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
votes
2answers
55 views

Differentiating $3\int_0^te^uf(u)du$

How can I differentiate this function - $$3\int_0^te^uf(u)du$$ Does differentiation cancel out the integration? And do I then have $$3(e^tf(t) - e^0f(0))$$ I am not sure if that is right and I'm ...
0
votes
2answers
66 views

How do I solve $980 = 98t + 1080e^ \frac{-t}{10}$ for t?

This came up in a differential equation, and I wondered if there is an algebraic way to solve this for $t$ without using WolframAlpha. Or is it a case of estimating with a power series?
1
vote
2answers
110 views

find $x,y,z$ such that $x+yz=M$, $y+zx=N$, $z+xy=K$

somebody asked me this. I don't know whether it is interesting but I hope someone can solve it. find $x,y,z$ such that $x+yz=M$, $y+zx=N$, $z+xy=K$, where $M,N,K$ are constants.
0
votes
1answer
113 views

Find the set of all natural number that make $(n+1)/(n+3)$ reducible

Lets assume $d$ is a natural number which makes $(n+1)/(n+3)$ reducible, then $d|n+1$ and $d|n+3$. $d|[n+3-(n+1)] = d|2$ which means $d=1$ or $d=2$. $n+1$ and $n+3$ must be divisible by $2$ so all ...
0
votes
1answer
33 views

How can I obtatain the regions for y?

In the following inequality: $$1+(y+s)^{2}>t(1+(y-s)^{2})$$,$t\in{R},s>0.$$$$$How can I obtain the regions for y? Sorry, I modified the second power due to mistake. $$$$ Thank you!
1
vote
1answer
37 views

Scaling sum of numbers to add up to particular value

I have a sum of real numbers $A_1 + A_2 + \cdots + A_N = A$ that add up to a known number $A$. All of the $A_1,\ldots,A_N$ are known as well. Is there a way of scaling the $A_1,\ldots,A_N$ so that ...
1
vote
1answer
47 views

Which statement regarding Lipschitz conditions is stronger?

Statement 1: A function $f$ satisfies a Lipschitz condition in the rectangular region $D$ if there is a positive real number $L$ such that $$|f(t, u) - f(t, v)| \leq L|u - v|$$ for all $(t,u) \in D$ ...
2
votes
1answer
444 views

Showing a function satisfies a Lipschitz condition

Have I got this right - $$ f(t,y) = 1 + t \sin(ty),\quad 0 \leq t \leq 2. $$ Here's as far as I have gotten - $|f(t, u) - f(t,y)|$ $= |1 + t\sin(tu) - 1 - t\sin(tv)|$ $= t\cdot |\sin(tu) - ...
2
votes
3answers
194 views

Proving that $\frac{n+1}{2n+3}$ is irreducible

I am trying to solve the following problem: Prove that the following fractions are irreducible for any n (n is a natural number and it cannot be null). $\frac{n}{n+1}$ $\frac{n+1}{2n+3}$ ...
5
votes
1answer
368 views

Upper bound of the sum $\sum_{i=2}^{N}{\frac{1}{\log(i)}}$

One of the questions in Sierpinski's book on number theory lead to finding a tight upper bound for the following sum: $$\sum_{i=2}^N {\frac{1}{\log(i)}}$$ The trivial upper bound like ...
9
votes
2answers
194 views

Determining the number $N$

Let $1 = d_1 < d_2 <\cdots< d_k = N$ be all the divisors of $N$ arranged in increasing order. Given that $N=d_1^2+d_2^2+d_3^2+d_4^2$, determine $N$. The divisors include $N$. It seems that ...
2
votes
2answers
165 views

how is $ 2\pi \int^R_{-R} \sqrt{R^2}dx = 4\pi R^2 $? (sphere area)

I am trying to understand the proof of the sphere area formula. In my math book they use the formula $y = \sqrt{R^2 - x^2}$ $-R \leq x \leq R$ They rotate the function above around the x-axis and ...
1
vote
1answer
53 views

Simple inequality on real numbers

Let $a, b > 0$ such that $a + b ≥ 1$. Show that $a^4 + b^4 ≥ \frac18$. What is the best possible approach on this problem?
9
votes
3answers
243 views

numbers' pattern

It is known that $$\begin{array}{ccc}1+2&=&3 \\ 4+5+6 &=& 7+8 \\ 9+10+11+12 &=& 13+14+15 \\\ 16+17+18+19+20 &=& 21+22+23+24 \\\ 25+26+27+28+29+30 &=& ...
2
votes
1answer
130 views

How do I solve $x\sin(x)=b$ for $x$?

How do I solve $x\sin(x)=b$ for $x$? This came up while I was trying to solve something else. It seems simple but I can't figure it out right now.
0
votes
1answer
105 views

How large does $t$ have to be for the exponentials in the solution to have decayed to $2\%$ of their original value?

Based on a solution given as: $x'' + 0.035x' + 0.00005x - 0.009 = 0$ Solve the characteristic equation. Based on your values of $r$, how large will $t$ have to be for the exponentials in the ...
8
votes
3answers
207 views

Other ways of solving $\cot^{-1}(x)=\sin^{-1}(x)$

Real solutions to $$\cot^{-1}(x)=\sin^{-1}(x)$$ I found this problem in an exam years ago and I solved it using geometry. The first mistake I made was assuming ...
-4
votes
1answer
109 views

strong equation!

Find $(x;y)$ sastisfied that: ...
3
votes
0answers
65 views

Existence of roots of a polynomial equation when coefficients have varying weights

I have two $n-$degree polynomials $f_{1}(p)$ and $f_{2}(p)$, where the domain of $p\in[0,1]$. I know that $\exists$ $0 < p_{1} < 1$ such that: $f_{1}(p_{1}) = f_{2}(p_{1})$. Let ...
0
votes
2answers
24 views

Trigonometric development help.

I need help with the following trigonometric development: $ x = r(\theta)\cos\theta$ $ y = r(\theta)\sin\theta$ this gives: $ x' = r'(\theta)\cos\theta - r(\theta)\sin\theta$ $ y' = ...
0
votes
3answers
96 views

Finding the ones digit for $2^{98}$

How can i find the ones digit for the number $$2^{98}$$
0
votes
2answers
31 views

Help with algebraic development within integral $k\int_{0}^{L}\sqrt{Lx-x^2}dx = k\int_{0}^{L}\sqrt{\frac{L^2}{4}-(x-\frac{L}{2})^2}dx$

I am having a bit of a hard time figuring this one out. Would be very nice if anyone would like to help me (no more forehead banging against wall :)) while: $ 0 \leq x \leq L$ The function is: ...
1
vote
1answer
381 views

Arrow-Debreu model of general equilibrium having many equilibria

I am just beginning to study some stuffs outside introductory/sophomore(?) micro/macroeconomics. And I met with a stuff called Arrow-Debreu model. The question is, 1) What would be the proof that ...
2
votes
2answers
409 views

Remainder Theorem with polynomial as divisor

I'm aware that in remainder theorem you take the divisor and make it equal to zero. For the result of that just plug in x into whatever the polynomial dividend is. But its different when the divisor ...
1
vote
2answers
166 views

Finding all real zeros of the polynomial

$$x^5 - 5x^4 +6x^3 -30x^2 +8x - 40 = 0$$ So far I have... $$r/s: +- 1, +- 40, +- 2, +- 20, +- 4, +- 10, +-5, +- 8$$ Only $+ 5$ works. Then I have $$(x + 5)( ) = x^5 - 5x^4 +6x^3 -30x^2 +8x ...
2
votes
1answer
137 views

Two cars meeting time

Between the cities A and B on the straight line there is city C. $|AC|= 180 \text{ (km)}, |BC| = 120 \text{ (km)}$. A=============C=========B Between A and C and B and C there are cars going back ...
0
votes
2answers
153 views

Evaluate $\lim_{n\to\infty} \frac{1}{n}\sum\limits_{k=1}^{2n}\frac{1-\ln(1+\frac{k}{n})}{(1+\frac{k}{n})^2}$

As shown in the title, I'm evaluating the following:$$\lim_{n\to\infty} \frac{1}{n}\sum\limits_{k=1}^{2n}\frac{1-\ln(1+\frac{k}{n})}{(1+\frac{k}{n})^2}$$ And I get stuck. Any ideas are welcome.
1
vote
1answer
106 views

Understanding revenue and profit math stuffs in labor theory of value

In http://wrongarithmetic.wordpress.com/2010/08/22/keen-i/, it talks about how economists Steve Keen's argument against Labor Theory of Value (LTV) is wrong. What I do not get is from This ...
1
vote
1answer
155 views

Finding the maximum area

If $P$ is a point inside quadrilateral $ABCD$ with $P A = 2$, $P B = 3, P C = 5$ and $P D = 6$, find the maximum possible area of $ABCD$.
2
votes
3answers
96 views

Is there any fast way to get the number of a certain day in a week

I'm realy sorry, if this question is a bit stupid... But this is my first time on mathematics stackexchange. Do you guys now for example, how to know the number of monday in a year?
1
vote
2answers
66 views

Polynomial degree problem

this problem is from Gelfand & Shen's Algebra book. Problem 171. The highest coefficient of $P(x)$ is $1$, and we know that $P(1)=0,P(2)=0,P(3)=0,\ldots,P(9)=0,P(10)=0$. What is the minimal ...
1
vote
1answer
309 views

Trig identity manipulation question

I'm working on manipulating trig identities and using Wolfram Alpha to check the identity still holds. I'm going from this: $$\frac{1-\cos x}{1+\cos x} = \frac{1}{tan^2x}-\frac{2}{\tan x \sin x} + ...
6
votes
2answers
292 views

Equating sums of square roots

I solved the following equation the hard way: $$\sqrt{x+1} +\sqrt{x+33}=\sqrt{x+6} +\sqrt{x+22}$$ The only solution is $x=3$. I am wondering if there is some easy observation that solves the equation ...
0
votes
1answer
77 views

Ordered set of integers

$\{x_i\}_{i = 1}^7$ is a set of 7 integers that satisfy $1≤ x_i ≤ 8$. How many such ordered sets of $7$ integers are there, such that $$x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 - x_1x_2x_3x_4x_5x_6x_7 ...
10
votes
7answers
2k views

Why is $\sqrt{8}/2$ equal to $\sqrt{2}$?

I am trying to help my daughter on her math homework and I am having some trouble on some equation solving steps. My current major concern relies on understanding why $\sqrt{8}/2$ equal to $\sqrt{2}$. ...
2
votes
3answers
160 views

How do I rearrange this formula? Circles around a larger circle.

My A-Level algebra is failing me. Can someone please tell me how to rearrange this formula to give $n$ when you know $R$ and $r$. $R \sin(180^\circ/n)/(1 - \sin(180^\circ/n)) = r$ This formula is ...
1
vote
0answers
30 views

How to prove that $\gcd(t^n-1,t^m-1)=t^{\gcd(n,m)}-1 $ [duplicate]

Possible Duplicate: Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$ Let $t$ be an element in any domain where gcd's exist. Then if $m,n$ are positive integer,prove that: ...
2
votes
1answer
100 views

Proving with factorials

Let x and y be the postive integers. Show that : $\displaystyle\frac{(x + y)!}{ (x + y)^{(x + y)}} < \frac{x! y!}{ (x^x + y^y)}$ Are there any identities we can use to easily prove this?
2
votes
1answer
159 views

The roots of $x^3+4x-1=0$ are $a$, $b$, $c$. Find $(a+1)^{-3}+(b+1)^{-3}+(c+1)^{-3}$

This is a question in A level Further Pure mathematics pastpaper Nov 2010. The roots of the equation $x^3+4x-1=0$ are $a$, $b$ and $c$. i) Use the substitution $y=1/(1+x)$ to show that the equation ...
2
votes
1answer
56 views

Finding the value of an expression

Given : $a + b + c = a^2 + b^2 + c^2 = a^3 + b^3 + c^3 = −2$, Find $a^4 + b^4 + c^4$
1
vote
1answer
46 views

Did I use Laplace correctly?

I haven't done Laplace transforms in a while and I wanted to know if I did this right. I start out with the expression $$\tau\frac{dT}{dt}+T(t)=T_{a}$$ I took the Laplace of this expression and got ...
0
votes
1answer
98 views

Solving for $x$ in $x(t) = \frac{-2}{3}\cos(10t) + \frac{1}{2}\sin(10t)$

A physics problem is asking me a to find when a weight on a spring crosses the equilibrium point. The equation of motion given is $$x(t) = \frac{-2}{3}\cos(10t) + \frac{1}{2}\sin(10t)$$ Basically, ...
0
votes
2answers
36 views

Please explain this distribution to me

I am refreshing algebra to get ready for calculus and i have a problem and my book doesn't explain it well. take this equation simplification -15(x/3) = -5 * 3 * (x * 1/3) Now, when i distribute ...
2
votes
4answers
224 views

What mathematical am I hearing?

I am currently hearing math at my university in germany and it is always difficult for me to translate the german topics into english. We are talking about limits , sequences and series ...
7
votes
2answers
167 views

Maximum of the difference

What is the maximum value of $f(… f(f(f(x_{1} – x_{2}) – x_{3})-x_{4}) … – x_{2012})$ where $x_{1}, x_{2}, … , x_{2012}$ are distinct integers in the set ${1, 2, 3, …, 2012}$ and $f$ is the absolute ...
4
votes
0answers
67 views

Digits of two irrational numbers, given their power with fixed number of digits

I have $a, b \in \mathbb{R} \setminus \mathbb{Q}$, I want to know the result of $a^b$, but I don't know exact $a, b$ because I write them in numeric form. My question is how many digits of $a, b$ have ...
1
vote
1answer
285 views

Struggling to solve a certain system of equations with unknown coefficients

$2 = A_0 + A_1 $ $0 = -A_0 + X_1A_1$ $2/3 = A_0 + X_1^2A_1$ I'm trying to find values for $A_0, A_1$, and $X_1$. If I knew $X_1$ this would be a pretty simple process (I'd probably just plug it ...
1
vote
1answer
324 views

Integer Partitions Formulas [duplicate]

Possible Duplicate: Identity involving partitions of even and odd parts. How would I go about to show the following: Let $pe(n)$ be the number of partitions of size n with an even number of ...
1
vote
1answer
155 views

How can I re-write an equation (or system of equations) in parametric form?

For the equation $y = 3x$ I need to re-write $x$ and $y$ in terms of a variable $t$. How can I find the value of each variable in terms of $t$?
3
votes
5answers
177 views

Polynomial proof

This is another problem from I.M Gelfand's book that I am stuck with. Problem 169. Assume that $x_1,\ldots,x_{10}$ are different numbers, and $y_1,\ldots,y_{10}$ are arbitrary numbers. Prove that ...