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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0answers
15 views

Are these two definitions of an affine subspace equivalent?

I've seen the notion of an affine subspace defined differently as follows: $S \subset \mathbb R^3$, non-empty, is an affine subspace if $(1-t)u + tv \in S$ whenever $u,v \in S$. $S$ is an affine ...
-2
votes
0answers
16 views

Calendar question [on hold]

A certain day, which is $x$ days before $17$th of August is such that $50$ days prior to that day it was $4x$ days since March $40$th of the same year. What is $x$?
-4
votes
1answer
58 views

Do you agree with this!

So yeah, I stuck with this theory. I thought -4 Square = 16
0
votes
1answer
28 views

Transpose exponential equation [on hold]

Could somebody please help with transposing the following equation to isolate x to the left side of the equation to solve for x? $$ y = 10^{1.830 \log(x)} + 2.686 $$
-2
votes
0answers
22 views

What is the difference between the middle factor and the middle term of permutation ? [duplicate]

What is the difference between the middle factor and the middle term of permutation ?
0
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1answer
21 views

Simple Harmonic Motion Formula

The SHM general formula is this: $y(t) = A\sin(\omega t + \alpha) +B$ I have two questions about it As far as I know, there is the cosine formula for when the particle starts at P and sine for ...
-3
votes
1answer
71 views

I've never seen this equation before [on hold]

everytime I try to solve this equation, (a= b+c÷4) I get the same answer as my equation. I'm not sure why, but I'm defitently having some difficulty. If anyone could help me, that would be greatly ...
0
votes
3answers
43 views

Simple interest problem

I'm new to this site. I'm not sure exactly what appropriate tag I should use, but if anyone could help me with the tags, thens thats great :-) Anyways, I'm having difficulty figuring out a solution to ...
4
votes
2answers
88 views

Evaluate $\lim_{n \to \infty} \int_{0}^1 \frac{n+1}{2^{n+1}} \left(\frac{(t+1)^{n+1}-(1-t)^{n+1}}{t}\right) \mathrm{d}t$

Evaluate $$\lim_{n \to \infty} \int_{0}^1 \frac{n+1}{2^{n+1}} \left(\frac{(t+1)^{n+1}-(1-t)^{n+1}}{t}\right) \mathrm{d}t$$ For this integral, I have tried using integration by parts and then ...
-1
votes
1answer
42 views

How can I prove by mathematical induction that $\sum_{i=0}^n i^4 = (\sum_{i=0}^n i)^3$?

How can I prove by mathematical induction that $$\sum_{i=0}^n i^4 = (\sum_{i=0}^n i)^3$$ ? I see easily that it holds for $i=0$. Using the inductive hypothesis, I get: $$\sum_{i=0}^{n+1} i^4 = ...
3
votes
4answers
53 views

$a^2b + abc + a^2c + ac^2 + b^2a + b^2c + abc +bc^2$ factorisation

I came across this from a university mathematics resource page but they do not provide answer to this. What I did was this: $(a^2+b^2+c^2)(a+b+c) - (a^3 + b^3 + c^3) + 2abc$ But I don't think ...
1
vote
1answer
14 views

Assign set in probability generating functional

say i have an integral $A = \lambda c_d \displaystyle\int_0^\sim (1-exp(-shr^{-\alpha}))dr^{d-1}\mathrm {d}r$ $A = \lambda c_d \displaystyle\int_0^\sim (1-exp(-shr^{-1/\delta}))\mathrm {d}r$ (subs. r ...
1
vote
1answer
36 views

I need to help to figure out a solution to this problem [on hold]

Jennifer drove at steady speed for $2$ hours, then reduced speed by $10$mph and drove for another $3$ hours. What was the initial speed if the total distance was $220$ miles?
-2
votes
2answers
27 views

Math Problem/Need Help with Solution [on hold]

Assign x, make an equation and solve the problem. A 56 ft cable was cut into 3 pieces. Every next piece is twice as long as the previous piece. Find the length of smaller piece.
1
vote
0answers
39 views

Total Ordering on the Rationals

I am trying to prove that $\leq$ is a total ordering on the set of all rationals. So I started with the definition of the $\leq$ in the rationals, which is that if $x$ and $y$ are in the rationals, ...
0
votes
1answer
16 views

Rotations of complex graphs

Let $c_1 = -i$ and $c_2 = 3$. Let $z_0$ be an arbitrary complex number. We rotate $z_0$ around $c_1$ by $\pi/4$ counter-clockwise to get $z_1$. We then rotate $z_1$ around $c_2$ by $\pi/4$ ...
0
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0answers
11 views

Inverse and asymptotes

Given $f(x) = (2x+3)/(3x+2)$: A) Find the formula for $f^{-1}(x)$. B) Find the vertical and horizontal asymptotes of $f^{-1}(x)$. I am having trouble solving for the inverse, because i get an xy ...
1
vote
1answer
24 views

Rotation in the complex plane

The function $f(z) = \frac{(-1 + i \sqrt{3}) z + (-2 \sqrt{3} - 18i)}{2}$ represents a rotation around some complex number $c$. Find $c$. Hello, I am having some trouble trying to do this problem. ...
0
votes
3answers
85 views

Why is $3^{n+1} = 3^n + 2\cdot 3^n$?

I don't see how this equation holds true $3^{n+1} = 3^n + 2\cdot 3^n$ especially when there is a $2\cdot 3^n$ on the right equation.
0
votes
0answers
6 views

Expressing weighted fraction as sum of weighted fractions

Suppose we are given $z = W \cdot \frac{\sum_i x_i}{\sum_i y_i}$ Is it possible to express $z$ in the following form?: $z \overset{?}{=} w \cdot \sum_{i}\frac{x_i}{y_i}$ How could we find $w$ in ...
1
vote
0answers
33 views

Is this chain of inequalities correct?

Is this chain of inequalities correct? If not how to make it works? $$\frac{\ln \left( 1+x^3+y^3 \right)}{\sqrt{x^2+y^2}} \le \frac{\left( x^3+y^3 \right)}{\sqrt{x^2+y^2}} \le \frac{ \left( ...
0
votes
1answer
58 views

Let $f(x) =\begin{cases} 0 & \text{if }x \leq 1 \\\ \log_2x & x > 1 \\ \end{cases}$ and let $f^{(2)}(x) =f(f(x)),f^{(3)}(x) =f(f^{(2)}(x))\ldots$ [on hold]

Problem : Let $f(x) =\begin{cases} 0 & \text{if }x \leq 1 \\ \log_2x & x > 1 \\ \end{cases}$ and let $f^{(2)}(x) =f(f(x)),f^{(3)}(x) =f(f^{(2)}(x)), \ldots$ and generally , ...
-1
votes
3answers
66 views

What is the easiest way to rearrange a formula to change the subject? [on hold]

I really fumbled at arranging formulas, reading and following lots of resources I got no nowhere. For simple formulas it's pretty much easy, but for $y = \frac{1}{1-x^2}$ this is really tough. I ...
0
votes
1answer
29 views

Closed under vector addition and scalar multiplication

If $S=\{[y\cdot z,y,z]^T \mid y, z \in \mathbb{R}\},$ is $S$ closed under addition and scalar multiplication? This one confuses me as there are not restrictions to what part of $\mathbb{R}^3$ our ...
0
votes
0answers
17 views

Finding the length from (0,0) to point 'B' of a quadratic function [on hold]

In my math assignment, I have been given a quadratic function on the graph with the vertex of $108$ metres from the $x$-axis and with $x$-intercepts of $(0,0)$ and $(72,0)$. From my calculation, I ...
0
votes
5answers
44 views

Cubic Equation. (Factorisation)

I'm given this question, factorise $4x^3-7x-3$. Is this answer acceptable? $(x+\frac{1}{2})(x-\frac{3}{2})(x+1)$.
-1
votes
0answers
30 views

Division-free differentiation of $\frac1x$

Given $a>0$, $~b=\frac1a$ and $c>0$ is it possible to calculate $\frac{1}{a+c}$ in such a way to avoid divisions? The solution can be approximated, but the percent error must be less than 1% ...
0
votes
4answers
40 views

Eliminating $t$ in the solution of a Differential Equation

My task is to show that the trajectories of the system: $\frac{dx}{dt}=y$, $\frac{dy}{dt}=x$ are hyperbolas given by $H(x,y)=y^2-x^2=c.$ Solving the above system I got: $x=c_1e^t+c_2e^{-t}$, ...
7
votes
10answers
1k views

Caden has 4/3 kg of sand which fills 2/3 ​​ of his bucket. How many buckets will 1kg sand fill?

I have already finished Calculus II but I go back and practice the basics on Khan Academy. This problem confuses me conceptually every time. I know what the answer is, but I am having a hard time ...
12
votes
5answers
1k views

Why aren't these negative numbers solutions for radical equations?

I was working on radical equations and I came across a few problems where I got answers that worked when I checked, but were not listed as solutions. My teacher's only explanation was, "just because." ...
0
votes
0answers
38 views

Basic algebraic simplification

How is this equation: $$\frac{1}{y+z} + \frac{z-x}{(y+z)^2} + \frac{-x-y}{(y+z)^2}$$ simplified to $\frac{2(z-x)}{(y+z)^2}$?
1
vote
1answer
61 views

Simplifying Sum

How would one show that $$ \sum_{i=0}^n\binom{n}{i}(-1)^i\frac{1}{m+i+1}=\frac{n!m!}{(n+m+1)!} ? $$ Any hint would be appreciated. Note: I tried to recognize some known formula, but since I don't ...
10
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5answers
2k views

How would multiplying money work?

This is a very silly question since nobody will actually do this because it makes very little sense in the real world but I just want to know how would it actually work if possible. For example let ...
0
votes
2answers
41 views

2xy-y=4+5x, what rule is being applied in the next step, that leads to (2x-1)y=4+5x

So the question says it all, I am sure that's the next step, and I also confirmed it with Wolfram Alpha, I am trying to calculate the inverse of a function, but I have a memory loss of what rule is ...
0
votes
1answer
32 views

Finding a basis for a subspace. Do i always need to test linearly independence?

Where the subspace is contained in {[5r-3s;2r;0;-4s] is an element of R^4: r and s are scalars} The generating set that can make up all of the input is {[5;2;0;0], [-3;0;0;-4]} This is only a ...
2
votes
2answers
58 views

Volume of a parallelepiped, given 8 vertices

Given the eight vertices $(0,0,0)$, $(3,0,0)$, $(0,5,1)$, $(3,5,1)$, $(2,0,5)$, $(5,0,5)$, $(2,5,6)$, and $(5,5,6)$, find the volume of the parallelepiped. I'm having trouble finding the 1 vertex ...
0
votes
3answers
45 views

Setting two equations equal to each other

I am trying to decipher the steps needed to determine the intersection point between curves represented by the two following equations: $$y= 2\;x+2$$ $$y= \dfrac{128}{(x+1)^2}$$ I know that they ...
2
votes
1answer
63 views

Proving a certain function is injective

I have found the following exercise on an exam for Geometry three dating to a past year. Let $F(u,v)=((2-v\sin\frac{u}{2})\sin u,(2-v\sin\frac{u}{2})\cos u,v\cos\frac{u}{2})$, with ...
0
votes
3answers
56 views

Confusing algebraic solution…

Hi everyone I'm a bit confused how they got that final value of M. Any ideas?
0
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1answer
30 views

An elephant and a lion are currently 1 mile apart. The elephant runs directly away from the lio

An elephant and a lion are currently $1$ mile apart. The elephant runs directly away from the lion at $19$ miles per hour, while the lion runs directly towards the elephant at $24$ miles per hour. How ...
-4
votes
1answer
35 views

Speed calculations [on hold]

Terance galloped 12 miles in 40 minutes. Then he trotted 2 miles in 40 minutes. By how much did his first rate exceeded his second rate?
0
votes
2answers
14 views

Expansion of Brackets

Case 1: $(a.b)(c.d)$ Case 2: $(e+f)(g+h)$ In both cases if you a value for the for each letter just calculate each bracket and multiply by the value of the second bracket. If no values are given ...
0
votes
1answer
24 views

Maximization of a function in an interval

I am writing a computer program where I have $x$ real positive varying in the domain $[\sqrt{U}, U]$. I want the value of $x$ which maximizes: $$ (1+ \sqrt{U}) - \frac{\sqrt{U}-1}{U-\sqrt{U}} x - ...
-3
votes
1answer
84 views

Why a line is said to have infinite number of points? [duplicate]

Why a line is said to have infinite number of points? Is this so because a line is ever lasting or we can not count how many points does it have? Finite means: Having an end. Infinite means: No end! ...
4
votes
1answer
44 views

Time and distance: Police and a thief with a twist.

A thief was given a head-start of 15 hour. The velocity of the thief being 4 km/hr and the police chasing after him be 5 Km/hr. A dog is moving to and fro between the police and the thief, starting ...
2
votes
5answers
88 views

How to sum $\sum_{k=1}^n (k+1)(k)(k-1)$

Is there an intelligent way to do this sum without using sums of cubes and sums of squares? $$\sum_{k=1}^n (k+1)(k)(k-1)$$
1
vote
2answers
26 views

Factorizing expressions

I am having trouble solving this problem $81f^2- \dfrac{9}{e^2}$. How do you begin when solving this problem? Do you move $f^2$ by replacing the $9$ and vice versa and does the minus change to plus?
-4
votes
1answer
23 views

Help with precalc homework | circles and radians [on hold]

Hi everyone, i need help with my homework, ive been trying to find out the answers for the last 2 questions but ended up with nothing, please help.
1
vote
2answers
51 views

Derivation for the general cubic formula

It's a long equation, and Wikipedia writes it to be $$x_k = -\frac{1}{3a}(b + u_kC + \frac{\Delta_0}{u_kC}), \quad k \in \{1,2,3\}$$ But there is no derivation of it. The sources I've read so far ...
3
votes
2answers
154 views

Simple maths and a typo: impossible answer?

Long story short: While doing some simple math exercises, I came across one that seemed impossible. Days later I decided to search the web for it, and found out there was a typo, putting the $^2$ ...