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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5
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5answers
57 views

Finding the sum of $\sin(0^\circ) + \sin(1^\circ) + \sin(2^\circ) + \cdots +\sin(180^\circ)$

I need help understanding the sum of $\sin(0^\circ) + \sin(1^\circ) + \sin(2^\circ) + \cdots +\sin(180^\circ)$ or $\displaystyle \sum_{i=0}^{180} \sin(i)$ This might be related to a formula to find ...
0
votes
0answers
6 views

Linear Algebra Analytical Exercise

This one has me stumped... $$H=C(sI-A)^{-1}B$$ and $$H_{CL} = C(sI-A+BK)^{-1}BG$$ Show that $$H_{CL} = H[I+K(sI-A))^{-1}B]^{-1}G$$ Any hints would be greatly appreciated!
0
votes
0answers
8 views

Geometric progression in annuity

I am working on the following problem that involves annuity which deposits form a geometric progression. Stan elects to receive his retirement benefit over $20$ years at the rate of $2,000$ per ...
-5
votes
4answers
43 views

Tea with grandmother

My mother still makes tea with the old saying: one spoon per person and one for the pot. We used to buy a packet of tea every week but since grandmother came to live with us we have to buy two packets ...
0
votes
2answers
33 views

Mathematics and logs

For this indirect utility function: v(y) = ln(1/3 y) + 2 ln (2/3 y) = 3 ln(y) + ln (4/27) How did they simplify to 3 ln(y) + ln (4/27)? Im a bit confused, if there is a site helping with this ...
-1
votes
3answers
32 views

Why examples of the order of algebraic computations do not agree with calculator results?

This my first lesson in Algebra I replaced the dot signs with x. I've never seen times done with a dot before Lesson taken from MathPlanet Operations in the correct order When you are faced with a ...
7
votes
3answers
46 views

Function composition: $f^{653}(56)=?$

Let $f(x) = \frac1{(1-x)}$. Define the function $f^r$ to be $f^r(x) = f(f(f(...f(f(x)))))$. Find $f^{653}(56)$. What I've done: I started with r=1,2,3 and noticed the following pattern: $$f^r(x)= ...
0
votes
0answers
9 views

Calculating the interest rate for an annuity (Exam FM)

I have been searching for a way to solve for the interest rate given the monthly payments of a loan. I would like to set up a problem as the following. $X$=monthly payment , $i$=effective ...
-1
votes
1answer
28 views

Factoring/Expansion explantion

Sorry if I call something by the wrong name since I didnt learn math in english. ok so for example this: (a+b)(a-b) if you break it down to the second "()" you will end up with this: a+-b could ...
0
votes
2answers
41 views

Why is $y{(\log_a(x))} = \log_a{(x^y)}$?

Why is $y{(\log_a(x))} = \log_a{(x^y)}$? I feel like I'm missing something here. Sorry if I put the title wrong..
1
vote
2answers
29 views

Trouble understanding algebra in induction proof

I'm on hour 20 of studying for the discrete math midterm tomorrow, and I've got to be honest I'm a little panicked. In particular I'm having trouble with induction proofs, not because I don't ...
1
vote
1answer
23 views

Remainder theorem thinking question given properties of the original equation

Consider a cubic polynomial function $y=f(x)$ with the following properties: $f(x) \ge 0$ only for $x=-1$ and $x\ge3$ when $f(x)$ is divided by $(x-4)$ the remainder is $50$. Find the equation ...
2
votes
4answers
611 views

Confusing algebra rule in discrete math

Math rule I don't understand. Hey guys, my discrete math midterm is tomorrow and I'm studying proof styles. I came across a rule (algebra maybe?) I don't quite understand and I was hoping someone ...
1
vote
2answers
22 views

How to solve higher grade polynomials of complex numbers $q^{10}-2q^5+2=0$

If I wanted to find the roots for $q^{10}-2q^5+2=0$, how would I go about doing that? I tried treating it like a quadratic equation, but couldn't get there. I also tried putting $q=(a+ib)$ but that ...
0
votes
1answer
16 views

between what two disjoint sections we can do a unification in order to get this group of solutions?

between what two disjoint sections we can do a unification in order to get this group of solutions? $0<|x+6|\leq{0.4}$ in other words, in what values should I fill the blankets: (____,____) ...
3
votes
3answers
50 views

How can I find the point where two algebraic equations, in the form $y=mx+b$, intersect without graphing?

Suppose I have these two algebraic equations in the format $y=mx+b$: $$ y=2x+4 \\ y=3x+5 \\ $$ Now, by graphing these two algebraic equations on a coordinate plane, I find that they intersect at the ...
5
votes
3answers
97 views

Solve $x^3 - x + 1 = 0$

Solve $x^3 - x + 1 = 0$, this cannot be done through elementary methods. Although, this is way out of my capabilities, I would love to see a solution (closed form only). Thanks!
-1
votes
2answers
93 views

What is the formula to generate this number sequence : 1 , 7 , 14, 30

What is the formula to generate this number sequence : 1 , 7 , 14, 30 I'm sure this is very simple for you guys. But it's got me alittle stuck. Thanks To clarify, I'm not an advanced maths student. ...
2
votes
1answer
32 views

Solving $z^2-2iz+1=0$ in complex numbers

Solve: $z^2-2iz+1=0$ I did: $$(z-i)^2-(i)^2+1=0$$ $$(z-i)^2+2=0$$ $$((z-i)-\sqrt{2})((z-i)+\sqrt{2})$$ but that's wrong. Why?
1
vote
5answers
104 views

Sum of $1+2+4+8+…$ [duplicate]

I was solving a recurrence problem which had a sequence such as $y = (1+2+4+8+...)\sqrt n$, and I wanted to find what $x = 1+2+4+8+...$ was. So consider $x = 1+2+4+8+...$ as an infinite series. $$x-1 ...
0
votes
2answers
22 views

Factoring Polynomial with Complex Coefficients - Cauchy's Theorem

I'm faced with another polynomial (with complex coefficients) that I seem to only know how to solve using wolfram alpha. Here is the original integral that I need to compute using algebra and Cauchy's ...
1
vote
1answer
27 views

Irrational numbers and proving constant functions

Let $f:\mathbb R \to \mathbb R$ be a function such that for any irrational number $r$, and any real number $x$ we have $f(x)=f(x+r)$. Show that f is a constant function. How would we go about solving ...
1
vote
4answers
33 views

Basic Math, exponents and algebra

I have the equation $$\frac{x_1^{-\frac{1}{2}}}{{x_2^{-\frac{1}{2}}}} = p_l/p_2$$ How do I get $x_2$ on its own? I have $$x_2^{-\frac{1}{2}} = \frac{p_2(x_1^{-\frac{1}{2}})}{p_1}$$ And if you have a ...
0
votes
1answer
13 views

Quick question on relative minima/maxima of a cubic function

I need help with #36 on this page. Is there a way to solve without looking at the graph? Thanks!
0
votes
1answer
45 views

Efficient way of calculating this?

Is there an efficient way for calculating the following? Find the sum of all integer values of $y$ where $y=\frac { 1+\sqrt { 4x+1 } }{ 2 } $ where $x$ is a positive integer in the range ...
4
votes
1answer
29 views

Efficiently solving algebraic equation

I would like to solve following equation: $$15 (x+2)^{-4} = 11(x+2)^{-2} +4$$ I would first remove the negative power by adding $(x+2)^4$ Then I get $$15 = 11(x+2)^2 + 4(x+2)^4\\ 11(x+2)^2 + ...
0
votes
1answer
13 views

Using descartes rule of sign

Use Descartes' rules of signs to discuss the possibilities for the roots of each equation. Do not solve equation. $$p(x)= x^3+5x^2+7x+1=0$$ $p(x)$ I saw no sign change $p(-x)$ I saw 2 sign ...
2
votes
1answer
19 views

Beautiful number theory numeral system problem.

The product of two consecutive positive integers was taken and converted into some numeral base-n system. In this base-n system it's written as a double-digit number with consecutive positive integer ...
13
votes
2answers
700 views

Explain $x^{x^{x^{{\cdots}}}} = \,\,3$

$$x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} =\quad 2$$ This equation has the answer $\sqrt{2}$ by taking $\log$ to both side. This answer is correct because I'd proved it by computing the equation repeatedly ...
1
vote
1answer
45 views

Solving $2^b=3^b -1$ where b is a natural number

Solving $2^b=3^b−1$ for natural b. I tried factoring out $3^b-2^b$, but what next then? It's obviously 1, but I have no idea how to prove it.
0
votes
4answers
56 views

How many boxes can be painted while respecting this restriction?

We have 30 boxes in a line: $x_1,x_2,...,x_{30}$. Some of them we can color in red. The rule is that if $x_k$ is colored red then $x_{k+2}$ can't be colored red and vice versa. What is the maximum ...
0
votes
1answer
39 views

need help calculating the interest “i”

A regular deposit of 120 dollar made at the beginning of each year for 20 years. Simple interest is Calculated at a rate of i per year for 22 years. At the end of the 22-year period, the total ...
1
vote
2answers
41 views

Solve for reals $x, y\in \mathbb R$ given system of two non-linear equations.

Solve for reals:- $$\begin{align} 5x\left(1+\frac{1}{x^2+y^2}\right)& =12\\ 5y\left(1-\frac{1}{x^2+y^2}\right)&=4\end{align}$$ I got this relation $$6x^{-1}+2y^{-1}=5$$ Now I substituted ...
-5
votes
0answers
44 views

need help . give answer clearly please.. [on hold]

A regular deposit of 120 dollar made at the beginning of each year for 20 years. Simple interest is Calculated at a rate of i per year for 22 years. At the end of the 22-year period, the total ...
20
votes
10answers
3k views

Can you cancel out a term if equal to zero?

quick question here: In my proofs class we had a problem that after a little work we end up with: $x(x-y)=(x+y)(x-y)$ where $ x = y $. Now, I know this is pretty basic, but my teacher said that for ...
0
votes
3answers
45 views

Simple real life problem

Let say my girlfriend makes $2000$ Euros per month and I make $3400$ Euros, let say all our living costs sum up to $1540$ Euros. How can i calculate how much each of us must pay (off the $1540$ Euros) ...
2
votes
3answers
23 views

Substitution with trigs (not integrals)

I am currently trying to find out the values of $x$ and $\theta$, given the two following statements. I have tried using substitution, but I am not quite getting it. I would be extremely glad if I ...
1
vote
1answer
33 views

Annuity/Finance

I'm trying to determine the question below: Mr. Learnwell wants to setup a scholarship of $4500 paid at the end of every six months. If the interest rate is 6.4% compounded semi-anually, how much ...
0
votes
2answers
19 views

Partial Fractions (3 Factors)

This document outlines a shortcut for partial fractions involving 2 factors in the denominator (P(x) + a) and (P(x) + b). At the end of the document it gives a challenge to find a similar shortcut ...
1
vote
0answers
20 views

Is shunting-yard algorithm needed if there is no parenthesis?

I am trying to find a permutation (with replacement) of operators(such as addition and multiplication) that makes numbers 1 2 3 , ..., 9 result in to some numbers. My guess is to find all possible ...
0
votes
2answers
35 views

current value of a forward contract

Suppose we hold a forward contract on a stock with expiration 6 months from now. We entered into this contract 6 months ago so that when we entered into the contract, the expiration was T=1 year. The ...
0
votes
2answers
45 views

Difficulty understanding addition of exponents

If you rewrite $(3^{12})(3^{-12})$ in the form $3^n$ what does it equal? What is the intuition behind it? Do exponents cancel each other out so it is just $3$? or do the negatives cancel out $((-x) ...
0
votes
2answers
32 views

Locus given by a pair of scissors sliding along the ground.

I came up with this problem when dragging a pair of scissors along the ground. The question is, more mathematically: Suppose there is a point (a,0) and a point (0,b) with a fixed distance m between ...
1
vote
3answers
32 views

Use a function to represent positive real numbers?

Is it correct to define the positive real numbers as $\{f(x) = x^2\mid x \in \mathbb R\}$?
0
votes
0answers
36 views

When Is This Equation an Integer?

I'm working on a math problem, but I've gotten stuck. When does $$-\frac{4(2^Q-3^Q)}{2^{Q+S}-2*3^Q}$$ produce an integer?
1
vote
1answer
14 views

Determine values of the constants a & b so the limit exists and is equal to f(2) in a piece-wise function

I am trying to determine the values of the constants a & b in a piece-wise function that has to satisfy these parameters: The limit f(x) as x approaches 2 does exist and is equal to f(2) The ...
-4
votes
1answer
35 views

please help me with this [on hold]

the cooling system of a car has a capacity of 15 liters if the system is currently filled with a mixture that is 10% antifreez how much of this mixture should be drained and replaced with pure ...
0
votes
1answer
15 views

Maximal value of $\vert r^2-n\vert$ with a special condition

let $M,n\in \mathbb{N}$ and $R=\lbrace r\in \mathbb{N} \mid \vert r- \sqrt{n}\vert <M<2\sqrt{n}\rbrace $. I have to show that the maximal value of $\vert r^2-n\vert $ for $r\in R$ is at most ...
1
vote
2answers
26 views

How is it sometimes helpful to use cross multiplication in order to complete proportions with a variable?

How can it be helpful to do cross multiplication with proportions with variables such as ${2\over 4}={3\over x}$? In this one, the value of x has to be found. It can be found this way: 1. Do the ...
1
vote
4answers
45 views

systems of linear equations intuition

I want to know why in a system of linear equations I'm allowed to sum or subtract the equations. I can't get the intuition of why I can do that to solve for the equations.