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
40 views

Calculator Shortcuts for Derivative?

I have a TI-84 and I am in a business calculus class. Our teacher doesn't give us much time on tests and I am a bit slow so I was wondering if there was any way to do some of these problems on the ...
0
votes
8answers
91 views

Least value of $a$ for which $4ax^2 + \frac{1}{x} \geq 1$

Find the least value of $a \in R$ for which $4ax^2 + \frac{1}{x} \geq 1$, for all $x>0$. The equation will transform into (Using $x>0$) $4ax^3-x+1\geq 0$ But I don't know how to deal with ...
0
votes
0answers
26 views

solving equation invloving both algebraic and trigonometric terms

$$ x\sin(3)+3\sin(x)-xlog(3)+3log(x)=10$$ I need to know a method that finds all the possible values of x that satisfy the above equation.
1
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2answers
24 views

sum of all positive integral values of $a\;,$ for which equation $\lfloor x \rfloor ^3+x-a=0$ has solution

The sum of all positive integral values of $a\;,$ Where $a\in \left[1,1500\right]$ for which the equation $\lfloor x \rfloor ^3+x-a=0$ has solution, Where $\lfloor x \rfloor $ Represent floor ...
0
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2answers
22 views

Prove that $aRb$ if $a = 2^kb$ is an equivalence relation.

Let $R$ be a relation on the set of integers given by $aRb$ if $a = 2^kb$, for some integer $k$. show that $R$ is an equivalence relation. I don't understand how it will be equivalence. Is it the ...
1
vote
2answers
67 views

$8^a=3$ and $3^b=5$ and $10^c=5$ then find $c$ in terms of $a$ and $b$.

if $8^a=3$ and $3^b=5$ and $10^c=5$ then find $c$ using $a$ and $b$. My Attempt: if $8^a=3$ and $3^b=5$ then we can say that $8^{ab}=5$ and then we have $2^{3ab}=10^c$ but i cant solve this ...
2
votes
3answers
63 views

How to solve the equation $(2x + 1)(2x + 3) = 143$ without using the Quadratic Formula?

I have been a bit stuck on this question. The product of two consecutive odd numbers is $143$. Find the next numbers. I have made this into: $$ (2x+1)(2x+3)=143. $$ I got $x_1 = -7$ and $x_2 = -5$ ...
0
votes
1answer
13 views

Is there a analytic algorithm to solve for a partially specified constant present in function & its derivative with the rate of change at an x value?

I have a Calculus problem that I am not entirely happy with how I solved it. Given the following information: $$y = x^{k} + x^{k-2}$$ $$(k \in \mathbb{N}) \wedge (k \mod{2} \neq 0) \wedge (k > ...
2
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2answers
45 views

Quadratic formula errors

I'm clearly making a silly mistake here, but I can't see it. EDIT: I missed brackets when typing out the expression to calculate. Apologies for timewasting. I have the equation $(2x + 3)(5x + 1)=0$. ...
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1answer
29 views

Kernel of a polynomial with matrix, $ker(p(A))$

Let $A\in Mat(3,3,\mathbb R)$ a matrix and $\chi_A(x)=p_1(x)\cdot p_2(x)$ the characteristic polynomial. Evaluate $ker(p_1(A))$.$$A=\begin{pmatrix} 0 & 0 & 2 \\ 1 & 0 & 1\\ 0 & ...
1
vote
7answers
89 views

Solve the following equation : $\log_2(x)*\log_4(x)*\log_8(x)=4.5$

I have the following equation : $$\log_2(x)*\log_4(x)*\log_8(x)=4.5$$ Usually, I do post what I made to do, but in this case a friend of mine tackle me with this question after I didn't mess with ...
0
votes
1answer
37 views

$1-x^2e^{-2x-y}+e^{-2x-y}\overset{?}=1-x^2e^{-2x-y}$ [on hold]

I'am looking at a solution of a problema that has this, it it correct? If so why? I'am looking at this and can't find out what step was made to simplify like that. ...
3
votes
2answers
69 views
+50

Jensen-like averaging inequality on integers

Let $\mathbb{Z}^*=\mathbb{Z}^+\cup\{0\}$. Let $f:\mathbb{Z}^*\rightarrow\mathbb{R}$ be a nondecreasing function such that $f(a+b)\leq f(a)+f(b)$ for all $a,b\in\mathbb{Z}^*$. Is it true that for all ...
4
votes
5answers
116 views

Roots of $x^{101}-100x^{100}+100=0$

I do not know how to prove that $x^{101}-100x^{100}+100=0$ has exactly two positive roots. Some can give me hint for solving this please. Thanks for your time.
2
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0answers
28 views

A congruence of sum of kth powers of first p-1 numbers [duplicate]

Problem: For $k < p-1$ where $p$ is an odd prime and $k$ is a natural number, prove that $$1^k+2^k+\cdots+(p-1)^k \equiv 0 \mod p.$$ My attempt: It's obvious for odd $k$, as we can pair the ...
5
votes
2answers
62 views

A tricky limit (indeterminate form)

While tutoring I came upon this limit I know that this limit is obviously 1, but how would I show this formally $$\lim_{\eta\rightarrow\infty}(2\eta + 5)^x-(2\eta)^x + 1$$ where $x\in (0,1)$ I've ...
1
vote
1answer
32 views

When do I use the 'plus-minus' sign when square rooting both sides of an equation? (example in main body).

Above is the image of an integration by substitution question that I was doing; the answer can only have a plus sign in front (if you were to differentiate the answer to check if it's correct). ...
0
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3answers
64 views

Multiplication of real and complex radicals

If I have, for example, the product $\sqrt{7+\sqrt{22}}\sqrt[3]{38+i\sqrt{6}} $ Can I perform the multiplication or this cannot be done and only remains to leave the product in this form?
0
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1answer
37 views

find the values of $a$ for which function is invertible

I had a question in which i was told to find the range of values of $a$ for which the function is invertible (for which inverse exist) and function is $f(x)=ax+3\sin x+4\cos x$ what i tried for ...
5
votes
1answer
44 views
+100

Maximum value of the smallest number of operations to obtain configuration from original configuration

Let $n$ be a positive integer. There are $n(n+1)/2$ marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing $n$ marks. Initially, each ...
0
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0answers
26 views

Number of solutions of a difference-of-two-squares congruence with prime moduli

Problem: Show that if $p$ is an odd prime then $p-1$ number of ordered pairs $x, y$(unique modulo p) satisfy $x^2-y^2 \equiv a\mod p$ (for some given $a$ coprime to p). When $a \equiv 0 \mod p$ then ...
-1
votes
1answer
20 views

How to find percentage change between original and final BPM

I am working on a random beat generator and one of the things it does is randomly select the BPM (beats per minute) from a database in between 50 and 200 BPM. The starting BPM of the files will ...
1
vote
2answers
24 views

How to simplify from this thing to this (double derivative, stuck in the alegba part)

In my maths class I am doing double derivative to find concavity of the equation if I graph it, and getting these big functions. Plugging in even on online calculators skips from this big thing to ...
0
votes
0answers
26 views

A simple Lagrange interpolation-type identity

I am unable to prove an identity that looks very much like the Lagrange interpolation identity, Problem: Given $f(x)$ is a monic, $n-1$ degree polynomial and $a_1, a_2, \cdots a_n$ distinct real ...
1
vote
0answers
25 views

Why did this incorrect partial fraction decomposition produce the correct answer?

I was reviewing a classmate (call him Bob)'s work on an integration of a rational expression (although integration is involved, it's beyond the scope of this question). The problem was: ...
2
votes
2answers
57 views

How to tell if the series $\sum_{n=1}^\infty \frac{ne^{-n^2}}{e^{-n}+4}$ converges?

$$\sum_{n=1}^\infty \frac{ne^{-n^2}}{e^{-n}+4} $$ Trying to figure out if this converges, trying to use the divergence test but I can't figure out how to simplify the problem.
1
vote
1answer
29 views

How to factor this equation?

$3x+2+\frac{1}{3x} = \frac{-4}3$ I was only able to find the solution by plugging it into a program. Never have encountered a problem like this before
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2answers
43 views

Find all the solutions of the polynomial x^3=3 [closed]

Please help me to find all the solutions of the polynomial $x^3=3$
0
votes
1answer
28 views

Workbook recommendation in preparation for Electrical Engineering

I'm currently preparing myself for starting my graduate degree in Electrical Engineering. The mathematics courses given are outlined as follows: Mathematics 1 Real functions Continuity, limits, ...
0
votes
4answers
37 views

How to isolate this variable?

$$m \cdot (a - b) + c = a$$ I can't figure out how to isolate $a$ in the equation above. I can't seem to detach $m$ so that $a$ can be left alone. What steps are required to isolate $a$?
0
votes
1answer
12 views

Algebra for a polytropic process, thermo

Polytropic process equation $p_1v_1^n=p_2v_2^n$ $pv=RT$ Where R is a constant, (Ideal gas law) How do you obtain the expression: $\frac{T_2}{T_1}=\left (\frac{p_2}{p_1}\right)^\frac{n-1}{n}$ ...
1
vote
2answers
84 views

Am I using sandwich theorem incorrectly?

I saw this question and wondered how OP of that question was able to do : $$0<\sin x+1<2$$ this $$\frac 0{|x|}<\frac{\sin x+1}{|x|}<\frac 2{|x|}$$ and when $x\to \infty$ he got the limit ...
2
votes
2answers
47 views

To show that the variables in the system are same in magnitude

I am stuck with this interesting problem, If for non-negative integers $a, b, \text{and} c$, $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are both integers then ...
0
votes
1answer
49 views

How do I evaluate $\displaystyle\prod_{r=1}^{\infty }\left (1-\frac{1}{\sqrt {r+1}}\right)$?

I am not being able to find the specific product $\prod_{r=1}^{k} \left(1-\frac{1}{\sqrt {r+1}}\right)$ so to evaluate the given problem when $k \to \infty $.
0
votes
4answers
44 views

Difference between squares and roots??? [duplicate]

Why does this happen?? $$ y = \sqrt9 \implies y=3$$ $$ y^2 = 9 \implies y=+3,-3 $$ While both equations are in same sense.
1
vote
0answers
23 views

A system of equations to find highest possible outcome

I am trying to solve a really simple system of equations, but its been about 10 years since Ive done this stuff. Can someone explain how to choose which stocks to buy based on cost, projected return, ...
0
votes
4answers
79 views

Prove that there is no term independent of $x$ in the binomial expansion of $\left(x-\frac 1x\right)^{19}$

I am dealing with a fairly simple question but I'm struggling a bit to come up with a formal demonstration on why the binomial expansion of $\left(x-\frac 1x\right)^{19}$ doesn't have a term ...
3
votes
2answers
63 views

$\left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right)=1$. Find $(x+y)$.

We know that $\left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right)=1$. Find the expression $(x+y)$. My work so far: $$\left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right)=1$$ ...
0
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0answers
34 views

Number theory/calculus/algebra etc. equivalents of Euclid's Elements?

Anybody know any books that tackle mathematical topics in a deductive, axiomatic structure akin to Euclid's Elements? Thanks.
0
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0answers
33 views

isolating for a variable

I'm trying to isolate for a variable (I) in a paper characterizing the operation of a solar panel, but I'm not sure what the most effective approach would be. Would this be a job for Mathematica, or ...
0
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1answer
40 views

What is the name of the given formula

Could anyone say what is the name of the given formula?
0
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1answer
80 views

Expand $(a-b)^3$ without formulas

When I solve $(a-b)^2$ I take $aa-ab-ab+bb$ I do not use formulas at all because I only forget them. To solve the above example all i do is to multiply one variable or constant at the time but when I ...
1
vote
1answer
43 views

The problem of surds and indices [closed]

If $$ \frac 12 \times \left(\sqrt[3] a + \sqrt[3] b + \sqrt[3] c\right) = \frac 1{\sqrt[3] 3 - 1}\;\;\;(a>b>c)$$ then what is $a - 2b - 3c $ ?
2
votes
1answer
40 views

Understanding Why Partial Fractions Works [duplicate]

I was wondering why, not how, partial fractions work the way we are normally taught to do. To be specific: We are told that, when we have a second degree expression on the bottom that can't be ...
1
vote
2answers
30 views

Given the volume of a stack of cubes, find the number of cubes

I was trying to solve a code challenge but could not wrap my head around the math. Your task is to construct a building which will be a pile of n cubes. The cube at the bottom will have a volume ...
3
votes
2answers
63 views

Sum of real values of $x$ satisfying the equation $(x^2-5x+5)^{x^2+4x-60}=1$

I have this equation from this paper (Q.63) Find the sum of all real values of $x$ satisfying the equation-$(x^2-5x+5)^{x^2+4x-60}=1$. My attempt- ...
1
vote
1answer
21 views

Find the positive difference of all possible values

This problem is quite challenging to me. It is highly appreciated if someone can help me with it or give me an hint. Thank you very much! Find the positive difference of all possible values of ...
0
votes
3answers
116 views

How to solve the equation $ x^{13} = 1$ by radicals?

Is there any elementary way to solve the equation $ x^{13}= 1 $ by means of radicals? If not, how to get all the solutions? Remark: The transcendental form of the solution by means of sines and ...
0
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1answer
50 views

Function Question: Where does the $3$ come from?

Can someone explain me how did ''3'' appear here ?
-1
votes
1answer
16 views

Working through summations to show equation

Given equation 1: $$E = \sum_{k=1}^N \tau x_k g(\frac{n_k}{\tau}) + \sum_{k=1}^N n_kh(\frac{n_k}{\tau})$$ equation 2: $$E = \frac{1}{2}\gamma X^2 + \epsilon \sum_{k=1}^N |n_k| ...