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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1
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2answers
85 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
27 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
28 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
33 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
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0answers
30 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
3
votes
6answers
537 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 ...
-6
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0answers
45 views

how to factor this expressiion [on hold]

How to factor $\frac{2x^2-4x}{x+10}$ ?
8
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1answer
55 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
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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
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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: $$z_{1,2} = \frac{3-4i\pm \sqrt{(4i-3)^2 -4(5i-1)}}{2}$$ ...
6
votes
1answer
102 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
19 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
59 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
97 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
27 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
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0answers
13 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
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2answers
36 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
169 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
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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
36 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
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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
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1answer
60 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
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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?
0
votes
2answers
18 views

Expressing a polar equation in rectangular form and then graphing it

$$\theta =-\frac {\pi}{ 2} $$ This question confuses me because the only way to find the Cartesian coordinates for this must be by using tangent. And this is where I get confused: $$ \tan\theta ...
2
votes
3answers
45 views

Arithmetic progression with deceleration

A train is travelling at $180 \text { km/h }$, $500\text { m }$ away from a train station, what is the constant deceleration needed to get to a complete stop at the station. A continued question ...
0
votes
2answers
84 views

How did they solve for a here?

Consider the following algebraic steps: $$ F - (M_1 a + \mu_k M_1 g) - \mu_k M_2 g = M_2 a $$ $$ F - \mu_k M_1 g - \mu_k M_2 g = (M_1 + M_2) a $$ $$ a = \frac{F - \mu_k M_1 g - \mu_k M_2 g}{(M_1 + ...
-1
votes
2answers
34 views

When to apply rules of logarithms, order of operation

Sometimes I get a little confused with what order to do things in when it comes to $ln$ being raised to the natural base. For example $e^{\int -A\ln{x} dx}$ where $A$ is an arbitrary constant. Should ...
3
votes
4answers
83 views

How to solve the system $x y^5=8000$ and $x y^4>4100$?

I need help getting this equation solved for a website I am building. I am pretty bad at math and am only in pre-algebra. I don't know how I would go about canceling out the ^5 and ^4 because I can't ...
2
votes
1answer
40 views

Periodicity of an infinitely differentiable function

Consider $f:[-\pi,\pi] \to \mathbb{C}$ be an infinitely differentiable function with $f^{(n)}(-\pi) = f^{(n)}(\pi)$ for all $n \in \mathbb{Z}^+$. Is this a periodic function ? I think it is a ...
-2
votes
0answers
32 views

algebra question MATH [on hold]

Find the indicated function and write its domain in interval notation. m(x) = , n(x) = x + 3, (m n)(x) = ? A) (m n)(x) = ; domain: [-5, ∞) B) (m n)(x) = (x + 3); domain: [-2, ∞) C) (m n)(x) = ...
-6
votes
1answer
22 views

Read properties of a function from its graph [on hold]

Use the graph of $y = f(x)$ to answer the questions. a. Determine $f(-1)$ b. Find all $x$ for which $f(x) = -4$ A) $f(-1) = -4$; $f(x) = -4$ for all $x$ on the interval ...
0
votes
4answers
93 views

$a_1^3+a_2^3+…+a_n^3=0 \Rightarrow a_1+a_2+…+a_n=0$ it is true or not? [on hold]

I have a question about this hypothesis/theorem : $a_1^3+a_2^3+\cdots+a_n^3=0 \Rightarrow a_1+a_2+\cdots+a_n=0$ It is true or not ? If it is true please give a reference .
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3answers
83 views

Is there any solution to this problem? [on hold]

$$\large 10^\alpha = \alpha^{50}$$
2
votes
2answers
51 views

Intersection of two circles.

Let $C_1$ and $C_2$ be the circles: $\rho=a\sin\theta, \rho=a(\cos\theta + \sin\theta)$ respectively. The graphs of these two circles are From the graphs, we see that the intersection points are ...
2
votes
2answers
76 views

How can I understand solving the equation?

$$\begin{align} &\left[(\sqrt[4]{p}-\sqrt[4]{q})^{-2} + (\sqrt[4]{p}+\sqrt[4]{q})^{-2}\right] : \frac{\sqrt{p} + \sqrt{q}}{p-q} \\ &= ...
0
votes
1answer
31 views

Complex roots of Complex polynomal

Apologies if this is a repeated thread I just couldn't quite find anything that helped. how do I go about finding the complex roots of a complex polynomial? such as $$x^3 + (1-i)x^2 + (1-i)x - i$$ ...
2
votes
1answer
32 views

Simple computation question about the limit of a function including little oh

Consider a sequence $$c_n:= t + o(t/n)n$$ where $o(\cdot)$ denotes little-oh I want to compute $\lim_{n\to \infty} c_n =?$ I guessing the result should be $\lim_{n\to \infty} c_n = t$ but not sure. ...
-5
votes
1answer
31 views

Boyle's Law Problem [on hold]

This question is confusing me as I don't know what I'm looking for "A popular size of scuba-diving tank holds the amount of compressed air that would occupy $71.2 \text{ ft}^3$ at a normal surface ...
1
vote
4answers
86 views

Rewrite $\sin(\cos^{-1}(x)-\tan^{-1}(y))$ as an algebraic function of $x$ and $y$.

Rewrite the expression as an algebraic function of $x$ and $y$: $$\sin(\cos^{-1}(x)-\tan^{-1}(y)).$$ I am unsure of how to change this into an algebraic function, yet I am able to simplify inso sin ...
0
votes
0answers
26 views

Find all solutions in the interval $[0, 2\pi)$: $5\cos(2\theta)=2$

Find all solutions in the interval $[0, 2\pi)$ rounded to five decimal places: $5\cos(2\theta)=2$. I began by using the double angle formula for $\cos(2\theta)$ and substituting with $1-\sin^2 ...
0
votes
2answers
21 views

How many of each ticket were sold in one day?

Child tickets - $\$7$ Adult Tickets - $\$10$ Senior Tickets - $\$5$ Day one sold $678$ tickets for $\$5,812$ Day two sold $535$ tickets for $\$4,541$ How many of each ticket were sold on day one ...
1
vote
2answers
18 views

Solved ODE, how did answer key rewrite solution to be in this form?

I was solving the ODE $\frac{dx}{dt} = 4(x^2+1)$ with initial condition $x(\frac{\pi}{4})=1$ I got $\tan^{-1}{x} = 4t+c$ Then I plugged in the initial value and rewrote to get ...