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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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
59 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
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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 ...
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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?
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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
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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
43 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 ...
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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
54 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
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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
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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 ...
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2answers
46 views

Long division of $X^3+2X^2+4 $ by $X+2$ produces $0$, why? [closed]

The problem is to divide the polynomials: $$\frac{X^3+2X^2+4 }{ X+2}$$ When I do this, on the second line I get a result of $0$. What did I do wrong?
1
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2answers
22 views

What does it mean when two variables are said to be proportional?

Assume we are dealing with two variables i.e. $x$ and $y$. And suppose that $x$ starts increasing and to a certain value of $x$, say $a$, $y$ is $Zero$ but starts increasing when $x>a$ and a ...
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votes
2answers
40 views

Prove this absolute value related inequality [closed]

$\left | |a+b|-|a|-|b| \right | \leq 2|b|$, $\forall a, b \in \mathbb{R}$. How can I prove it?
1
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1answer
21 views

Finding the value of $y=b^2(3a^2+4ab+2b^2)$ if $a^2(2a^2+4ab+3b^2)=3$ and $a$ and $b$ are distinct zeros of $x^3-2x+c$

If $a$ and $b$ are distinct zeroes of the polynomial $x^3-2x+c$ and $$a^2(2a^2+4ab+3b^2)=3$$ $$b^2(3a^2+4ab+2b^2)=y$$ Evaluate $y$ I tried for many hours but couldn't solve this question. ...
1
vote
1answer
19 views

Using the triangle inequality to show that if $|x| < 4$ then $|x^2-2x+3| < 27$

I'm starting school soon and doing some review problems to prep for Calculus. I'm a bit stuck on this problem: Show that if $|x| < 4$ then $|x^2-2x+3| < 27$. I know that I have to use the ...
12
votes
2answers
144 views

Solve $x^7-5x^4-x^3+4x+1=0$ for $x$

Solve for $x$ $$x^7-5x^4-x^3+4x+1=0$$ This equation has been bugging me since the past few days. I have found, using the Rational Root Theorem that $x=1$ is a root of this equation. However, ...
0
votes
4answers
55 views

How does one go about simplifying $\sqrt{72} $

In my book I am reading I sometimes see that the writer simplifies most of the answers most of the time. Take the following example. I calculated an answer to the following $\sqrt{72}$, the book has ...
0
votes
1answer
33 views

Show that these two expressions are equivalent [closed]

How would I show that: $$\frac{\displaystyle\sum_{i=1}^n(y_i-\bar{y})}{\displaystyle\sum_{i=1}^n(x_i-\bar{x})} = ...
0
votes
2answers
55 views

Showing that the function is less than $\frac{2}{n+1}$

Showing that $g(t)=t\left(1-\frac{t}{2}\right)^n \leq \frac{2}{n+1}$ for every natural $n$ and $t$ in $[0, 1]$. How is this done? Is there a simple way to prove this? I tried putting in numbers and ...
0
votes
1answer
41 views

$8:4\times2=1$ or $8:4\times2=4$? [duplicate]

A simple question but I know two conflicting rules on this: Multiplication is stronger that dividing: $8:4\times2=8:8=1$ Dividing is the same as multiplication with the inverse: ...
4
votes
1answer
59 views

Recurrent problem about polynomials

Given is a sequence of polynomials $P_n$, defined as follows: $P_0(x)=0, P_{n+1}(x) = P_n(x) + \frac{x-P_n^2(x)}{2}. $, n= 0,1,2,..., and x is real. Proving that for all non-negative integers n and ...
3
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4answers
55 views

To prove $(\sin\theta + \csc\theta)^2 + (\cos\theta +\sec\theta)^2 \ge 9$

I used the following way but got wrong answer $$A.M. \ge G.M.$$ $$ \frac{\sin \theta + \csc \theta}{2} \ge \sqrt{\sin \theta \cdot \csc \theta}$$ Squaring both sides, \begin{equation*} (\sin\theta + ...
0
votes
1answer
48 views

Is there a summation formula for this equation (contains square roots, and functions within the square root)?

I am trying to solve a summation formula that is quite complex. However, to make the "answering" process for you guys easier I'll isolate the part I am having trouble with... The equation is as ...
4
votes
1answer
54 views

Solving cubic with a nice real solution

Solve the cubic for $x\in\mathbb{R}$ $$x^3-9 x^2-15x-6 =0$$ The only real solution is $x=3+2\sqrt[3]{7}+\sqrt[3]{7^2}$. Given the regularity of this solution, can we solve this constructively ...
2
votes
2answers
45 views

Equivalence of trigonometric identity

Is writing $$ \cot{2\theta}=\frac{a-c}{2b} $$ equivalent to $$ \cot{\theta}=\frac{a}{b},\tan{\theta}=\frac{c}{b} $$ becuase of the trigonometric identity $$ ...
2
votes
1answer
39 views

Equivalence of geometric and algebraic definitions of conic sections

I have not been able to find a proof that the following definitions are equivalent anywhere, thought maybe someone could give me an idea: A parabola is defined geometrically as the intersection of a ...
2
votes
7answers
281 views

Prove that $1+ \frac{1}{x^4} \geq \frac{1}{x} + \frac{1}{x^3}$

Prove That $$1+ \frac{1}{x^4} \geq \frac{1}{x} + \frac{1}{x^3}$$ where $x \in \mathbb Z^{+}$
1
vote
3answers
58 views

Prove $1+ (\frac{1}{x}) \geq (\frac{1}{x^4}) +(\frac{1}{x^3})$ [closed]

Prove That $$1+ \frac{1}{x} \geq \frac{1}{x^4} + \frac{1}{x^3}$$ where $x \in \mathbb Z^{+}$
6
votes
2answers
80 views

Evaluate this Trigonometric Expression

Evaluate $$ \sqrt[3]{\cos \frac{2\pi}{7}} + \sqrt[3]{\cos \frac{4\pi}{7}} + \sqrt[3]{\cos \frac{6\pi}{7}}$$ I found the following $\large{\cos \frac{2\pi}{7}+\cos \frac{4\pi}{7} + \cos ...
4
votes
2answers
98 views

When the quadratic formula has square root of zero, how to proceed?

Is there an easier way to solve the following equation? $$x^2=2x-1$$ I think I know how to find $x$, using the quadratic formula: I get $$x^2-2x+1=0$$ then $$x=\frac{2 \pm \sqrt{4-4})}2= ...
1
vote
1answer
25 views

How to write this expression about unit digits symbolically?

From a GRE book: "The units digit of a product of positive integers is equal to the units digit of the product of the units digits of those integers." I read this and was thinking... why would you ...
3
votes
1answer
47 views

How many mappings $\phi:\Bbb{N}\cup\{0\}\to\Bbb{N}\cup\{0\}$ exist such that $\phi(ab)=\phi(a)+\phi(b)$?

How many mappings $\phi:\Bbb{N}\cup\{0\}\to\Bbb{N}\cup\{0\}$ exist such that $\phi(ab)=\phi(a)+\phi(b)$? My book says that the answer is finite. However, I am getting infinite as the answer. Let ...
0
votes
1answer
36 views

Find any number that can be square rooted and cube rooted.

Anyone can help with this? It is like asking: $z^2=x $ , $ y^3=x$ where $y,z$ are integers. we want to find $x$
0
votes
1answer
47 views

How can I find this limit? $\lim_{x\to0}\left(\frac{x\csc(2x)}{3\cos(5x)}\right)$

$$\lim_{x\to0}\left(\frac{x\csc(2x)}{3\cos(5x)}\right)$$ My attempt was just turning csc to 1/sin, how can I solve this
2
votes
1answer
31 views

easy calculus hw question on computing work from spring compression

hw question reads: A 5-kg mass is attached to a spring that hangs vertically and is stretched 3 m from the equilibrium position of the spring. Assume a linear spring with F(x) = kx. How much work is ...
1
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0answers
41 views

A problem related to complex polynomial

Let $$P_{t}(z) =a_{0}(t) + a_{1}(t)z + ...+a_{n}(t)z^n$$ be a polynomial where the coefficients depend continuously on a parameter $t \in (−1, 1)$. Assume that there exists $\text{t}_{0} \in (−1, 1)$ ...
3
votes
3answers
136 views

AHSME 1981 #22 - Number of lines that pass through four distinct points

How many lines in a three dimensional rectangular coordinate system pass through four distinct points of the form $(i, j, k)$ where $i$, $j$, and $k$ are positive integers not exceeding four? ...
3
votes
2answers
44 views

How to identify this function? $y = \log_2(y^{-1} + 4y)$

$$y = \log_2(y^{-1} + 4y)$$ How can I deal with the $y^{-1}$ and $4y$, also does identify mean find the domain, range and symmetry?
2
votes
2answers
50 views

Separable equation

I had $y=e^{4\ln|x|}+e^{4C}$, then simplified to $y=e^4 \cdot e^{\ln|x|} +e^{4C}=A\ln|x|+C_2$. This seems to be wrong and should've been $e^{\ln|x^4|}+e^{4C}$. Why is what I initially did wrong? ...
1
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3answers
36 views

How to graph $g(x)=4^x-1$ and find its domain and range? [closed]

How to graph $$g(x)=4^x-1$$ and give its domain and range using interval notation? I have no idea what to do.
1
vote
1answer
37 views

Find the diagonal matrix A that satisfies the equation

Find the diagonal matrix $A$ that that satisfies the equation: $$A^{-3}=\pmatrix{-27&0&0\cr0&8&0\cr0&0&-1}$$ Attempted solution: my intuition tells me that the inverse of ...
0
votes
1answer
28 views

Basic algebra problem for weighted averages

This question has me completely stumped for some reason, I would appreciate a bit of help. If you have 20 pounds of coffee for $1.80 a pound And then add X amount of coffee for $1.44 a pound How ...
0
votes
1answer
30 views

Why is decomposition failing me here

Why is decomposition failing me here, and what can I do about it in the future $$8x^2+10x+3 = 8x^2-2x+12x+3 = -2x(-4x+1)+3(4x+1) $$ see how one $4x$ is negitive? thats what I mean by failing
1
vote
1answer
40 views

Find the Viète formula

I know that the Viète formula for the equation $ax^2+bx+c=0$ is: $$x_1+x_2=-\frac{b}{a}$$ $$x_1x_2=\frac{c}{a}$$ But I didnt know which are the formula for the equation $ax^3+bx^2+cx+d=0$. Please ...
2
votes
1answer
27 views

If $a_n = \sum^n_{r=0} \frac{(\ln10)^n}{r! (n-r)!}$ for $n \geq 0$ …

Problem: If $a_n =\sum^n_{r=0} \frac{(\ln10)^n}{r! (n-r)!}$ for $n \geq 0$ then find the value of $a_0+a_1+a_2+\cdots \infty$ My approach: $a_n = \sum^n_{r=0} \frac{(\ln10)^n}{r! (n-r)!}$ $= ...
2
votes
3answers
84 views

I am working on proving or disproving $\cos^5(x)-\sin^5(x)=\cos(5x)$

True or false? $$\cos^5(x)-\sin^5(x)=\cos(5x)$$ for all real x. I have no idea how to prove or disprove this. I tried to expand $\cos(5x)$ using double angle formula but I wasn't sure how to go from ...