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.

learn more… | top users | synonyms (2)

-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
84 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
41 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
79 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
34 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 [closed]

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
90 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
63 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
197 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
56 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
67 views

Using differentiation [closed]

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
32 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 ...
0
votes
2answers
52 views

derivatives $\frac{dy}{dx}$

Sand is falling from a rectangular box whose base measures $40$ inches by $20$ inches at a constant rate of $300$ cubic inches per minute. (Include units in your answers.) a) How is the ...
3
votes
1answer
30 views

Question about recursive algorithm

I have following problem: $$f(n)=\frac{1}{1^2+1}+\frac{2}{2^2+1}+\frac{3}{3^2+1}+\cdots+\frac{n}{n^2+1}$$ Write recursive algorithm for $f(n)$ Prove that recursive algorithm is correct ...
0
votes
2answers
40 views

How to prove by induction that $2^{2^n} + 1$ has $7$ in unit's place?

It must be true,first of all, for $n \geq 2$ . So, firstly I proved that for $n = 2$, it's true. Taking for $n = m$, is true, how can I prove that for $n = m + 1$ ie. $2^{2^{m + 1}} + 1$ has $7$ in ...
2
votes
3answers
65 views

If $|z-3i|+|z-4|=5$ then find the minimum value of $|z|$

Question : If $|z-3i|+|z-4|=5$ then find the minimum value of $|z|$ What I did : $$|z-3i| \leq |z|+3 \tag i$$ Also $$|z-4| \leq |z| +4 \tag{ii}$$ Now adding (i) and (ii) we get $$ ...
2
votes
4answers
185 views

How to prove a right angle if i have two tangents?

I would appreciate your help, it is long time since I solve trigonometric, like if I have the tangent of angle B equal to $\sqrt{2}-1$ and the tangent of angle C equal to $\sqrt{2}+1$, how can I prove ...
1
vote
1answer
29 views

Let $|z|=1, $ prove that $|z^2-3z+1|\leq 5$ …

Problem : Let $|z|=1, $ prove that $|z^2-3z+1|\leq 5$ My approach : Let $z = x +iy$ $ \Rightarrow (x^2+y^2)=1$ $\Rightarrow |z| =1 $ represent a circle with centre at (0,0) and radius 1 ...
0
votes
5answers
42 views

Equation of the straight line equidistant from $(2,-2)$ & $3x - 4y + 1 = 0$?

I have tried this as: $$\sqrt{(2 - h)^2 + (k + 2)^2} = \dfrac{3h - 4k + 1}{5}$$ where $(h,k)$ is the point on the required straight line. But on expanding, the equation contains terms of $h^2$ & ...
5
votes
2answers
77 views

Rules for whether an $n$ degree polynomial is an $n$ degree power

Given an $n$ degree equation in 2 variables ($n$ is a natural number) $$a_0x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+a_{n-1}x+a_n=y^n$$ If all values of $a$ are given rational numbers, are there any known ...
1
vote
0answers
41 views

Where is the fault in this approach for transforming this Dirichlet series?

Mathematica knows that: $$\lim_{s\to 1} \, \zeta (s)\left(-2^{1-s}-3^{1-s}+6^{1-s}+1\right)=\sum _{n=0}^{\infty } \left(\frac{1}{6 n+1}+\frac{-1}{6 n+2}+\frac{-2}{6 n+3}+\frac{-1}{6 n+4}+\frac{1}{6 ...
0
votes
3answers
58 views

Explain sandwich theorem

I was reading my math book trying to understand "limits and derivatives". I understood almost everything till this. Below is the statement from my book. Can anyone please explain this to me. If ...
0
votes
2answers
35 views

Interesting problem about abc being a perfect power

For natural a,b,c this equality holds: $a^3c^2+b^3a^2+c^3b^2=3 \sqrt[3]{(a^5b^5c^5)}.$ Show that abc is a perfect fifteenth power. I got to this point so far in my problem. I tried to make a ...
-4
votes
2answers
71 views

How do I mathematically explain this relationship?

At 40cm, 1.96N was produced At 46cm, 1.47N was produced At 56cm, 0.98N was produced At 80cm, 0.49N was produced. Is it inverse?
0
votes
1answer
24 views

Train overtake time

I am having trouble solving this problem. In particular I am having difficulty translating this problem into an equation. Question: A train leaves a station and travels east at 75 km/h. Two and ...
3
votes
1answer
61 views

If $f(x)+2f(1/x)=3x$, find all $y$ such that $f(y)=f(-y)$.

The function $f(x)$ is not defined when $x=0$. This function has the property that $f(x) + 2f\left(\frac 1x\right) = 3x$. Find all such values of $y$ such that $f(y) = f(-y)$. (This means it is an ...
5
votes
7answers
123 views

Good Pre-Calculus book?

I was reading this article and the author mentioned I should come here and get some advice. I'm 17, currently taking Pre-Calc in high schooling doing really good, but I feel like I'm not getting the ...
0
votes
1answer
32 views

Convert the following: $\frac{-1^{k}(k+1)}{2}(-k-2)$ to $\frac{-1^{k+1}(k+2)}{2}$

I am teaching myself induction proofs and stepping through the algebra for the sample problems. But I got stuck on this part, can't get rid of $(k+1)$. Can someone please step me through the process ...
0
votes
1answer
24 views

Find the equation of parabola tangent to a line

I know how to find the equation of the line tangent to a parabola through a certain point. But how do I find the equation of the parabola from the point and the tangent line? For example, how do I ...
2
votes
3answers
42 views

System of equations $x^2=y^3, x^y=y^x$

Solve the system of equations $x^2=y^3, x^y=y^x$ in positive real numbers. Taking $\ln$ of the second equation, we have $\ln x/x=\ln y/y$. This function is increasing in $(0,e)$ and decreasing in ...
-2
votes
0answers
15 views

Value of K that produces same remainder with two divisors? [closed]

$f(x)=3x^3+6x^2+Kx−4$I am having difficulty finding $K$ such that $f(x)$ has the same remainder when divided by $x-1$ and $x+2. Any help would be greatly appreciated.
0
votes
1answer
53 views

How to fully factor a polynomial of 4th degree?

How to fully factor this polynomial? $$ 2x^4+3x^3-32x^2-48x$$ Can anyone describe the full steps to factor it? Thanks for the help.
1
vote
2answers
46 views

Avoiding extraneous solutions

When solving quadratic equations like $\sqrt{x+1} + \sqrt{x-1} = \sqrt{2x + 1}$ we are told to solve naively, for example we would get $x \in \{\frac{-\sqrt{5}}{2},\frac{\sqrt{5}}{2}\}$, even though ...
0
votes
3answers
52 views

Other method show that $ A(x)=x^2+x+1=0$ has a zeros in $\mathbb{R}$ but why this contradiction?

Let $ A(x)=x^2+x+1$ be a quadratic polynomial equation and $ x \in\mathbb{R}$. It is well known that $ A(x)=x^2+x+1=0$ hasn't a roots in $\mathbb {R}$ , we choose another way to solve this equation ...