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
17 views

Find the largest segment

I have seven lines with different measures. The length of each line it's a positive integer and the shortest length is equal to 1 cm. It is known that's impossible to choose three of them that makes a ...
-3
votes
1answer
56 views

Can someone help me why this equation equals zero?

I played around with some numbers and stuff and made this weird equation: $$\huge x^{- \frac{n}{n^{-x}}}$$ So the thing is, with every number I tried typing this into a calculator, I got 0. Can ...
-1
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1answer
22 views

Multiplying logarithms of different bases

How do you multiply the following logs... $$\log_5(n) * \log_2(n)$$
-1
votes
2answers
29 views

fifth order Differential EQuation

find the general solution of higher order linear differential Equation? find the general solution of Differential equation using auxiliary equation? $2y^{(5)}-7y^{(4)}+12y"'+8y"$=0
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votes
2answers
24 views

How to prove that eventually $(x^p/e^{x^q}) < 1/(x^2) $ for $p,q>0$

How to prove that eventually $x^p/\exp(x^q) < 1/(x^2) $ for $p,q>0$. I tried showing that $x^{p+2} > \exp(x^q)$ by using the Taylor expansion of e but this didn't really work.
2
votes
2answers
84 views

Simple 2 equations and 2 unknowns

I am reading the second partial derivative test example, but I am suck on the following step: $$f(x,y) = -x^3 + 4xy - 2y^2 + 1$$ And we have the partial derivatives as follow... $$f_x(x,y) = -3x^2 ...
4
votes
2answers
28 views

Roots of unity, where $\omega^3 = 1, \omega \neq 1$.

Say that $\omega^3 = 1$ and $\omega \neq 1$. Find the value of $(1 - \omega + \omega^2)(1 + \omega - \omega^2)$. I'm not very good at the roots of unity. May I have a couple of hints to get started? ...
4
votes
1answer
145 views

Partial fractions - different results when done in steps than not

We have: $\frac 1 {(1-x)(1+x)(1-2x)}$ If I do the partial fractions straight: $\frac 1 {(1-x)(1+x)(1-2x)}= \frac a {1-x} + \frac b {1+x} + \frac c {1-2x}$ I get: $a=-\frac 12, b = \frac 1 6, c=\frac ...
1
vote
0answers
16 views

Is there any solution to this quadratic Diophantine 3 variables equation?

Is it possible to find all positive integer triplets $(x,y,z)$ satisfying the parametric equation : $$x^2 + 2ax + y^2 + 2by = z^2 + 2cz$$ Here $a, b, c$ are fixed positive integers.
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votes
1answer
26 views

Precalculus: Velocity addition

A boat is rowed 6.4 km up a river and back again and this takes in total 2 hours. The stream velocity is 2.4 km/h. What velocity would the boat have been moving in if the water was standstill? I ...
4
votes
1answer
28 views

If $|x|\leq 1\;,|ax^2+bx+c|\leq 1\;,$ Then $\bf{Max.}$ possible value of $|2ax+b|\;$

If $a,b,c\in \mathbb{R}$ and If $|x|\leq 1\;,|ax^2+bx+c|\leq 1\;,$ Then $\bf{Max.}$ possible value of $|2ax+b|\;$ is, Where $-1 \leq x\leq 1$ $\bf{My\; Try::}$ Put $x=1$ in ...
9
votes
7answers
458 views

How do we prove this logarithm?

Given: $$\dfrac{\log x}{b-c}=\dfrac{\log y}{c-a}=\dfrac{\log z}{a-b}$$ We have to show that : $$x^{b+c-a}\cdot y^{c+a-b}\cdot z^{a+b-c} = 1$$ I made three equations using cross multiplication : ...
1
vote
1answer
16 views

Finding orthonormal basis using orthogonal basis

I am very confused how to go about finding an orthonormal basis using a orthogonal basis. My book says to just normalize the vectors but it doesnt further explain. When i look at answers for ...
4
votes
3answers
48 views

If $f(x) = \sin^4 x+\cos^2 x\;\forall x\; \in \mathbb{R}\;,$ Then $\bf{Max.}$ and $\bf{Min.}$ value of $f(x)$

If $f(x) = \sin^4 x+\cos^2 x\;\forall x\; \in \mathbb{R}\;,$ Then $\bf{Max.}$ and $\bf{Min.}$ value of $f(x).$ My Solution:: Let $$\displaystyle y = \sin^4 x+\cos^2 x \leq \sin^2 x+\cos^2 x=1$$ ...
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0answers
9 views

Generalization of minimisation problem

First I would like indtroduce my problem ! There is an easy way to solve this one : Find the value of $$ \inf_{(a,b)\in \mathbb{R}^2} \int_0^1 (t^2-at-b)^2 dt $$ and precise for which values $a$ ...
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votes
0answers
45 views

Mathematics doubt. [on hold]

What is mathematics. I need a proper definition mathematics.
0
votes
3answers
29 views

When is $\theta$ obtuse or acute in sin, cosine, tan when they are positive, negative or both?

My textbook gives a non intuitive answer and tells us to memorize when the ratios are positive or negative or both based on some arbitrary rule that I don't understand. I know how to do both of ...
0
votes
1answer
24 views

$b^{\frac{m}{n}}=(b^{\frac{1}{n}})^m=(b^m)^{\frac{1}{n}}$ except $b$ is not negative when $n$ is Even.

The following property, known as Rational number property, is taken from the book (I am following now a days) College Algebra by Raymond A Barnett and Micheal R Ziegler I restate, ...
2
votes
1answer
20 views

given the following two conditions, find $f(x,y)$

Suppose that a function $f$ defined on $\mathbb R^2$ satisfies the following conditions: $f(x+t,y)=f(x,y)+ty$; $f(x,t+y)=f(x,y)+tx$; $f(0,0)=k$; then for all $x,y \in\mathbb R$, $f(x,y)=$ a) ...
-3
votes
1answer
51 views

What grade does Bob need on his final to pass his math class?

Bob's teacher has a syllabus where the grading breakdown is as follows: Homework: 10% Tests (2): 60% Final Exam: 30% Bob receives all 10% from the homework Bob receives 78/100 on Test 1 (78%) ...
0
votes
2answers
26 views

Vector Magnitude problem

I do not understand how to set up the following problem: "Forces of 20 lb and 32 lb make an angle of 52 degrees with each other. find the magnitude of the resultant force." An actually picture would ...
3
votes
1answer
23 views

How to find the numbers that the product of it's digits is equal to ten times the sum of them.

I formulated this so that the number be in range $[111,999]$, it was narrowed so that $a,b,c$ is not equal to zero. $$a\cdot b\cdot c = (a+b+c)\cdot 10$$ With this we can see that $\frac{a\cdot ...
-2
votes
1answer
43 views

An algebra word problem in one variable on percentages - assistance required [on hold]

I'm new here. I'm not sure how this works, but I was wondering if someone could help me with this problem. I'm having difficulty solving it, and I was hoping someone could help me. Thank you! Assign ...
-1
votes
2answers
70 views

Word problem on the perimeter and sides of a triangle [on hold]

Having difficulty answering this question. I'm not sure how to do it, but if anyone could show me the steps; so I could answer it myself, that would be great. Thanks Assign x, make an equation and ...
16
votes
3answers
126 views

Intriguing Indefinite Integral: $\int ( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1})^2 \mathrm{d}x$

Evaluate $$\int \left( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1}\right)^2 \mathrm{d}x$$ I tried using partial fractions but the denominator doesn't factor out nicely. I also substituted ...
-2
votes
0answers
13 views

Central orbits ellipse change of centre of force [on hold]

a body is decribing an ellipse of eccenticity e under the action of central force directed towards the focus, and when at the nearer apse, the centre of force is transferred to the other focus, Find ...
2
votes
2answers
49 views

Advanced (for me) algebra and mean

I have been struggling over questions like these which my maths teacher has been throwing into our weekly papers for about a week now, and it has stumped all of us. Can you help? Question: There are n ...
-6
votes
1answer
55 views

A trick and interesting math SUM [duplicate]

If $x=(a/b)^{2ab/(a^2-b^2)}$ I want to prove that $$( (ab)/(a^2+b^2) )(x^{a/b} + x ^ {b/a})=(a/b)^{(a^2+b^2)/(a^2 - b^2)}.$$
3
votes
2answers
33 views

Stuck in a problem in permutation and combination.

I am solving problems in permutation & combination and stuck in this problem. Two players $P_1$ and $P_2$ play a series of $2n$ games. Each game can result in either a win or a loss for $P_1$. ...
-4
votes
0answers
61 views

Prove this enticing equation [on hold]

If $$x=\left(\frac{a}{b}\right)^{\frac{2ab}{a^2-b^2}}$$ I want to prove that $$\left(\frac{ab}{a^2+b^2}\right)\left(x^{\frac{a}{b}} + x ...
1
vote
1answer
25 views

Total no. of Transitive Relation on $A = \{a,b,c\}$

Calculation of total no. of Transitive Relation on $\displaystyle A = \left\{a,b,c\right\}.$ $\bf{My\; Try::}$ First We will calculate Total no. of Relation on $A$, Which is $\displaystyle = 2^{3^2} ...
7
votes
2answers
379 views

Is a circle classified as an ellipse?

I read that an ellipse had $2$ focal points. So, I thought if a circle had $2$ points that were simply infinitesimally close together wouldn't it be classified as an ellipse? Help would be ...
1
vote
3answers
44 views

Show that: $\sinh^{-1}(x) = \ln(x + \sqrt{x^2 +1 } )$

could someone Please give me some hint of how to do this question thanks
1
vote
3answers
49 views

Precalculus unit circle with imaginary axis.

(a) Suppose $p$ and $q$ are points on the unit circle such that the line through $p$ and $q$ intersects the real axis. Show that if $z$ is the point where this line intersects the real axis, then $z = ...
1
vote
1answer
81 views

Express $C_n = \cosh(0) + \cosh(1) + \cosh (2) + \dots + \cosh(n)$

Could someone give me some hint of how to do this question please. I've been stuck for more than $3$ hours on this question.
2
votes
3answers
72 views

Could you explain the expansion of $(1+\frac{dx}{x})^{-2}$?

Could you explain the expansion of $(1+\frac{dx}{x})^{-2}$? Source: calculus made easy by S. Thompson. I have looked up the formula for binomial theorem with negative exponents but it is confusing. ...
-3
votes
1answer
20 views

Solving for a variable where the variable appears twice in the equation [on hold]

Introduction: Lean body mass (LBM) can be calculated by the formula $$x=z-(z\cdot y),$$ where $x=$LBM, $z=$total weight and $y=$percent body fat. Problem: Solve for $z$ if $x$ and $y$ are known. ...
0
votes
2answers
58 views

How to find minutes?

Need help solving this real life problem, I have an SD Card of $4$GB(gigabyte), and a $32$ second video occupies $6.12$MB(megabyte), I need to know how many minutes or seconds can this $4$GB SD Card ...
3
votes
2answers
35 views

Proving $1+5+9+\cdots+(4n+1) = (n+1)(2n+1)$ by induction (is there a typo?)

Using mathematical induction, prove that $$1+5+9+\cdots+(4n+1) = (n+1)(2n+1).$$ I understand the steps to take in order to prove by induction. It is also to my understanding that step 1 would be ...
4
votes
2answers
62 views

Proving that $3^n<n!$ when $n\geq 7$

It's been 10 years since my last math class so I'm very rusty. How would I go about proving $$3^n < n!$$ where $n \geq 7$? I understand that factorials grow faster than set values with a variable ...
0
votes
1answer
35 views

Basic function manipulation and simplification question for $f((x-f(x))^2)$

I've run into a bit of a wall trying to understand why the following two equations are equivalent: $$f((x-f(x))^2) = f(x^2)-f(x)^2$$ I'm running into this with calculating population variance in ...
1
vote
4answers
39 views

How does $\frac{-1}{x^2}+2x=0$ become $2x^3-1=0$?

Below is part of a solution to a critical points question. I'm just not sure how the equation on the left becomes the equation on the right. Could someone please show me the steps in-between? Thanks. ...
1
vote
2answers
121 views

Simplifying $\frac{\log(x)}{x}=y$.

I am trying to find the value of $r$ where the Rule of 72 will accurately estimate an investment's doubling time. Put simply, the Rule of 72 requires that 72 be divided by the interest percentage per ...
0
votes
2answers
71 views

Show that point you found for $x^2 - 2y^2 = 1$ using line through$ (1, 0)$ has integer coordinates.

The curve $x^2 - 2y^2 = 1$ includes the point $(1, 0)$. Let $L$ be the line through $(1, 0)$ with slope $m$. Find the other point where $L$ intersects the curve. Suppose that you take $m = v/u$, where ...
0
votes
3answers
54 views

I can't understand how to factor a quadratic

I can't figure out why some actions are possible in factoring $\frac{r–3}{r^2+11r–42}$. Factor the quadratic out of the denominator $\frac{r-3}{(r-3)(r+14)}$. This does not make sense how can $+11$ be ...
0
votes
0answers
12 views

moment generating function with Taylor series simplification

Denotes $a(0,r_1,r_2)$ as the annulus with radii $r_1<r_2$ centered at the origin $0$ Consider two bands $a(0,s,t)$ and $a(0,u,\sim)$ for $1\leq s\leq t\leq u$ Suppose a variable (call it an ...
2
votes
4answers
74 views

Rewrite $\sum_{i=0}^{n-1} (2i+1)=n^2$ to start induction from $k = 0$?

I'm trying to learn mathematical induction. The text asks for being totally rigorous i.e start induction from $k=0$. I want to prove that $$\sum_{i=0}^{n-1} (2i+1)=n^2,$$ i.e. the sum of the first ...
2
votes
1answer
47 views

Expansion of $(a+b+c+d)^n$

How to expand when $(a+b+c+d)^n$? Can I allow it to be $[(a+b)+(c+d)]^n$ and then use binomial theorem to expand it?
1
vote
1answer
47 views

Algebraic solution for a equation involving radicals?

$$f(x)=-\frac{5-x}{\sqrt{(5-x)^2+4}}-\frac{5-x}{\sqrt{(5-x)^2+9}}+1$$ I'm having trouble solving $f(x)=0$. I know there is a root for this function, I took the steps to set $f(x)$ to be equal to ...
0
votes
2answers
36 views

Solving for $x$ after simplifying, please check my work! [duplicate]

I was presented with the question: $$(\frac{x}{a})^{3.2} + (\frac{y}{b})^{3.2} = 1$$ While: $$\frac{a}{b} = \frac{174.1}{86}=c$$ I began to simplify and reached: $$x = c(b^{3.2}-y^{3.2})^{1/3.2}$$ ...