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)

1
vote
1answer
16 views

Adding conditions on double sums

Suppose now we have $a_{jk}$ where $1 \leq j,k \leq n$. If $a_{jk} = a_{kj}$, then we have \begin{equation} \sum_{1 \leq j < k \leq n} a_{jk} = \frac{1}{2} \sum_{j=1}^n \sum_{k=1}^n a_{jk} ...
0
votes
1answer
27 views

Combining two fractions involing powers of x

Is there any way i can write $x^a+x^b$ as d$x^c$ Im considering writing letting $a=a-1$ and partial fractions but im getting really confused.
1
vote
0answers
65 views

Prove the equality that $4x^{4} + 4y^{3} + 5x^{2} + y + 1$ >or equal to $12xy$ if $x$ and $y$ are real and positive

Prove this inequality: $$4x^4 + 4y^3 + 5x^2 + y + 1 \geq 12xy$$ if $x$ and $y$ are real and positive. Please , I am beginner and have no idea how to solve this, so don't use any strange theorems.
2
votes
1answer
12 views

Another way to prove that if n$^{th}$-degree polynomial $r(z)$ is zero at $n+1$ points in the plane, $r(z)\equiv 0$?

The original problem is as follows Let $p$ and $q$ be polynomials of degree $n$. If $p(z)=q(z)$ at $n+1$ distinct points of the plane, the $p(z)=q(z)$ for all $z\in \mathbb{C}$. I attempted ...
1
vote
3answers
31 views

How did we get $ p\sum_{n=1}^{∞} (1-p)^{n-1}=\frac{p}{1-(1-p)}$

I am not sure how we got below expression.. $$\sum_{n=1}^{∞} P(X=n)= p\sum_{n=1}^{∞} (1-p)^{n-1} = \frac{p}{1-(1-p)} = 1$$ I understand that we calculate expected value for n trials using linearity ...
3
votes
5answers
84 views

Solving $\sin\theta -1=\cos\theta $

Solve$$\sin\theta -1=\cos\theta $$ Steps I took to solve this: $$\sin^{ 2 }\theta -2\sin\theta +1=1-\sin^2\theta $$ $$2\sin^{ 2 }\theta -2\sin\theta =0$$ $$(2\sin\theta )(\sin\theta -1)=0$$ ...
0
votes
4answers
57 views

Applying a function to both sides of an equation doesn't change it?

Why is it that applying a function to both sides of an equation doesn't change it? Can this be proven? Can you point to some material to read more about this?
0
votes
1answer
22 views

Solving a given equation using a trigonometric identity

$2\sin2\theta -3\sin\theta =0$ Steps I took: $$2(2\sin\theta \cos\theta )-3\sin\theta $$ $$4\sin\theta 2\cos\theta -3\sin$$ $$\sin\theta (8\cos\theta -3)=0$$ $\sin\theta =0$ so, $\theta =0+\pi k$ ...
0
votes
0answers
34 views

Application of Rado's Theorem

Find the smallest positive integer n that satisfies the following: We can color each positive integer with one of n colors such that the equation $w + 6x = 2y + 3z$ has no solutions in positive ...
1
vote
3answers
52 views

Solving $3t^2-\frac{12}{3}t+\frac{4}{3}=0$

I need to to solve: $$3t^2-\frac{12}{3}t+\frac{4}{3}=0$$ The solution manual factorizes this to $\dfrac{1}{3}(3t-2)^2$. How can you do this easily?
0
votes
0answers
18 views

Simplest way to calculate the width of a segment of a convex shape

A convex shape $C$ is cut using a a chord. What is the width of the resulting segment? This is the length of the green thick short line in the figure below: Here is my current solution: Mark the ...
2
votes
2answers
43 views

Bijection $f:\mathbb{N} \to \mathbb{N}.$

Prove that there is only one monоtone bijection $f:\mathbb{N} \to \mathbb{N}, 0 \in \mathbb{N}.$ Is $n \mapsto n$ the answer?
2
votes
1answer
34 views

Justification of steps

I'm taking Real Analysis starting in January and I'm getting a head start now so as to make the class somewhat easier. In the text we are using (Bartle and Sherbert, 4th ed.), it has examples where ...
5
votes
3answers
214 views

Integral of Rational Functions

$$\int \frac{dx}{ax^2 + bx + c} \quad \text{for} \quad 4ac-b^2 >0$$ then $$\begin{align} ax^2 + bx + c &= a\biggl(x+\frac{b}{2a}\biggr)^2 + \frac{4ac-b^2}{4a} \\ &= ...
0
votes
1answer
41 views

The range of a>0 and a<0 [closed]

Determine the range of $$y=\frac{x^2 -x -a}{(x-1)^2}$$ in the following two cases: (i) $a>0$ (ii) $a<0$ Express the endpoints of each interval in terms of $a$.
0
votes
1answer
21 views

Magnitude of vector $\vec{v}$? [closed]

The initial is $(-6,1)$ and the terminal is $(2,5)$. I was just wondering how to find the magnitude.
0
votes
1answer
13 views

Magnitude and direction angle?

What is the magnitude and direction angle of $v=4i+4j$? I have no idea how to start this problem, so any hints/formulas/tips and tricks would be useful, thank you!
0
votes
2answers
19 views

Find the projection of $U$ onto $V$?

I'm really stuck on this precalc problem: Find the projection of $u$ onto $v$ if $u=(-3,3)$ and $v=(-2,5)$.
2
votes
2answers
52 views

If $\tan x = -1$, simplify $\tan(\pi/3+x)$

If $\tan x = -1$, simplify $\tan(\pi/3+x)$. Progress so far: See new expression in top right. What do I do to rationalize this fraction's denominator? Do I multiply the fraction by ...
-2
votes
1answer
22 views

Determine the values of these constants and deduce [closed]

We would like to express $x^2 +7y^2 +20z^2 +8yz -2zx +4xy$ in the form $A(x+py+qz)^2 + B(y+rz)^2 +Cz^2$, where $A,B,C,p,q,r$ are constants. Determine the values of these constants and deduce that the ...
0
votes
1answer
44 views

Finding the values of $a$ and $b$. [closed]

If $\dfrac{a}{10^x -1} + \dfrac{b}{10^x +2} = \dfrac{2\cdot 10^x +3}{(10^x -1)(10^x +2)}$ is an identity for positive rational values of $x$, find the values of $a$ and $b$.
5
votes
4answers
415 views

question about double sums

Suppose we have an expression $$ \sum_{1 \leq k < j \leq n } f(k)f(j) $$ Can we express this as a double sum like $$ \sum_{k=1}^n \sum_{j=1}^n f(k)f(j) $$ ???
0
votes
0answers
26 views

finding all values of m such that.

Find all values of "$m$" such that $x^2 + 3xy + x + my − m$ has two factors, with integer coefficients, which are linear in $x$ and $y$.
3
votes
1answer
58 views

Simplifying the sum $\sum\limits_{i=1}^n\sum\limits_{j=1}^n x_i\cdot x_j$

How can I simplify the expression $\sum\limits_{i=1}^n\sum\limits_{j=1}^n x_i\cdot x_j$? $x$ is a vector of numbers of length $n$, and I am trying to prove that the result of the expression above is ...
0
votes
1answer
32 views

Straight-line distance as a function of camera angle

At a swim meet, a parent is videotaping his son from a seat in the stands that is $20$ meters past the starting line and $8$ meters away from his son's lane. Let $x$ represent the distance the son has ...
2
votes
1answer
44 views

Applying Newton-Raphson method to $a\cdot b^{-2}=c\cdot d^4+e\cdot f(d)$

I am familiar with the method and it's application in classic problems, but I have troubles tackling the function I need to solve with it. So, variables in problem: Real numbers, all are known ...
0
votes
1answer
30 views

A problem of calculus.A fly walks with a speed of $1$ cm/sec in every direction of a piece of paper which is considered to be $xy$ plane…

A fly walks with a speed of $1$ cm/sec in every direction of a piece of paper which is considered to be $xy$ plane. If the fly walks away from the origin in the direction of $u^{\to} = i − 7j$, the ...
1
vote
1answer
15 views

If $a_0,a_1,a_2 \cdots a_{99} \in R$ and $f(x) =x^{100}+a_{99}x^{99}+a_{98}x^{98} +\cdots +a_0$ be such that $|f(0)|=f(1)$..

Problem : If $a_0,a_1,a_2 \cdots a_{99} \in R$ and $f(x) =x^{100}+a_{99}x^{99}+a_{98}x^{98} +\cdots +a_0$ be such that $|f(0)|=f(1)$ and each root of f(x) =0 is real and between 0 to 1. If product ...
2
votes
1answer
29 views

What is the answer for this aptitude question?

If 13!/2^x is an integer, which of the following represents all possible values of x? a) 0 <= x <= 10 b) 0 < x < 9 c) 0 <= x < 10 d) 1 <= x <= 10 e) 1 < x < 10 The book ...
-1
votes
2answers
65 views

Find the Locus of the Orthocenter

Vertices of a variable triangle are $$(3,4)\\ (5\cos\theta,5\sin\theta) \\ (5\sin\theta,-5\cos\theta) $$ where $\theta \in \mathbb R$. Given that the orthocenter of this triangle traces a ...
0
votes
1answer
59 views

Is it true :$f(f^{-1}(C))=C? $

Let $f: X \to Y$ and $C$ be a subset of $ Y.$ Is the following hold: $f(f^{-1}(C))=C? $ If it is wrong then what is right answer? As for me $f(f^{-1}(C)) \subseteq C.$
2
votes
12answers
156 views

Simplifying $\frac{x^6-1}{x-1}$

I have this: $$\frac{x^6-1}{x-1}$$ I know it can be simplified to $1 + x + x^2 + x^3 + x^4 + x^5$ Edit : I was wondering how to do this if I didn't know that it was the same as that.
6
votes
4answers
103 views

How to solve $x^{2/3}=4$?

OK I know this sounds pretty stupid, but I am stuck on solving $x^{{2}/{3}}=4$. I rewrote it to $\sqrt[3]{x^2}=4$, but I don't know what to do next. Would the radical go away if I took the ...
3
votes
2answers
54 views

Solving $2\cos^2 x-2\sin^2 x-2\cos x=0$

$$f(x) = 2\cos^2 x-2\sin^2 x-2\cos x$$ Need values of x that which make $f(x) = 0$ Tried $a^2-b^2 = (a+b)(a-b)$ with no luck Really just need a hint that could bring me in the right direction ...
1
vote
1answer
32 views

If $g(x) = \max(y^2-xy)(0 \leq y\leq 1)\;,$ Then minimum value of $g(x)$

If $g(x) = \max\limits_{0 \leq y\leq 1}(y^2-xy)$, then minimum value of $g(x)$ $\bf{My\; try::}$ We can write $\displaystyle f(y) = y^2-xy = y^2-xy+\frac{x^2}{4}-\frac{x^2}{4} = ...
-2
votes
1answer
59 views

Solving $\sqrt{2}\times\sqrt{15}$ [closed]

Solve $$\sqrt{2}\times\sqrt{15}$$ Please explain in easy to follow steps
1
vote
4answers
46 views

What is the value of the expression $2x^2 + 3xy – 4y^2$ when $x = 2$ and $y = - 4$? [closed]

What is the value of the expression $2x^2 + 3xy – 4y^2$ when $x = 2$ and $y = - 4$? I'm not good at algebra so please explain in easy to understand steps. Thanks
0
votes
2answers
15 views

Help creating equation for parabola word problem?

The cables of a suspension bridge create a parabola. The towers are 600 feet apart and 80 feet tall. If the cable touches the road halfway between the towers, what is the height of the cable at a ...
0
votes
2answers
38 views

Solving for $x$ using $\ln$ or any possible way.

$$ 12.46x=1-(1+x)^{-20} $$ I tried solving for $x$ using $\ln$ and other methods but the only answer i got was 0.8. The correct answer is approximately to $0.05$.
2
votes
4answers
67 views

Show that $\sin45°+\sin15°=\sin75°$

Steps I took: 1) Finding the value of the left hand side $$\sin45=\sin\frac { 90 }{ 2 } =\sqrt { \frac { 1-\cos90 }{ 2 } } =\sqrt { \frac { 1 }{ 2 } } =\frac { \sqrt { 2 } }{ 2 } $$ ...
0
votes
3answers
45 views

Finding equation of a line that is perpendicular to another line

I need to find the equation of a line that is perpendicular to $2x-3y=1$, with the point $(1/4,-3/5)$. The answer I've got it $60x+40y+9=0$, but I can't figure out how to get there. The one page I ...
0
votes
1answer
17 views

Is my algerbra correct for my work dealing with the alternating series test for: $a_n = {(-4)^n \over n4^n} $

I have two questions (see "(1)/(2) Is this valid" sections) below. Given the series: $a_n = {(-4)^n \over n4^n} $ I'd like to see if this converges and think this is an alternating harmonic series. ...
5
votes
9answers
124 views

How to construct a bijection $\mathbb{N} \to \mathbb{N} \times \{0, 1\}.$

How to construct a bijection from $\mathbb{N}$ to $\mathbb{N} \times \{0, 1\}?$ My first idea was $n \mapsto (n, n \mod 2)$ but it is wrong. Any hint?
3
votes
2answers
42 views

Find all integers $a$ for which $x^2-x+a$ divide $x^{13}+x+90$.

Find all integers $a$ for which $x^2-x+a$ divide $x^{13}+x+90$. The answer is $a=2$.
0
votes
2answers
43 views

Solve for reals: $a(b+c-a^3)=b(c+a-b^3)=c(a+b-c^3)=1$

Solve for reals: $a(b+c-a^3)=b(c+a-b^3)=c(a+b-c^3)=1$ I found cyclic relation $c=(a+b)(a^2+b^2)$ and a solution $a=b=c=1$ But now I am not getting anything.
4
votes
1answer
22 views

Using the half angle formula

Find $\sin\frac { x }{ 2 }, \cos\frac { x }{ 2 }$, and $\tan\frac { x }{ 2 }$ from $\sin x=\frac { 3 }{ 5 } ,\quad 0°<x<90°$ What I did: $$\sin\frac { x }{ 2 } =\sqrt { \frac { 1-\frac { 4 }{ ...
2
votes
3answers
61 views

Prove using De Moivre's formula,that $\sum\limits_{k=0}^{n}\sin(kx)=\frac{1}{2}\cot(x/2)-\frac{\cos(nx+(x/2))}{2\sin(x/2)}$

I've been asked to prove that: $$ \sum\limits_{k=0}^{n}\sin(kx)=\frac{1}{2}\cot(x/2)-\frac{\cos(nx+(x/2))}{2\sin(x/2)} $$ When $0<x<2\pi$. I know there are many similar posts on this site, but ...
0
votes
2answers
102 views

4th grade word problem [closed]

I need some help figuring out this math problem for my friend's nephew! ...
0
votes
1answer
22 views

Visual way to understand Mixture Word Problem

Consider the following problem: How much antifreeze which is 30% alcohol must be removed from a 48-ounce container and replaced with water to make 48 ounces of a solution which is 20% ...
2
votes
3answers
51 views

How to isolate $n$ in the inequality $ 3n + 7n^3\gt c(17 + 34n^2) $?

I have an equation $$ 3n + 7n^3\gt c\left(17 + 34n^2\right) $$ and I want to turn this inequality into something like $$ n \gt c(\mbox{something that does not have}\ n) $$ I don't know why but ...