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
39 views

Find the domain of $g(x) = \ln \left( {\frac{x}{{x - 1}}} \right)$

I know the argument of the function has to be greater than 0, so: $\eqalign{ & \left( {\frac{x}{{x - 1}}} \right) > 0 \cr & x > 0 \cr} $ however in this case $x \ne 1$, $x \ne 0$ ...
0
votes
1answer
41 views

Is there an absolute value sign missing here? Factoring a term out of a square root.

$$\int_{1}^{8}\frac{x^2}{27x^2/8}\sqrt{1+x^{-2/3}}dx=\frac{8}{27}\int_{1}^{8}x^{-1/3}\sqrt{1+x^{2/3}}dx$$ Shouldn't $x^{-1/3}$ be $|x^{-1/3}|$ because $\sqrt{x^2}=|x|$?
0
votes
2answers
33 views

The range of $b - c \sin x$

The function $f(x)$ is defined by: $$f(x) = b - c\sin x, 0\le x \le 360$$ where $b$ and $c$ are constants, $c > 0$. Find the range of $f(x)$, in terms of $b$ and $c$. I ...
1
vote
4answers
70 views

$a^{1/n}$ - How do you explain it cannot be $0$ for $a > 0$?

How do you explain formally, $a^{1/n}$ cannot be equal to $0$ for every $a > 0$? Thanks
0
votes
1answer
97 views

How do you find the equation of motion in the absence of damping (x(complementary) and x(particular) solved)?

Let $x(0) = 0$, $x'(0) = 0$, and take a particular solution $x_p = e^{-2t}(\frac{1}{2}\cos(4t) - 2\sin (4t))$ and homogeneous solution $x_c = c_1 e^{8t} + c_2 e^{8t}$. So I put those two together to ...
0
votes
2answers
436 views

Show that the equation of the folium of Descartes in terms of $x$ and $y$ is $x^3+y^3=axy$

I'm given that the parametric equations are $x=\frac{at}{1+t^3}$ and $y=\frac{at^2}{1+t^3}$ and that $a>0$ Here's my attempt at a solution: Find $x^3$ and $y^3$ in terms of $t$.. ...
3
votes
2answers
196 views

Find a simplified form for $n!+(n-1)!+(n-2)!+(n-3)!+ \dots +1!$ .

Find a simplified form for $n!+(n−1)!+(n−2)!+(n−3)!+\dots+1!$ . By simplified, I mean that there should not be "..." in the equation. If there isn't one, prove it. Thanks! Edit 1: Is it possible ...
0
votes
1answer
99 views
1
vote
0answers
66 views

How to use an exponent that contains a variable

I am trying to understand a problem that uses mathematical induction to prove the validity of a statement. This is how one section moves to another: $$ 2k + 3 = 2^{k + 1} $$ $$ 2k + 3 = (2k + 1) + 2 ...
0
votes
2answers
48 views

Find the period of $y= 3\sin(8(x+4))+5$

I am doing a math problem for my homework and I know I got the answer wrong by looking at the back of the book. I am just trying to find out how to get that answer for future reference. The question ...
0
votes
2answers
165 views

Function transformations: reflections and stretches

I don't understand how to get this answer. $1)$ Suppose that the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f$. $$y=-2f(x)$$ Answer: Reflect in ...
0
votes
1answer
37 views

computing and algebra

I have a question: Let $f(x)=\sqrt{x^2 +1} - 1$. When $x=10^{-3}$ compute $f(x)$ working to 5 sf. Show algebriacally $f(x)=\frac{x^2}{\sqrt{x^2+1}+1}.$ After desperately rearranging I'm just going ...
0
votes
1answer
40 views

$-ia(1\pm \sqrt{1-1/a^2})$, $a>0$ inside unit circle?

Given $a>0$ I would like to know whether: $\alpha=-ia(1+ \sqrt{1-1/a^2})$ and $\beta =-ia(1- \sqrt{1-1/a^2})$ are inside the unit circle. How can I check that?
1
vote
0answers
85 views

Color adjustment with reference pixel values

First of all, I apologise if this is the wrong place to post this question, but it seems highly mathematical for me so I'm giving it a shot: I'm trying to build an application where a couple things ...
12
votes
6answers
3k views

Factorize the polynomial $x^3+y^3+z^3-3xyz$

I want to factorize the polynomial $x^3+y^3+z^3-3xyz$. Using Mathematica I find that it equals $(x+y+z)(x^2+y^2+z^2-xy-yz-zx)$. But how can I factorize it by hand?
0
votes
1answer
93 views

Geometric sequences interest question

Winston invests a sum of money at 6% per annum. How many years does it take him to double his money? I let the initial sum of money be $£a$. Then at the end of the first year, he has $£1.06a$ since ...
0
votes
1answer
37 views

Show that a simple inequality holds.

I'm trying to show that $\dfrac{a}{b} < \dfrac{a + 1}{b + 1}$ given $a < b$ and $b$ is positive. Ideas?
0
votes
1answer
286 views

Expressing logarithms as ratios of natural logarithms

$$\frac{\log_2 x}{\log_3 x}=\frac{\ln x}{\ln 2} \div \frac{\ln x}{\ln3}$$ Why can logarithms be written as ratios of natural logarithms? Can you explain it abstractly, please? Example of an ...
22
votes
6answers
848 views

If $(\sqrt{y^2-x}-x)(\sqrt{x^2+y}-y)=y$ then $x+y=0$.

Let $x,y$ be real numbers such that $$\left(\sqrt{y^{2} - x\,\,}\, - x\right)\left(\sqrt{x^{2} + y\,\,}\, - y\right)=y$$ Show that $x+y=0$. My try: Let ...
0
votes
1answer
31 views

Question about basic exponential/logarithm properties

Solve for $k$: $$e^{k/2}=a$$ Solution: $$e^{2k}=a$$ $$ k/2 = \mathbf{ln}a$$ $$ k=2\mathbf{ln}a$$ $$= \mathbf{ln}a^2$$ My question is: why does $2\mathbf{ln}a = \mathbf{ln}a^2$? Why can you ...
1
vote
1answer
1k views

Intersection of a plane with an infinite right circular cylinder by means of coordinates

So, I started studying analytic geometry and I must say I'm finding it much harder than "classic" geometry, because of the equations without help from diagrams... Still, I wanted to see how to use it ...
1
vote
1answer
49 views

Important polynomial identities?

I've learnt that there is $x^n-y^n$ and $x^n-1$ but are there any more I should learn for aiding with factorization? Also if you have a series such as $s_n = a + ar + ar^2 + ar^3 + ... + ar^{n-1}$ if ...
3
votes
2answers
609 views

$\sin ^6x+\cos ^6x=\frac{1}{8}\left(3\cos 4x+5\right)$, Any quick methods?

How to prove the following equation by a quick method? \begin{eqnarray} \\\sin ^6x+\cos ^6x=\frac{1}{8}\left(3\cos 4x+5\right)\\ \end{eqnarray} If I use so much time to expand it and take extra care ...
0
votes
1answer
59 views

You can write ${\left( {\frac{1}{2}} \right)^x}$ as ${2^{ - x}}$ , can the same be done with ${\left( {\frac{2}{3}} \right)^x}$?

You can write ${\left( {\frac{1}{2}} \right)^x}$ as ${2^{ - x}}$ as: ${\left( {\frac{1}{2}} \right)^x} = {({2^{ - 1}})^x} = {2^{ - x}}$ But what about ${\left( {\frac{2}{3}} \right)^x}$? Can it be ...
1
vote
5answers
60 views

Finding the image of $f(x)=\frac{1}{1+x^{2}}$

$f(x)=\frac{1}{1+x^{2}}$ and $x\geq0$ To find the image: $y=f(x)$ $y=\frac{1}{1+x^{2}}$ $x=y^{-1}$ $x=\sqrt{y^{-1}-1}$ $y\geq1$ Then the image of $f(x)=\frac{1}{1+x^{2}}$ is $y\geq1$ Is this ...
0
votes
1answer
38 views

Find the minimum value of $\operatorname{Im}(z^5)/(\operatorname{Im}(z))^5$ [closed]

If $z$ is a complex number, then find the minimum value of $$\frac{\operatorname{Im}(z^5)}{(\operatorname{Im}(z))^5},$$ where $\operatorname{Im}(z)$ denotes the imaginary part of z.
1
vote
0answers
120 views

How to factorize an equation?

How can I quickly factorize an equation without using Ruffini's rule or polynomial division? Take this equation as an example: x^4-9x^2+20x=0
3
votes
2answers
108 views

A theory of equation question from my exam paper

Consider The equation $x^3+3x^2+3x+3=0$ Then the sum of it's non-real roots is A) is equal to $0$ B)lies in $0$ and $1$ C)lies in $-1$ and $0$ D)Greter that $1$ Which one is correct , plz explain ...
0
votes
1answer
63 views

how to prove that $f(x^k)=k . f(x)$ for all $k\in R^+$

I am trying to prove that $f(x^k)=k. f(x)$ where $f$ is continues and $f(xy)=f(x)+f(y)$. For $k\in N$ is straightforward, however I am struggling when $k=\frac{p}{q}\in Q^+$ and moreover when its ...
0
votes
1answer
30 views

find infimum of $(n+1)^2 / 2^n$ (using Bernoulli's inequality)

What is the infimum for: $$(n+1)^2 / 2^n$$ I've tried to simplify $2^n$ with Bernoulli's inequality $((1+1)^n \ge (1+n))$ but it didn't work out..
0
votes
1answer
45 views

recurrence relation question.

I have this expression: $I_{n} = \int_0^1 \frac{x^{n-1}}{2-x} dx$ for $n=1,2,3,\ldots$ I have been asked to show that by writing $x^n = x^{n-1} (2-(2-x))$ that the recurrence relation $I_{n+1} = 2I_n ...
-1
votes
1answer
46 views

Simplify equation - Wolfram providing different answers

I have an equation $-n = m-x(a-w(x-\frac{2l}{3})) - w(x-\frac{2l}{3})(\frac{l}{3}+\frac{x}{2})$ I have simplified this to $n = -m + ax - w(\frac{x^2}{2}+\frac{2l^2}{3})$ However Wolfram and my ...
1
vote
1answer
87 views

A single transferrable vote question

I'm preparing for the Oxford TSA, and am using past papers as a way to practice. One of the questions, I thought I had right, but turned out incorrect. Would love it if you were to analyze my ...
2
votes
2answers
73 views

Fast algebraic expansion

Is there an algebraic trick to expand the following expression without multiplying each term with another, expanding the standard way it gives $6+6+6=18$ terms and then cancelling the same terms with ...
-1
votes
2answers
93 views

How do I read this table?

Image: I'm preparing for the Oxford TSA, and doing the previous papers as practice. One question was a bit wacky for me though. Attached is the image of it. It talks about the mock examination ...
0
votes
2answers
34 views

How can I force this expression into a given form?

This problem has been irritating me, and conceptually, it should be very straight forward! This equation is derived from a circuit with a dependent voltage source; it describes the gain. $ ...
1
vote
1answer
20 views

How do I get an answer of $14$ using simpsons rule for $\frac{152e}{180n^4}<.0001$

I must have the algebra wrong somewhere but here is the original equation: $$\frac{152e}{180n^4}<.0001$$ If I then multiply like this: $$152e<.0001(180)n^4$$ Which then gives: $$152e < ...
0
votes
2answers
40 views

Deduce an unspecified polynomial?

I'm having a little trouble with problems that have an unspecified polynomial, for example $p(x)$, and having to get properties of them. A problem I ran across had something along the lines of $p(x) ...
0
votes
2answers
56 views

Which quadrant does -1.326 rad go? [closed]

Which quadrant does -1.326 rad go? Please tell me how to find out by using the unit circle. Thank you.
1
vote
3answers
51 views

How to find the inverse of a exponential function?

The function is $f(x) = 3^{-x}$. I have tried by finding the inverse, i.e. $-\log_3x$, but I am not so sure.
5
votes
3answers
242 views

Error in finding sum of $1\cdot 2+3\cdot 4+ \cdots \text{to}\space n\space \text{terms}$

To find sum of the series $1\cdot 2+3\cdot 4+ \cdots \text{to}\space n\space \text{terms}$ My approach, Let S=$1^2+2^2+3^2 + \cdots +n^2$ If $n$ is even S=$(1-2)^2+(3-4)^2+ \cdots +[(n-1)-n]^2+2(1 ...
-1
votes
1answer
58 views

I am not how they got characteristic equation from the given equation.

![can someone tell me they got characteristic equation from the given recursive equation.][1] i know how to do the rest of problem but getting characteristic equation stopped me. The recurrence is ...
-2
votes
2answers
54 views

Simplifying this Further

$$2x(x^2-3)^{10} + 20x(x^2+3)(x^2-3)^9$$ I would like to double check my answer (if anyone can double check this please) Please simplify the top
2
votes
2answers
50 views

Solve $\left(x^{2010}+1\right)\left(1+x^2+x^4+x^6+…+x^{2008}\right)=2010x^{2009}$

Solve for $x$ $\left(x^{2010}+1\right)\left(1+x^2+x^4+x^6+.......+x^{2008}\right)=2010x^{2009}$ solution should be by hand
1
vote
0answers
26 views

Finding the totient functionlike function for an irrational number like (a+b*sqrt(5)) where a and b are whole numbers mod M where M is a whole number.

I need to find if a value $T$ exists for irrational number of the form $(a+b\cdot \sqrt{5})$ such that $(a+b\cdot \sqrt{5})^T = 1 \pmod M$. Also ,is it possible to find out upper bound for T .
1
vote
3answers
111 views

How is $\sqrt{\frac{\sqrt{3}+2}{4}} = \frac{\sqrt{2}(\sqrt{3}+1)}{4}$

How is $\sqrt{\frac{\sqrt{3}+2}{4}} = \frac{\sqrt{2}(\sqrt{3}+1)}{4}$? (Prove by using algebraic manipulation not by calculation) I've tried to come up with something myself but I can't find a ...
1
vote
2answers
947 views

Find all the values of $(1+i)^{(1-i)}$

The question says to find all the values of $(1+i)^{(1-i)}$ I have trouble figuring out firstly, exactly what values are being looked for. I can toy around with the equation a bit to try to make it ...
0
votes
1answer
115 views

Proving that roots of a quadratic lie between two values

To prove that one of the roots of a quadratic $f(x) = ax^2 + bx + c$ with real coefficients lies between two values $x_1, x_2$ is it enough to prove that: $$f(x_1) < 0 < f(x_2) $$ Can this be ...
6
votes
3answers
240 views

Solve $\lfloor{x}\rfloor$+$\lfloor2x\rfloor+\lfloor4x\rfloor+\lfloor16x\rfloor+\lfloor32x\rfloor=12345$

Solve for $x$ $$\lfloor{x}\rfloor+\lfloor2x\rfloor+\lfloor4x\rfloor+\lfloor16x\rfloor+\lfloor32x\rfloor=12345$$ I tried to put $x$=$I$+$f$ where $I$ is integer part and $f$ is fractional part ...
1
vote
1answer
73 views

Changing order of integration of $\int_0^{\infty}\int_{a}^{\infty} f_{Y}(x) dx\ dy $?

I am trying to follow a proof where $Y$ is a continuous random variable with probability density function $f_{Y}$. What I don't get is how or why changing order of integration of the following ...