showing that $ \frac{2a_1^2}{a_1+a_2}+\frac{2a_2^2}{a_2+a_3}+...+\frac{2a_n^2}{a_n+a_1}\geq a_1+a_2+...+a_n$ holds for positive $a_i$s. I've tried adding $a_1+a_2, a_2+a_3,...,a_n+a_1$ respectively ...
3
votes
0answers
61 views

If $A = \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{999}}+\frac{1}{\sqrt{1000}}.$ Then $\lfloor A \rfloor$ is,

If $\displaystyle A = \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots\cdots\cdots+\frac{1}{\sqrt{999}}+\frac{1}{\sqrt{1000}}.$ Then $\lfloor A \rfloor$ is, where $\lfloor A\rfloor = A-\{A\}.$ ...
0
votes
2answers
28 views

Help with system of linear equations

I am in need of help solving for $x$ and $y$. $$ \begin{cases} 10x-8y=-5x \\ 5x=4y-20 \end{cases} $$ I've tried but don't really get what I am doing wrong. Thanks in advance.
0
votes
1answer
26 views

How do I compute the coordinates of the center and the radius of a circle.

$$9y^2+7x^2+35=8x−2x^2−36y $$ I am having trouble with this problem, and I managed to do some of it. but I got stuck here. My steps so far: $$9y^2+7x^2+35-8x+2x^2+36y = -35$$ $$(9y^2 + 36y)+(9x^2 ...
0
votes
0answers
10 views

Spacing of points in closed curves

Given an (implicitly defined by $F(x,y) = 0$) closed curve $C$ lying on $\mathbb{R}^2$ and the set of points inside $C$ be named $I$ for $I \in \mathbb{R}^2$, and $\sigma$ points, is it possible to ...
0
votes
3answers
30 views

Exponential inequality to Different Bases

How do I solve this exponential equation if I can't make 6 in base 2? $$ 6 - 2^x \geq 4^x $$ I know that the solution is $ x \in ]-\infty,1] $ because it just makes sense that $ 6-2=4$ I just don't ...
1
vote
1answer
41 views

Evaluation of $\sum_{i=0}^{100}\sum_{j=0}^{100}\binom{100}{i}\cdot \binom{100}{j}\cdot\binom{200}{i+j}^{-1}$

Evaluation of $\displaystyle \sum_{i=0}^{100}\sum_{j=0}^{100}\frac{\binom{100}{i}\cdot \binom{100}{j}}{\binom{200}{i+j}} = $ $\bf{My\; Try::}$ Let $\displaystyle i+j=n\;,$ Then Sum convert into ...
-1
votes
0answers
19 views

Finding the roots of fourth degree polynomial [duplicate]

$$ax^4 + bx^2 +cx + d = 0$$ How do I find just the real roots not even complex roots ?
0
votes
0answers
17 views

Do we recognize higher degree asymptotes beyond Horizontal and Oblique?

I am reading a textbook, and it talks about doing synthetic division in order to rewrite a function into the quotient $$R(x)=\frac{p(x)}{q(x)}= f(x) + \frac{r(x)}{q(x)}$$ Since $\frac{r(x)}{q(x)}$ ...
0
votes
1answer
25 views

Solve denominator so quotient is whole number?

I have a simple equation. road_length = ROADLENGTH / ROADSPACING The problem is, I really need road_length to be a whole number because it's used in FOR loop in ...
0
votes
1answer
48 views

A minimum Value Sum [on hold]

The minimum value of $\sqrt{x^4 - x^2 - 24x + 145} + \sqrt{x^4 - 23x^2 - 2x + 145}$ can be expressed in the form ($a\sqrt{b}$), where $a$ is an integer, $b$ and is not divisible by the square of any ...