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
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2answers
316 views

Find all natural numbers n such that $7n^3 < 5^n$

Find all natural numbers $n$ such that $7n^3 < 5^n$. I drew a graph which showed that $n \geq 4$ wolfram How can I prove that? I guess I need to use induction with the base case $n=4$? But I am ...
-1
votes
3answers
151 views

If b and d are positive real numbers, then prove that $\frac{b+d}{2}>(bd)^{\frac 12}$.

Prove the following statement: If $b$ and $d$ are positive real numbers, then $\frac{b+d}{2}\geq(bd)^{\frac 12}$. Thank you.
0
votes
4answers
203 views

How do I find the center and radius of this circle? [closed]

How do I find the center and radius of this circle? $$4x^2+4y^2+24x-16y+41=0$$
2
votes
2answers
93 views

How to solve these simultaneous equations: $y=10x-3$ and $y=x^2-3x$?

Okay so I am normally good at these kinds of things but I received this problem that even the top people in my class had trouble solving. The problem is that everyone is getting different results. We ...
0
votes
1answer
892 views

How do I find the equation of the line that passes through this circle?

Find the equation of the line that has x-intercept and passes through the center of the circle that has equation $$x^2 + y^2-4x+10y+26=0$$
-1
votes
1answer
62 views

How do I simplify this expression?

How do I simplify this expression? $$\frac{(4-x)^2 (1/3) (6x+1)^{-2/3} (6) - (6x+1)^{1/3} (-2x)}{(4-x^2)^2}$$
0
votes
1answer
224 views

How do I simplify the expression (a^-1 + b^-1) ^-1?

How do I simplify the expression ... $$(a^{-1} + b^{-1}) ^{-1}$$ ?
0
votes
2answers
81 views

Find the minimum value of a function with additional condition

Find the minimum value of a function $f(x,y) = x^{2} + 4y^{2} - 4xy + 3x - y + 6$ if $x+y=1$.
1
vote
2answers
107 views

How do I simplify $p^8-Q^8$?

How do I simplify the expression $ p^8-q^8$? Also, and explanation of how you got/why that's the answer would be great
0
votes
1answer
43 views

Simplifying exponentials of the form $\,a^x \cdot b^y$

I am given the exponential $\left(\dfrac{1}{2}\right)^x\cdot 4^{(x/2)}$. While my intuition screams that this can be simplified to $\dfrac{2^x}{2^x} = 1$, I am unable to see a concrete mathematical ...
1
vote
3answers
106 views

How to solve equation?

How to solve this equation in the set of real numbers? $$(x^{2}+3y^{2}-7)^{2} + \sqrt{3-xy-y^2}=0$$ I tried to solve $x^{2}+3y^{2}-7=0$ and $\sqrt{3-xy-y^2}$=0 for x. But it did not help.
1
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2answers
22 views

How do I reach this form

I just want to check with you guys how to do this correctly: I have to reduce: $$ \frac{1}{N}+3=\frac{2R+2}{5R+2}$$ ...to the following form: $$ N=\frac{aR+b}{cR+d}$$ Any leads on how I reach this ...
0
votes
1answer
573 views

How to prove “If $R$ is transitive, then $R^n$ is transitive.”?

I can understand $R^n$ is $R$'s subset, but I can't understand why $R^n$ is transitive,too. I used mathematical induction: Basis step: Let $n = 2$. If $a R^2 b$, $b R^2 c$, I need to prove $a R^2 c$. ...
0
votes
3answers
588 views

Methods to show linear independence of functions, polynomials

There are apparently many methods to show that a set of functions are linearly independent. Forexample, there were some cases, where I saw use of derivative. At least to prove Lgrange polynomials are ...
0
votes
1answer
74 views

How to read the notations in the proof of the inclusion-exclusion principle?

This question is a follow up to this question. How do I create a sequence from this formula? $\sum\limits_{i=1}^{n} P(A_i) - \sum\limits_{1\le i < j \le n-1}P(A_i \cap A_j) + \dots + (-1)^n ...
0
votes
1answer
116 views

A curve such that all lines on the plane intersect it : cont..

Further to this question (which appears more or less settled); "Is there a curve on plane such that any line on the plane meets it (a non zero ) finite times ?" I ask now the upper bounds of the ...
0
votes
1answer
162 views

Simplify the following expressions using fractional exponents

Simplify the following expressions using fractional exponents. Display your answer using fractional exponents. $${ \sqrt[ 3 ]{x^{ 7 }} = }\text{ and }{ \sqrt[ 7 ]{x^{ 3 }} = }$$ Thanks..not sure ...
2
votes
3answers
108 views

Show $\binom{k+1}{r}+\binom{k+1}{r+1} = \binom{k+2}{k+1} $

I have been attempting to show $$\binom{k+1}{r}+\binom{k+1}{r+1} = \binom{k+2}{r+1} $$ and my work is $$\binom{k+1}{r}+\binom{k+1}{r+1} = \frac{(k+1)!}{r!((k+1)-r)!} + ...
3
votes
2answers
146 views

Lines, slopes, linear equations

Give an equation of a line with slope zero which passes through $(5,1)$. I know that $y=1$ But I don't understand the next two questions: b) Give an equation of a line perpendicular to the first ...
5
votes
3answers
4k views

Number of surjective functions from A to B

Am I on the right track? I am not sure about my reasoning... Number of surjective functions from $A$ to $B$ $$A = \{1,2,3,4\} ; B = \{a,b,c\}$$ We must count the surjective functions, meaning the ...
1
vote
1answer
185 views

Rewriting a sum

I'm stuck proving this result... Let $a$ be a positive even integer and $m$ a positive integer. Show that $$a^m\sum_{k=0}^{m}\frac{(-1)^k {2k \choose k}a^{-k}}{2^{2k}} $$ can be rewritten as ...
6
votes
2answers
438 views

How do I solve for $x$ in $\ln(x)\ln(x) = 2 +\ln(x)$

How do I solve for $x$? $$\ln(x)\ln(x) = 2 +\ln(x)$$
0
votes
3answers
2k views

Determine the number of real roots in the equation…

Determine the number of real roots in the equation $2x^3 + x^2 = 3$. I know about finding the different roots, and solving giving that it has (for example) 2i as a root, but I'm not sure how to just ...
-1
votes
1answer
269 views

undo/inverse a function

I have a smallish math function, and I don't have access to the change the function itself, only undo it via reverse functions. I have the output var5 at the end, and I know the function to get to it. ...
0
votes
1answer
116 views

Upper bound to Relative error

The focal length $f$ of a lens is given by the formula: $$\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$$ The values of $u$ and $v$ may be in error by as much as $2$ percent of the true corresponding ...
1
vote
1answer
128 views

Please help to complete proof of inclusion and exclusion principle

I want to complete the following proof: So continuing where the author left off, I did the following: $\begin{array} {cc} & \sum\limits_{i=1}^{n-1} P(A_i) - \sum\limits_{1\le i < j \le ...
6
votes
1answer
72 views

real solution of $x$ in $\{x\} = \{x^2\} = \{x^3\}$

Calculation of real values of $x$ in $\{x\} = \{x^2\} = \{x^3\}$ , where $\{x\} = $fractional part of $x$ My Try:: Let $\{x\} = \{x^2\} = \{x^3\} = k$ and we know that $0\leq \{x\} <1$ and $0\leq ...
1
vote
2answers
171 views

Percentage to absolute value within another range?

I'm not sure if my question is well formulated or not (the title), but basically. I have a min and max value, min is 50 and max is 200. To calculate the percentage 50 is of 200, I do: ...
1
vote
3answers
63 views

Implicit differentiation issue

I am trying to implicitly differentiate $$\sin(x/y) = 1/2 $$ The solution manual says Step 1. $$\cos(x/y)\cdot\frac{y-x\frac{dy}{dx}}{y^2} = 0 $$ But I don't understand how they arrive at this next ...
1
vote
3answers
74 views

Value and simplify

I want to find the value and simplify square root 36 ? Square root of 36 is 6 But I would know how to find the value and simplify it .
1
vote
0answers
110 views

Recommendation for a good book on equation solving theory from the basics

I'm relearning calculus but I often find myself applying algebraic operations to equations mechanically without having a solid understanding of the side-effects of those operations. Such as extra or ...
0
votes
1answer
850 views

functions word problems

The owner of a luxury motor yacht that sails among the 4000 Greek islands charges $\$ 470$ per person per day if exactly 20 people sign up for the cruise. However, if more than 20 people (up to the ...
1
vote
1answer
74 views

simplifying this basic equation

I don't quite get how to simplify this equation correctly. Could someone show me the steps please? Big thanks! So if... $$f(x) = x^3$$ then: $$f(x+\Delta x) = (x+\Delta x)^3$$ and can be ...
1
vote
2answers
39 views

Determine t such that the triangles have same area.

I have three points $A=(2,3), B=(6,4)$ and $C=(6,6).$ Given $\vec{AB}=\vec v$ and $\vec{BC}={0 \choose 2}$. I have also that for every $t\in [0,1]$ there is a point $D$ given as $\vec{AD}=t\vec{v}.$ ...
9
votes
2answers
192 views

A problem regarding rational and irrational numbers.

Let $n$ be a positive integer greater or equal to $2$. Prove that there are infinitely many irrational numbers $a$ such that $a+a^2+\cdots+a^n$ is rational. Well, let $p$ be a prime. We consider ...
0
votes
0answers
26 views

Determining Range

$X_i\sim^{iid}N(0,1);\quad i=1,2$ so $x_i$ ranges from $-\infty$ to $\infty$. Now $Y=X_1^2+X_2^2$ so $y$ ranges from $0$ to $\infty$. But how $Z=X_2$ is ranges from ($-\sqrt y$) to $\sqrt y$ ?
0
votes
1answer
47 views

Partial Differentiation.

$$\frac{\partial}{\partial y}(y-z)^{\frac{1}{2}}=???$$ I can differentiate it $$\frac{d}{d y}(y-z)^{\frac{1}{2}}=\frac{1}{2}(y-z)^{\frac{-1}{2}}$$ But i don't know when it is Partial.
1
vote
2answers
35 views

a - b > 0 algebra correction

Simple Algebra $\frac{1}{2}-\frac{1}{5} > 0$ $\frac{1}{2} > \frac{1}{5}$ $5 >2$ Looks correct, but where am I wrong in this, $\frac{1}{2}-\frac{1}{5} > 0$ $-\frac{1}{5} > ...
3
votes
3answers
826 views

condition for a cubic polynomial to have a real root

Let $a,b \in R$ and assume that $x=1$ is a root of the polynomial $p(x)= x^4+ax^3+bx^2+ax+1$. Find the ranges of values of $a$ for which $p$ has a complex root which is not real. Here first I ...
-1
votes
1answer
193 views

Calculating error tolerance for approximate values

If $a$ and $b$ are nearly equal,show that $$ \frac{a+b}{2}-\frac{(a-b)^2}{4(a+b)}$$ is the approximate value of $\sqrt{ab}$ with an error in excess less than $$\frac{(a-b)^4}{4(a+b)^3}$$
1
vote
1answer
293 views

How is algebra and geometry related to calculus?

Probably a too trivial question but here goes: What is the relationship of algebra and geometry with calculus? Are they pre-required knowledge for calculus, is calculus a different type of theory on ...
0
votes
1answer
49 views

Marginal distribution of $P$

Joint distribution of $P$ & $Q$ is $$f_{P,Q}(p,q)=\frac{1}{2\sqrt{(2\pi)}\sigma}\exp[-\frac{1}{2}{(\frac{\frac{p+q}{2}-\mu}{\sigma})^2}] \times\theta\exp[-\theta(\frac{p-q}{2})],\quad ...
3
votes
1answer
939 views

Roots of $ x^3-3x+1$

How can I find the roots of $x^3-3x+1$ using Cardano's formula? So far, I found that $$x = \sqrt[3]{\dfrac{-1+\sqrt{-3}}{2}} + \sqrt[3]{\dfrac{-1-\sqrt{-3}}{2}}$$ $$x = ...
7
votes
1answer
108 views

Find the sum of the first ten terms

How do I find the sum below? $$\sum_{i=1}^{10}\frac{2i+1}{i^2(i+1)^2}$$ I think there should be a simpler way instead of just adding the ten terms up using brute force, since it's impossible to do ...
3
votes
2answers
650 views

How to prove $\cos 36^{\circ} = (1+ \sqrt 5)/4$?

Given $4 \cos^2 x -2\cos x -1 = 0$. Use this to show that $\cos 36^{\circ} = (1+ \sqrt 5)/4$, $\cos 72^{\circ} = (-1+\sqrt 5)/4$ Your help is greatly appreciated! Thanks
2
votes
3answers
142 views

Factor $(x+y)^7-(x^7+y^7)$

So I was doing some practice problems to prepare upcoming math contests. This is one of the problems: Factor $(x+y)^7-(x^7+y^7)$ I got zero for $(x+y)^7-(x^7+y^7)$, however, the solutions ...
1
vote
2answers
76 views

factoring a quadratic equation

I know that if a quadratic equation has roots s and t, you can factor it as (x-s)*(x-t) and I used that property countless times, but I have never quite understood why that should be. Any ...
5
votes
1answer
102 views

Inductive Proof of Inequality

$\textbf{Question:}$ Prove using induction that the following inequality holds for all positive integers $n$: $$\dfrac{(1+a_1)(1+a_2)\cdots(1+a_n)}{1+a_1a_2\cdots a_n}\leq 2^{n-1},$$ where ...
2
votes
1answer
93 views

Sum of squares of power of two sums and differences

What is the sum $$\sum\left(\frac12-\frac1{2^n}\pm\frac14\pm\frac18\pm\ldots\pm\frac1{2^n}\right)^2$$ where the sum is taken over all $2^{n-1}$ combinations of plus and minus signs? If the sum is too ...
0
votes
2answers
199 views

Deriving the Kinematic equation without time? $x = (v^2 - v_0^2)/2a + x_0$

Ok so this is a short question (maybe a little bit more about my inability to do math). I need to derive $x = \frac{v^2 - v_0^2}{2a} + x_0$ from the position function $ = v_0t + \frac{1}{2}at^2 + ...