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
25 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 ...
6
votes
1answer
64 views

Product of numbers $\pm\sqrt{1}\pm\sqrt{2}\pm\cdots\pm\sqrt{n}$ is integer

Prove that the product of the $2^n$ numbers $\pm\sqrt{1}\pm\sqrt{2}\pm\cdots\pm\sqrt{n}$ is an integer. I want to consider the polynomial $P(x)=(x-a_1)(x-a_2)\ldots(x-a_{2^n})$, where the $a_i$'s are ...
1
vote
1answer
25 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: $$z_{1,2} = \frac{3-4i\pm \sqrt{(4i-3)^2 -4(5i-1)}}{2}$$ ...
5
votes
1answer
93 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 ...
1
vote
1answer
27 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 ...
3
votes
2answers
17 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 ...
4
votes
1answer
52 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)\\ ...
2
votes
6answers
88 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
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2answers
25 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 ...
0
votes
0answers
8 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
32 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 ...
3
votes
9answers
159 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 ...
0
votes
1answer
16 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 ...
1
vote
1answer
35 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 ...
0
votes
1answer
42 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
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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
58 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
46 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 ...
7
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0answers
44 views

Summation of cosine terms

I got stuck on the following problem: Let $q\in \mathbb{N}$ be a fixed odd number and $k,n \in \{ 1,…,\frac{q-1}{2}\}$. I want to show that $$ \left|1 + 2\sum_{j=1}^k \cos (\frac{2\pi n}{q}j) \right| ...
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
29 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
33 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
80 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 .
-5
votes
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
73 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 ...
1
vote
1answer
61 views

Series and Sequences Train Question

There's a question here that put me off, it differs from the normal AP/GP questions asked. A train is travelling at $180 \text { km/h }$, $500\text { m }$ away from a train station, what is the ...
0
votes
2answers
27 views

Rules regarding exponents

Given the following algebra problem: $$2^{n+1}-1+2^{n+1}=2^{n+1+1}-1$$ I know $2^{n+1}=2^n2^1$ but just to confirm the truth of the problem above, I just assumed the left hand side is $2^{n+2}-1$ ...
8
votes
1answer
157 views

How to solve $y^2=3x^4+3x^2+1$ for integers.

If $x,y \in \mathbb Z$ , then find all the solutions of $$y^2=3x^4+3x^2+1$$ I was asked this question by my friend who said that he encountered this while solving another problem. I have ...
1
vote
2answers
23 views

Question about converting a polar equation to a rectangular equation

$$\sec\theta =2$$ So I went through all the steps and got: $$\cos\theta =\frac { 1 }{ 2 } $$ $$\sin\theta =\pm \sqrt { 1-\frac { 1 }{ 4 } } $$ $$\sin\theta =\pm \frac { \sqrt { 3 } }{ 2 } $$ ...
0
votes
5answers
53 views

Why sometimes we get only one root of quadratic equations?

What is logic behind getting (sometimes) only one root of a quadratic equation which satisfies the equation?
-6
votes
4answers
62 views

Using differentiation [on hold]

The curve shown below has its equation: $y=3x^5-5x^3$ Find algebraically the coordinates of the points $A$ and $B$. ($7$ mark question)
3
votes
2answers
30 views

Converting a polar equation to a rectangular one

$$r=\frac { 4 }{ 1+2\sin\theta } $$ Steps I took: $$(1+2\sin\theta )r=\frac { 4 }{ 1+2\sin\theta } (1+2\sin\theta )$$ $$r+2r\sin\theta =4$$ $$r+2y=4$$ $$(r+2y)^2=16$$ ...
0
votes
0answers
29 views

AoPS Intermediate Algebra vs. Higher Algebra by Hall and Knight? And some more questions about learning math.

Ok. I'm learning algebra at the level of AoPS algebra 2, and I want to quickly progress through math. Allow me to explain the situation. I am highly interested in artificial intelligence/computer ...