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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-2
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2answers
46 views

If $\sqrt{n}+ 8= n+1$, what is $n$? [on hold]

If $\sqrt{n}+ 8= n+1$, what is $n$? Please show as many steps as possible so I can understand the process.
0
votes
3answers
65 views

Discriminant of the polynomial $f(x)=4x^3-ax-b$

Definition. The discriminant of the polynomial $f(x)=4(x-x_1)(x-x_2)(x-x_3)$ is the product $16\{(x_2-x_1)(x_3-x_2)(x_3-x_1)\}^2$. How to prove that the discriminant of $f(x)=4x^3-ax-b$ is ...
1
vote
1answer
78 views

Finding the points of intersection of a circle and a line

In a test (of math in arabic language) we we're asked to find the points of intersection of a circle and a line. Their equation is given. In the test I solved system of equations made of their ...
2
votes
1answer
38 views

Is the simplest form of a quadratic equation factored form or standard form?

I've done a bit of research about what defines simplest form, but I could not find a clear answer. Suppose we had to choose: $$(x - 4)(x + 2) \quad\text{or}\quad x^2 - 2x - 8$$ A question asked me, ...
-2
votes
0answers
113 views

Is My Professor Wrong or I Am? [on hold]

In a test (of math in arabic language) we we're asked to find the points of intersection of a circle and a line. Their equation is given. In the test I solved system of equations made of their ...
0
votes
2answers
60 views

How to find all solutions of the equation $\sin x+\cos x=0$ which belong to $(-\pi, \pi)$?

Could you please help me understand and answer this question? Find all  the  solutions of this equation $$ \sin x+\cos x=0 $$ which belong  to  the interval $(-π; π)$ Progress Divided by ...
2
votes
1answer
57 views

Why this approach to differentiate $\log_{10}(x+1)^x$ does not work?

I am trying to differentiate $\log_{10}(x+1)^x$ but I don't get the correct answer, could you please help me? I know that one correct solution is the following: \begin{align} ...
-8
votes
0answers
50 views

hard question, please help [on hold]

11) Assume a sorted array (A) of size n. Propose an algorithm for finding two elements x and y in A that minimize |x-y|. Your algorithm should run in O(n) time for full credit. (Note: |x-y| represents ...
1
vote
3answers
46 views

How to answer the question “what is the domain of this function”?

Could you please help me understand and solve this problem about domain of function? All that is written for the question is: What is  the  domain of this function? $$ 2\sin\sqrt{2x-1}+1 $$ ...
-2
votes
2answers
72 views

If we know $a^{1 / 2} + a^{-1/2}$, how can we calculate $a + a^{-1}$?

Someone could help me with this? Thanks so much! Knowing the value of $a^{1/2}+a^{-1/2}$, calculate $a+a^{-1}$.
5
votes
1answer
55 views

What are some remarkable and interesting uses of AM-GM Inequality ? Cite and explain with problems.

There are really lot of problems on AM-GM inequality because of its elementary nature and powerful applications. What I want is a collection of questions/problems which look very complex but get ...
1
vote
1answer
58 views

2014 Fall OMO #28

Here is a problem from this year’s OMO: Let $S$ be the set of all pairs $(a,b)$ of real numbers satisfying $1+a+a^2+a^3 = b^2(1+3a)$ and $1+2a+3a^2 = b^2 - \frac{5}{b}$. Find $A+B+C$, where $$ A = ...
3
votes
1answer
21 views

inequality on real numbers.

Suppose one is given real numbers $\alpha_1, \lambda_0 \ge1$ and $ \lambda_1$ such that $\alpha_1^2\le1$ and $\lambda_0^2=1+\lambda_1^2$. Then it is easy to show that ...
-8
votes
0answers
37 views

need a pre calc answer ASAP on tangent functions [on hold]

My pre calculus teacher is making us use her way of tangent function graphing using Undefined, (-1, 0, 1, Undefined so basically finding your maximas and minimas which is easy enough for sin/cos ...
0
votes
1answer
24 views

$\mathbb E[\bar X_n]=0$

A conditional normal rv sequence, does the mean converges in probability, in this question how can i get $\mathbb E[\bar X_n]=0$? Here is my attempt; $$\mathbb E[\bar ...
8
votes
3answers
136 views

How to solve the differential equation $(2x^3y)\:\text{dy}+(1-y^2)(x^2y^2+y^2-1)\:\text{dx}=0$?

Solve $$(2x^3y)\:\text{dy}+(1-y^2)(x^2y^2+y^2-1)\:\text{dx}=0$$ I tried the substitution $y^2=t$ ; $2y\:\text{dy}=\text{dt}$ to get $$(x^3)\:\text{dt}+(1-t)[(x^2+1)t-1]\:\text{dx}=0$$ ...
1
vote
2answers
43 views

Closed form for $\sum_{t=0}^{n} t^2x^t$

I am trying to come up with a formula for $\Sigma_{t=0}^{n} t^2x^t$ I understand that $$\sum_{t=0}^{n} x^t=\frac{1-x^{n+1}}{1-x}$$ I also was able to find that $$\sum_{t=1}^{n} ...
-3
votes
0answers
19 views

Finding a quadratic function to determine how fast someone travelled [on hold]

I need to determine a quadratic function that would determine how fast a person was cycling given that they ran $6$ miles and then bicycle $20$ miles. If the bicycle speed is $8$ miles per hour faster ...
4
votes
1answer
53 views

Inequality $\frac{x^3+y^3}{x-y}>4$

Let $x>y>0$ and $xy\geq 1$. Prove that $$\frac{x^3+y^3}{x-y}>4.$$ Of course we can factor $(x^3+y^3)=(x+y)(x^2-xy+y^2)$, but it is not very useful. For fixed $x-y$, we can try to find the ...
1
vote
2answers
92 views

How does $n < 2^n$ become $\log n < n$ by taking log of both sides?

How does $n < 2^n$ become $\log n < n$ by taking the log of both sides? I understand the left side but I do not understand the right side of the inequality. The once was $\log 2^n$ becomes $n$ ...
3
votes
1answer
29 views

Project Motorola: setting up and solving an equation

Stuck on a homework project in a highschool college algebra question. I'm given the following information: Tact time is the average time to pick and place one part. Throughput is the number of ...
0
votes
2answers
29 views

Can this be rewritten as the following? [on hold]

Can $x(x^2-1)-1(x-1)$ be rewritten as $(x-1)(x^2-1)(x-1)$ ? It is during decomposition in factoring. Thanks.
1
vote
1answer
43 views

Partial Fraction Decomposition of Exponential Generating Functions

I want to see if it is possible to write $$ \left(\frac{x}{e^x-1}\right) \left(\frac{x^2/2! }{e^x-1-x}\right) \left(\frac{x^3/3!}{e^x-1-x-x^2/2}\right)$$ as a linear combination of the factors ...
2
votes
2answers
65 views

How did Sir Newton develop and formulate the famous binomial theorem?

After completing combination, I have started to read Binomial Theorem. My book only mentioned about Pascal's Triangle. And the formula was then given straightforward. But how did Sir Issac Newton ...
0
votes
0answers
31 views

how to add supremums

I need to prove that $$\sup(S)+\sup(T)=\sup(S+T)$$ I don't understand what $\sup(S+T)$ means, can you show me examples for groups $S$ and $T$ so this equation works. Thanks
2
votes
7answers
565 views

How to solve these equations for x and y..

equations are $(x-y)(x+2y)(2x+y) = 20$ and $x^2+xy+y^2 = 7$ i want the METHOD not the solutions
0
votes
1answer
31 views

Defining a list using set theory?

EDIT: Changed set If I have the following set of numbers: $\{1, 2, 4, 8, 16,...\}$ where the universe of discourse is natural numbers $\{0, 1, 2, ...\}$ How can I define this? I note that ...
-7
votes
0answers
48 views

how to factor this expressiion [on hold]

How to factor $\frac{2x^2-4x}{x+10}$ ?
8
votes
1answer
58 views

$(1-a)(1-b)(1-c)(1-d)\geq abcd$ for $a^2+b^2+c^2+d^2=1$

Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$. Prove that $$(1-a)(1-b)(1-c)(1-d)\geq abcd.$$ I thought about substituting $a=\sqrt{w},b=\sqrt{x}$, etc. (assuming first that $a,b,c,d$ ...
1
vote
1answer
27 views

Solution for a complexed equation

Find $z$ for the equation $e^z + e^{-z} = 0$. So $$e^z + e^{-z} = 0 \iff e^z = -e^{-z} \iff e^z = e^{\pi i - z} \iff z = \pi i -z + 2\pi ik$$ I understand all expect the $2\pi ik$. Can you ...
1
vote
1answer
26 views

find roots in the complexes

Find the roots of: $$ z^2 -3z +4iz = 1-5i $$ Rearranging the terms: $z^2 + z(4i-3) + 5i - 1 $ Solving by using the quadratic formula: $$z_{1,2} = \frac{3-4i\pm \sqrt{(4i-3)^2 -4(5i-1)}}{2}$$ ...
9
votes
1answer
109 views

Prove that $ ax^2+bx+c=0 $ has at least one root in $(0,1)$ if $10a+12b+15c=0$

If $10a+12b+15c=0$, Prove that $$ ax^2+bx+c=0 $$ has at least one root in $(0,1)$. Progress I tried to solve this by Rolle`s theorem ($f'$ has a root between any two roots of $f$), but could not ...
3
votes
1answer
30 views

Inequality $a^2b^2+2(a+b)\geq 4ab+1$

Let $a,b\geq 1/2$. Prove that $$a^2b^2+2(a+b)\geq 4ab+1.$$ We know that $(ab-1)^2\geq 0$ implies $a^2b^2+1\geq 2ab$, so the inequality reduces to $2(a+b)\geq 2ab+2$, or $a+b\geq ab+1$. But this is ...
4
votes
2answers
20 views

Evaluating $\sum_{i=a+1}^{N}\frac{i(i-1)}{i-a}$

I am trying to solve the German Tank Problem. There might be numerous ways of finding the expected value of N. However, the way in which I am proceeding, I need to find this sum. However I am stuck at ...
6
votes
1answer
63 views

Given $f(x)$ and $g(x)$, find $(fg)(x)$

I've attempted to solve the problem below, and here is what I got for a solution: Given $f(x)=x^2-9$ and $g(x)=x^2+3x-1$, find $(fg)(x).$ $$ \begin{align} (fg)(x)&=(x^2-9)(x^2+3x-1)\\ ...
5
votes
6answers
104 views

Solve $\sin2x +\sin x = 0$ algebraically

I am studying for a final and came across a review question that I have no idea how to do. The question is "Solve the equation $\sin(2x) + \sin(x) = 0$ on the interval $[0, 2\pi)$. I can graph it ...
1
vote
2answers
31 views

Is parametric form of a given function unique? [on hold]

Can we say that for any given function in single/multivariable, it is always possible to have a parametric form? (Elementary functions, complicated functions?) Given any function, is parametric form ...
1
vote
0answers
14 views

Macaulay duration for a coupon bond. Proof

I am working on showing the following. There is a coupon bond redeemable at par with annual coupon rate $r$ per year. The yield to maturity is $i$. The total number of coupons is $n$. Show ...
1
vote
2answers
38 views

How much thrust is required to move a boat of 120 kg / 265 pounds to a speed of … [on hold]

How much thrust is required to move a boat of 120 kg / 265 pounds to a speed of 10 km / 6 miles per hour in 7 seconds. I found the following: http://en.wikipedia.org/wiki/Thrust-to-weight_ratio ...
4
votes
9answers
171 views

Why doesn't $e=1$?

I'm sure that this is a very basic question, but it has been bothering me for a while: If $e=\lim\limits_{x\to \infty} (1+x^{-1})^x$, shouldn't $e=1$? If $x$ is tending towards infinity, why ...
1
vote
1answer
17 views

Airplane Wind problem

Airplane flying at 400 mph at an angle of 30 deg encounters a wind. The resultant velocity of the airplane is 475.3 mph at an angle of 27.18 deg. What was direction of the wind. I set this up as ...
2
votes
1answer
38 views

The function f is defined as follows: $f:A \to A$

The function f is defined as follows:$f:A$ to $A$ where$$ f(x)=\frac{3(x +1)}{x^2-1}$$ Along my proof in showing that show that there exists an x ∈ A with $f(x) = y$ (showing f is onto) ,I ran into ...
1
vote
1answer
44 views

Show that $f(x,y,z)=0$ if and only if $(\sqrt {x^2+y^2}-1)^2+z^2=r^2$.

Define $f(x,y,z)=(x^2+y^2+r^2-z^2-1)^2-4(x^2+y^2)(r^2-z^2)$, where $0<r<1$ Show that $f(x,y,z)=0$ if and only if $(\sqrt {x^2+y^2}-1)^2+z^2=r^2$. Here is what I have tried: Let ...
0
votes
2answers
45 views

Reverse an equation with ln and power

I'm trying to solve for $x$ in the following equation: $\ln(y) = a \cdot (\ln(x)) ^ b + c$ $a = 0.0838 b = 2.6275 c = 0.2506$ but my results look bad. Can anybody show me his demonstration ? Thanks ...
0
votes
0answers
32 views

Algebra Questions-Academic

ı must solve this ı only know question 2)s' b) option is Euler's prime-generating polynomial.But ı dont know to show that,too.please help :(
1
vote
1answer
61 views

No real $x,y$ such that $(x+y)^2+(x-2)^2+(y-2)^2=4$

Here's the context of this problem. Solve: $x^2=y^3-3y^2+2y$ $y^2=x^3-3x^2+2x$ We subtract the second equation from the first and obtain $$(x-y)(x^2+y^2+xy-2x-2y+2)=0$$ The first ...
0
votes
1answer
47 views

Reasoning behind multiplying by conjugates

What is the reason behind multiplying by conjugates? I am currently studying single variable calculus and throughout the lessons from the text I'm using, the reasoning as to why one would multiply by ...
0
votes
0answers
20 views

Factoring a Polynomial to Find Tangent Line

I have a polynomial equation $ x^n + a x^{n-1} + bx^{n-2} ... + z =0$ for which the coefficients depend on a parameter $ t $. The equation has one real root that I am interested in. For this real ...
0
votes
1answer
27 views

How would I graph this polar equation?

$$r=-2cos\theta $$ Steps I took: $$r^{ 2 }=-2x$$ $$x^{ 2 }+y^2=-2x$$ $$x^{ 2 }+y^{ 2 }+2x=0$$ Usually I can complete these problems by completing the square in order to find the equation of the ...
-2
votes
2answers
30 views

Calculus question on radioactive decay help [on hold]

A radioactive substance decays by $88.1\%$ every $3$ years. What is the half-life of this substance, in years?