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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4
votes
2answers
90 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= ...
0
votes
4answers
52 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 ...
3
votes
2answers
30 views

Function notation meaning: $f: \{a,b\} \to a$ - Zorich - MA I - p18

I have some notation I haven't seen before: $$f: \{a,b\} \to a\text{ and } g:\{a,b\}\to b$$ What does this mean? We are mapping from some $X=\{a,b\}$ to some $Y=a$? So pretty much we are always ...
0
votes
1answer
45 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 ...
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
33 views

Show that these two expressions are equivalent [on hold]

How would I show that: $$\frac{\displaystyle\sum_{i=1}^n(y_i-\bar{y})}{\displaystyle\sum_{i=1}^n(x_i-\bar{x})} = ...
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 ...
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: ...
3
votes
4answers
52 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 + ...
6
votes
2answers
79 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
1answer
53 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 ...
0
votes
2answers
310 views

What annual installment will discharge a certain debt?

What annual installment will discharge a debt of $ 717.60 due in 4 years at 20% p.a. simple interest, if the installments are paid at the each end of each year? I tried the following: $ 717.60 ...
2
votes
7answers
280 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 $x \in \mathbb Z^{+}$
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
38 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 ...
1
vote
3answers
57 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 $x \in \mathbb Z^{+}$
1
vote
1answer
144 views

Is an algebraic formula to test real numbers equality?

Is there a formula to test numbers equality ? Let $x$ and $y$ real numbers. If $x=y$ the formula will results $1$. Else the formula will results $0$. I'm not searching for an algorithmic solution ...
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 ...
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
54 views

Signing $y''$ from $\log(\frac{x+y}{x})=x+y$

Suppose that $x,y>0$ are positive reals such that $y$ is defined implicitly in terms of $x$ via: $$ \log\left(\frac{x+y}{x}\right)=x+y.\tag{$\star$} $$ I would like study the sign of $y''$. ...
0
votes
4answers
55 views

Applying a function to both sides of an equation doesn't change it?

Why is it that applying a function to both sides of an equation doesn't change it? Can this be proven? Can you point to some material to read more about this?
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
3
votes
2answers
42 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
49 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? ...
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 ...
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? ...
1
vote
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 $\text{t}_{0} \in (−1, 1)$ ...
2
votes
1answer
81 views

Determine when the system has a) no solution, b) 1 solution and c) infinitely many solutions

This question is not for an assignment, it was on the midterm and I am interested in figuring out how to solve it before the final exam. cheers, Determine when the system has a) no solution, b) 1 ...
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 ...
1
vote
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
6answers
125 views

Are there numbers such that A + B = 10A+B? [closed]

I was just wondering, apart from zero,are there numbers where $A+B=10A+B$ (the number AB)?
0
votes
1answer
27 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
2answers
37 views

Minimize the surface area of a $3$-dimensional object consisting of a ball on top of a truncated cone

Suppose I want to produce some special objects consisting of a ball with a truncated cone as bottom (the ball is placed on top of the truncated cone). The whole object has a volume of $325$ ...
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 ...
0
votes
3answers
66 views

Simple question on exponentiation

I know this one is trivial, but I was wondering: if I have something like $$a^{b^c}$$ then i know that it should be read as $$a^{\left(b^c\right)}$$ if no other parenthesis is present. Question: if ...
0
votes
0answers
11 views

Convert to all applicable forms (polar, rectangular, exponential) [closed]

(8-6j) 6.5e^(pi/4)j NOTE: I dont know how to make the angle symbol. 20(angle)300° Please show detailed steps on how you arrived to your answer. These are question from a practice test for a ...
14
votes
2answers
8k views

How to prove a limit exists using the $\epsilon$-$\delta$ definition of a limit

I understand how to find a limit. I understand the concept of the $\epsilon$-$\delta$ definition of a limit. Can you walk me through what we're doing in this worked example? It is from my student ...
3
votes
3answers
150 views

Computing the infinite sum, $\sum_{n=0}^\infty \frac {5^n}{25^n + 1} $

So I'm trying to compute $$ \displaystyle\sum_{n=0}^\infty \dfrac {5^n}{25^n + 1}. $$ The closed form is not very nice and I don't see any immediate telescoping. Any ideas?
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: $a_n = \sum^n_{r=0} \frac{(\ln10)^n}{r! (n-r)!}$ $= ...
0
votes
1answer
386 views

Calculating how many pieces fit into a given area

Is there any program which calculates how many pieces of an item with different sizes you can put in one area? For example, I have a sheet of glass with size $3210 \times 2210$mm. Now I have ...
4
votes
2answers
442 views

Solving an equation with exponentials

$$2^x+4^x+12=0$$ How exactly am I supposed to solve this? Am I supposed to get $x$ alone or solve it another way?
12
votes
3answers
561 views

FoxTrot Bill Amend Problems

So I found this on the Wolfram website today: So I was wondering about how one might be able to (if possible) solve those four problems by hand. Here are the problems, $\LaTeX$ed: $ \lim_{x \to ...
2
votes
3answers
83 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 ...
1
vote
6answers
170 views

How to find roots of $4t^3-18t^2+24t-10=0$

Disclaimer: I am not a student trying to get free internet homework help. I am an adult who is learning Calculus from a textbook. I am deeply grateful to the members of this community for their time. ...
5
votes
5answers
221 views

Show: $(x+y)^4 \leq 8(x^4 + y^4)$

I wish to show the following statement: $ \forall x,y \in \mathbb{R} $ $$ (x+y)^4 \leq 8(x^4 + y^4) $$ What is the scope for genralisaion? Edit: Apparently the above inequality can be shown ...
0
votes
2answers
33 views

Derivative Inverse of a function

I have a question: $\begin{array}{lrl} \mbox{If :} & f(x) & = x^5 + 3x^3 + 2x + 1 \\ \mbox{And :} & g(x) & = f^{-1} (x) \\ \mbox{What is :} & g'(7)&\mbox{?} \\ \mbox{What I ...
0
votes
1answer
32 views

Precalc word problem

Can someone to point me in the right direction for this math problem I have on my homework? I don't know where to begin on this. The elk population in a certain region is given by the function ...
6
votes
2answers
47 views

Solving the equation: $3\cos x - \sin 2x = \sqrt{3}(\cos 2x + \sin x)$

Solving the equation: $$3\cos x - \sin 2x = \sqrt{3}(\cos 2x + \sin x)\tag{1}$$ I tried to write $(1)$ becomes $$\sqrt{3}\sin \left(\frac{\pi}{3}-x\right)=\sin \left(\frac{\pi}{3}+2x \right)$$ Now, ...