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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1answer
86 views

Proof of floor of division

I got stuck proving $$\left\lfloor\frac{x/a}b\right\rfloor = \left\lfloor\frac{\lfloor x/a\rfloor}b\right \rfloor$$ This is what I got: Using the division algorithm we can write $x = qa+r$, where ...
2
votes
5answers
105 views

Is it more practice and intuition or rather algorithmic to solve third degree polynomials of this type?

Consider $x^3 - 6x^2 + 11x - 6 = 0$ I can not reasonably factor this intuitively in any short amount of time with my skill level. Is this the only hope to solving such equations by hand? What tools ...
2
votes
3answers
633 views

How do I simplify this fractional expression?

$\frac{1}{\sqrt{x+h}} - \frac{1}{\sqrt{x}}$ and these numbers are all above the denominator of $h$. Can someone please help me to understand how to simplify this expression?
1
vote
4answers
70 views

Question regarding differentiating with $e$

What is $\left[h(x) = \dfrac{3x-2}{e^x} \right]'?$ My textbook tackles this problem in this way: $h'(x) = \dfrac{e^x\cdot3-(3x-2)e^x}{(e^x)^2} = \dfrac{3-(3x-2)}{e^x}$ etc... I however don't ...
1
vote
1answer
442 views

Dividing the lateral area of a cone into two equal parts.

"The height of a cone is h. A plane parallel to the base intersects the axis at a certain point. How far from the vertex must this point be if the plane divides the lateral area into two equal parts? ...
6
votes
0answers
367 views

Series sum formula

Is there any general formula to sum following series: $$S = 1^1 + 2^2 + 3^3 + \dotsb+(n - 1)^{n - 1} + n^n, n \in N$$ I mean for $S = f(n)$, is there a formula to compute $f(n)$? Regards, vishal.
1
vote
2answers
133 views

Simplifying the sum of terms that are “polynomial fractions”

I need to simplify: $$\left(\frac{z^2-2}{1+2z}\right)^2 \;+ \;\left(\frac{z^2-2}{1+2z}\right)$$ I get $2z^4-8z^2+8$ in the numerator when I know it should be $((z^2-2) (z^2+2 z-1))$.
6
votes
6answers
346 views

How to show that $\frac{x^2}{x-1}$ simplifies to $x + \frac{1}{x-1} +1$

How does $\frac{x^2}{(x-1)}$ simplify to $x + \frac{1}{x-1} +1$? The second expression would be much easier to work with, but I cant figure out how to get there. Thanks
2
votes
1answer
61 views

Why is this reasoning of factor wrong?

Since $$x^3 - y^3 = (x - y)(x^2 + xy + y^2)$$ then $$x^5 - y^5 = (x^{5/3})^3 - (y^{5/3})^3= (x^{5/3} - y^{5/3})((x^{5/3})^2 + x^{5/3}y^{5/3}+(y^{5/3})^2)$$ It looks very wrong to me, but I can't find ...
0
votes
2answers
56 views

Fit screen resolution given ratio and total number of pixels

Given: width: 1920 height: 1080 total pixels: width * height = 2073600 aspect ratio: 1920 / 1080 ~= 1.8 How do I calculate a new resolution (width and height) ...
1
vote
1answer
96 views

Analyzing function $\;y=\large\frac{(x-a)^2(x+a)^2}{(x-a)}$

Which of these statements about this function is true? $$y=\frac{(x-a)^2(x+a)^2}{(x-a)}$$ There is a hole at $x=a$. There is a repeated zero at $x=-a$. The function "behaves" like $y = x^4$ (is ...
0
votes
3answers
156 views

Showing that $X_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}$ is not bounded above. [duplicate]

Possible Duplicate: Why does the series $\frac 1 1 + \frac 12 + \frac 13 + \cdots$ not converge? Prove that the sequence converges I have to show that $X_n$ is not bounded above, ...
3
votes
2answers
134 views

Choosing convenient limits of integration on Basel problem

I have recently discovered ...
4
votes
4answers
178 views

Proof of tangent half identity

Prove the following: $$\tan \left(\frac{x}{2}\right) = \frac{1 + \sin (x) - \cos (x)}{1 + \sin (x) + \cos (x)}$$ I was unable to find any proofs of the above formula online. Thanks!
1
vote
1answer
215 views

How to solve this (difficult) simple equation in one variable?

I have a rather convoluted equation in one variable that I am trying to solve, in terms of many other parameters. Let $$w(x) = Ax^2 - Bx^4, \quad A,B > 0$$ and $$\varphi = \arccos\left(\rho ...
2
votes
2answers
649 views

finding the intersection of two line segments in 2d (with potential degeneracies)

I am trying to write an algorithm that finds the intersection of two line segments in the plane. The line segments are given as pairs of points, which I'm writing as $C_1$ and $C_2$, where $C_1 = ...
3
votes
4answers
134 views

Number of solutions for sixth order equation

I have a problem, from Gelfand's "Algebra" textbook, that I've been unable to solve, here it is: Problem 268. What is the possible number of solutions of the equation $$ax^6+bx^3+c=0\;?$$ Thanks ...
0
votes
2answers
997 views

Find the smallest positive integer $k$, such that product of $420$ and $k$ is a perfect square.

Find the smallest positive integer $k$, such that product of $420$ and $k$ is a perfect square. Please help me in this question.
1
vote
2answers
109 views

Inequality problem algebra?

How would I solve the following inequality problem. $s+1<2s+1<4$ My book answer says $s\in (0, \frac32)$ as the final answer but I cannot seem to get that answer.
2
votes
2answers
441 views

Find the equation of the sphere $ x^2+y^2+z^2-2x-4y+8z=15$

I'm not sure how you get from this: $x^2+y^2+z^2-2x-4y+8z=15$ To: $(x^2-2x+1) + (y^2-4y+4) + (z^2+8z+16)-1-4-16=15$ How do you get the $1,4,16$?
3
votes
2answers
78 views

If $r = \min \left\{ a , \frac{b}{ a+ a^2 b^3} \right\}$ , find $ r_{\max}$

Studying differential equations I came cross through this: Let $ \displaystyle{ r = \min \left\{ a , \frac{b}{ a+ a^2 b^3} \right\} } $, where $ a,b >0$. Find $ r_{ \max} $. Here is what I did: ...
1
vote
0answers
50 views

Specific solution of a sequence of equations

Consider a sequence of equations $$ \exp\left( \frac{k}{1+\varepsilon k}\right) = \frac{\text{e}^{-k}m_k}{n_k+\text{e}^{-2k}m_k}, \;\;\; k=1,2,\ldots $$ on numbers $m_k, n_k >0$. Here ...
4
votes
6answers
168 views

How to solve $|x-5|=|2x+6|-1$?

$|x-5|=|2x+6|-1$. The answer is $0$ or $-12$, but how would I solve it by algebraically solving it as opposed to sketching a graph? $|x-5|=|2x+6|-1\\ (|x-5|)^2=(|2x+6|-1)^2\\ ...\\ ...
1
vote
2answers
64 views

the existence of this number $(a_{1}+a_{2}\ldots +a_{n})^{2}-(n^2-n+2)a_{i}a_{j} \geq 0$

Let $a_{1}, a_{2}, \ldots, a_{n}$, $n \geq 3$. Prove that at least one of the number $(a_{1}+a_{2}\ldots +a_{n})^{2}-(n^2-n+2)a_{i}a_{j}$ is greater or equal with $0$ for $1 \leq i < j \leq n$. I ...
1
vote
1answer
78 views

$\forall \space a \space (6a^2 \geq6 \Rightarrow a \geq 1)$

My workbook have this question: Let be the following expression. What is its logical value on $\mathbb{Z}$ and $\mathbb{R}$? $$\forall \space a \space (6a^2 \geq6 \Rightarrow a \geq 1)$$ First, ...
0
votes
1answer
233 views

Circle intersection in radial coordinates?

We have two circles in the plane described by $C_0 = (x_0, y_0, r_0)$ and $C_1 = (x_1, y_1, r_1)$ We know that they intersect but one does not completely overlap the other. That is to say their ...
2
votes
2answers
1k views

Two circles overlap?

If we have two circles in the plane described by $(x_1, y_1, r_1)$ and $(x_2, y_2, r_2)$ we can determine if they are completely disjoint by simply: $$(x_1 - x_2)^2 + (y_1 - y_2)^2 < (r_1 + ...
1
vote
1answer
68 views

Question on an algebra/calculus word problem?

A typical monthly water bill in Bellingham consists of a fixed fee, plus a charge for each $100$ cubic foot (ccf) of water used. $1$ ccf ($100$ cubic foot) = $748$ gallons. A household using $10$ ...
0
votes
2answers
528 views

calculate the volume of water in a portion of a cone

Imagine we have a cone filled with water, if we were to take the upper portion of that cone how would we calculate the volume of water present. For example: So, in this example we have a surface ...
2
votes
2answers
299 views

Solve an equation involving the sine and the inverse tangent

The equation is $$ \sin\left(\frac{x}{x-1}\right) + 2 \tan^{-1}\left(\frac{1}{x+1}\right)=\frac{\pi}{2} $$ The answer is $0$, but I do not know how they got that.
1
vote
5answers
90 views

Simple Calculus Function

I am doing a course in calculus and I was given this problem : Given that $f(x)=3x^4−6x^3+4x^2−7x+3$, evaluate $f(−2)$. The answer is meant to be 129 according to the tutor, but no matter how many ...
5
votes
4answers
186 views

$(\tan^2(18^\circ))(\tan^2(54^\circ))$ is a rational number

Assuming $$\cos(36^\circ)=\frac{1}{4}+\frac{1}{4}\sqrt{5}$$ How to prove that $$\tan^2(18^\circ)\tan^2(54^\circ)$$ is a rational number? Thanks!
3
votes
2answers
75 views

Algebra question involving fractions

How would I perform the indicated operation. $$\frac{t+2}{t^2+5t+6}+\frac{t-1}{t^2+7t+12}-\frac{2}{t+4}.$$ I simplified it to $$ \frac{t+2}{(t+3)(t+2)} + \frac{t-1}{(t+4)(t+3)}-\frac{2}{t+4}. $$ ...
-2
votes
1answer
79 views

Slope whats the error in this problem

Kyle graphed a line, given a slope of 43 and the point (2, 3). When he used the slope to find the second point, he found (5, 7). What error did Kyle make
0
votes
1answer
79 views

When $(\sum_{i=1}^nk_i < \prod_{i=n}^ni^{k_i}k_i!)$?

Consider $\Omega \subset \mathbb{N}$ a finite subset of $\mathbb{N}$, $\phi: \Omega \rightarrow \mathbb{N}$ an enumeration of $\Omega$ such that $\phi(\omega)=i$ and $|\Omega|=n$, $$ ...
1
vote
1answer
316 views

Let $x,y,z>0; ~ xyz(x+y+z)=1$. Show that $(x+y)(y+z)\ge 2$

Assuming positive real values of $x,y,z$, and that $ xyz(x+y+z)=1$, how can we prove that $(x+y)(y+z)\ge 2$ I tried using the AM-GM inequality but as if I were looping. I'm not sure if C-S or other ...
2
votes
1answer
2k views

Derivative of $\sin^4(x) -\cos^4(x)$

Find the derivative of $\sin^4(x) -\cos^4(x)$. My attempt: $\frac{d}{dx}(\sin^4(x) -\cos^4(x)) = 4\sin^3(x)\cos(x) +4\cos^3(x)\sin(x). $ The problem is I need to simplify this to its simplest form ...
19
votes
6answers
2k views

Helping my daughter with her homework: solving an algebra word problem.

Three bags of apples and two bags of oranges weigh $32$ pounds. Four bags of apples and three bags of oranges weigh $44$ pounds. All bags of apples weigh the same. All bags of oranges weigh the ...
2
votes
1answer
112 views

Establishing an inequality using principal convergents and continued fraction representation.

If $\theta$ is irrational with continued fraction representation $[0;a_1,a_2,\ldots]$, $\lbrace \frac{m_k}{n_k} \rbrace$ is the sequence of principal convergents of $\theta$ and $\lbrace b_k\rbrace$ ...
0
votes
3answers
268 views

Common method of calculating zero places of quadratic and linear function.

Very basic stuff from school we know that we can calculate zero places of quadratic function which has form $ax^2 + bx + c$ and we assume that $a \neq 0$, now what if $a=0$? Why can't we use delta to ...
1
vote
2answers
44 views

proving the formula for x and y in the extended euclidean algorithm

I found this on the wikipedia page of the Extended Euclidean Algorithm: It states: "Suppose $d_i = d_{i-2} - k_{i-1} \cdot d_{i-1}$. Then it must be that $x_i = x_{i-2} - k_{i-1} \cdot x_{i-1}$ and ...
5
votes
2answers
106 views

How is $\sqrt{n^2+2n}-\lfloor\sqrt{n^2+2n}\rfloor=\frac{2}{\sqrt{1+\frac{2}{n}}+1}$.

I have trouble seeing how $$\sqrt{n^2+2n}-\lfloor\sqrt{n^2+2n}\rfloor=\frac{2}{\sqrt{1+\frac{2}{n}}+1}.$$ I can't see where to start even.
3
votes
2answers
284 views

How do you work out the sum of the series: $\cos{x}+\cos{2x}+\cdots+\cos{(n-1)x}$ by multiplying through by $2\sin(x/2)$?

How do you work out the sum of the series: $\cos{x}+\cos{2x}+\cdots+\cos{(n-1)x}$ by multiplying through by $2\sin(x/2)$? I am supposed to find the sum using only this method and I'm not completely ...
1
vote
3answers
101 views

Is $ \forall\space x \in \mathbb{R} \space \exists \space y \in \mathbb{R} \space(x+y^2=10)$ true or false?

Given the expression $\forall\space x \in \mathbb{R} \space \exists \space y \in \mathbb{R}\space(x+y^2=10)$ tell what is its logical value? When I look to an expression with quantifiers, I try ...
-1
votes
4answers
56 views

Limit Function Continuous [closed]

How to solve this question: Compute the value of A that will make the function f(x) continuous everywhere. $$f(x)=\begin{cases}x^2-2&,\;\;\text{if}\;\;x=-1\\Ax-4&,\;\;\text{if}\;\;x\neq ...
0
votes
3answers
62 views

How does one show using algebra or basic mathematical prowess to show that ψ = 1 - φ

Where φ is the golden ratio. $\frac{1 + \sqrt 5}{2}$ How can I use basic mathematical skills to show that: $ψ = 1 - φ$ This is from wikipedia. I do not think I have done anything quite like this ...
0
votes
1answer
104 views

How to Simplify this to the given answer? (matrix equation, trig functions)

Can someone help me to simplify $$\Bigg( \begin{matrix} 5\cos t &5\sin t \\2\cos t+\sin t & 2\sin t-\cos t\end{matrix} \Bigg) \Bigg( \begin{matrix} u_1 \\u_2\end{matrix} \Bigg) ...
13
votes
4answers
759 views

Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$ [duplicate]

Yesterday, my uncle asked me this question: Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$. How can we do this? Note that this is not a diophantine ...
-2
votes
2answers
96 views

math problem solve [closed]

You are the advertising chief for a video game company. You have a budget of \$200,000.00 to spend each week on television advertising time. Advertising time is \$5,000 per minute for prime time and ...
-1
votes
1answer
114 views

What is $0^0$? Indeterminate or 1? [duplicate]

Possible Duplicate: Zero to zero power Sorry for asking this simple question, but googling this question yields conflicting answers. Some say it's indeterminate, other's say it's $1$.