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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25
votes
2answers
363 views
+50

Prove $|P(0)|\leq 2n+1$

Let $P(x)$ be a polynomial with degree $\leq n$ and $|P(x)|\leq\frac{1}{\sqrt{x}}$ for $x\in(0,1]$. Prove that $|P(0)|\leq 2n+1$. The idea should be that if $|P(0)|$ is too large, then the polynomial ...
1
vote
3answers
70 views

Is $f(x) = \frac{5x^2}{1+x^2}$ bounded?

Show that $$f(x) = \frac{5x^2}{1+x^2}$$ is a bounded function? I know that if $x=0$ the function is undefined, but how can you prove that it is bounded? Help is much appreciated!
-1
votes
1answer
17 views

Write an equation for a rational function with:

Write an equation for a rational function with: Vertical asymptotes at $x = -5$ and $x = 5$ $x$ intercepts at $x = -2$ and $x = -6$ Horizontal asymptote at $y = 6$ $y =$ ? I have ...
0
votes
1answer
22 views

Bounds on sum of entries of an idempotent symmetric matrix

Suppose that $M$ is symmetric and idempotent, dimensions $n\times n$, and trace $n-k$. Let $e$ ($n\times 1$) be a column of $1$'s. Let $$ S_1\equiv e'Me,\quad ...
1
vote
6answers
97 views

$f(x) - f'(x) = x^3 + 3x^2 + 3x +1; f(9) =?$

Given the following $f(x) - f'(x) = x^3 + 3x^2 + 3x +1$ Calculate $f(9) = ?$ I have tried to play with different number of derivatives. Also tried to solve it by equations. Maybe there is some ...
-1
votes
2answers
62 views

Solve a system of two equations with cubic radicals

Solve the following system of equations ($x,y \in \Bbb R$): $$\begin{cases} (8x-13)y&=(x+1)\sqrt[3]{3y-2}-7x \\ (y-1)x^2+(8y+7)x&=y^2+12y+(x+1)\sqrt[3]{3y-2}. \end{cases}$$ I think this ...
0
votes
5answers
71 views

Prove that $a^2+b^2+16\ge ab+4a+4b$ for all $a, b$.

How to prove that for all numbers $a, b$, $$a^2+b^2+16\geqslant ab+4a+4b .$$ I got an idea that I can move everything to the left side: $$a^2+b^2+16-ab-4a-4b\geqslant 0$$ and then treat it as ...
-2
votes
0answers
12 views

problem on conical frustum [closed]

A glass of conical frustum shape has its two radius 5 cm and 2 cm and height of the frustum is 10 cm.. that means it has a capacity of 408.2 ml. now if we fill the glass up to 200 ml then find the ...
2
votes
0answers
45 views

If $a+b=8$ and $ab+c+d = 23$ and $ad+bc=28$ and $cd=12\;,$ Then $abcd$

If $a+b=8$ and $ab+c+d = 23$ and $ad+bc=28$ and $cd=12\;,$ Then value of $(1)\;\; a+b+c+d=$ $(2)\;\; ab+bc+cd+da = $ $(3)\;\; abcd=$ My attempt: Let $x=a\;,b$ be the roots of ...
1
vote
1answer
24 views

How to find the most general value of $\cos(A-B) = 1/2$ and $\sin(A+B) =1/2$?

I'm learning Trigonometry right now with myself and at current about general solution. I have a question which is confusing me from some time. The question is - $If \cos(A-B) = 1/2$ and $\sin(A+B) ...
1
vote
1answer
23 views

Solve for $b$ in $(\frac{1}{a} + b(\frac{n}{1/a})^{\frac{1}{b}}) * log_2({\frac{1}{\delta}}) = \frac{1}{a}log_2({\frac{1}{\delta}}) * log_2(n)$

Need to get $b$ from $$(\frac{1}{a} + b(\frac{n}{1/a})^{\frac{1}{b}}) \times log_2({\frac{1}{\delta}}) = \frac{1}{a}log_2({\frac{1}{\delta}}) \times log_2(n)$$ and values for $b$ for $$(\frac{1}{a} + ...
0
votes
2answers
27 views

Limit on a five term polynomial

it has been two years since I have taken or used calculus, and I am having some trouble with factoring a polynomial in order to take a limit on it. I have searched for previous similar questions here, ...
-2
votes
0answers
22 views

Exponential growth or exponential decay? [closed]

Can I get a step by step answer to this question? Write the function below in the form $P = P_0a^t$. Is this exponential growth or exponential decay? $$P = 2e^{-1.1 t}$$ Round the base of the ...
0
votes
1answer
55 views

Put $A\cos(x) + B\sin(x)$ into form : $A\sin(x+ \theta)$ [closed]

The task is to manipulate $$\cos(x) + \sqrt{3}\sin(x)$$ into the form $$A\sin(x+ \theta)$$ My question is : why is $\pi$ in the numerator and denominator both divided by $6$? I am familiar with ...
0
votes
3answers
47 views

Geometric interpretation of $ \frac{x^2+y^2}{y}=\text{constant} $.

I would love help in interpreting the following expression geometrically $$ \frac{x^2+y^2}{y}=\text{constant} $$ for simplicity, let $ c = \text{constant} $, and then through rearrangement we have ...
1
vote
2answers
65 views

Common factor in $2\sin(x)\cos(x) + \sin(x) = 0$ [closed]

I am stuck on part of a question : The 1st line of work is : $$2\sin(x)\cos(x) +\sin(x) = 0$$ The next line is : $$\sin(x) \cdot (\sin(x)\cos(x) +1)$$ I see that $$\sin(x) \cdot 1 $$ gives ...
0
votes
1answer
18 views

Proof for Theorem of Upper and Lower Bounds On Zeroes of Polynomials

I'm currently a high school Pre-Calculus student and my textbook presents the following theorem without proof: Let $f(x)$ be a polynomial with real coefficients and a positive leading coefficient. ...
0
votes
1answer
31 views

Can a denominator of fraction be multiplied by -1 without affecting the numerator ? and if so why?

I have been presented with a solution for solving trigonometric identities. However I would like to see further proof that one of the lines of work are valid. \begin{align*} \frac{2\sin x\cos x}{1 + ...
28
votes
6answers
2k views

What is the order when doing $x^{y^z}$ and why?

Does $x^{y^z}$ equal $x^{(y^z)}$? If so, why? Why not simply apply the order of the operation from left to right? Meaning $x^{y^z}$ equals $(x^y)^z$? I always get confused with this and I don't ...
0
votes
1answer
29 views

Solving the exponential equation $4^x - 10 \cdot 4^{-x} = 3$

I was gliding through some problems using substitution like my book was using and I ran into this problem. I'm not quite sure where to start with it. $$4^x - 10 \cdot 4^{-x} = 3$$ What would be ...
-1
votes
3answers
97 views

Inequalities with quadratics [closed]

$$\frac{12}{x^2 + 2x} < \frac{3}{x^2 + 4x + 4}$$ I am confused. Can someone help me? Update : you can see my work in the comments. i figured out the answer but the answers other people gave ...
2
votes
2answers
39 views

How to solve for x/z and y/z here?

I got stuck solving these two equations: $$a_1(x/z) + b_1 (y/z) + c_1 = 0$$ and, $$a_2(x/z) + b_2 (y/z) + c_2 = 0$$ for $$x/z$$ and $$y/z$$. The desired result would be: $${x \over z} = {b_1c_2 - ...
0
votes
1answer
26 views

What is the difference between base vectors and vector components.

I understand that you add them to create a vector. But are these just alternate names? In a math investigation I am using the two of them and I want to stay consistent in the way i name things. Thank ...
6
votes
5answers
311 views

Absolute value and max/min function: why $a + b + |a - b|=2\max(a,b)$? [duplicate]

I am being told that $a + b + |a - b|$ is equal to $2\max(a,b)$. What is the reasoning behind this?
11
votes
2answers
181 views

How do I calculate this limit: $\lim\limits_{n\to\infty}1+\sqrt[2]{2+\sqrt[3]{3+\dotsb+\sqrt[n]n}}$?

I have seen this question on the internet and was interested to know the answer. Here it is : Calculate $\lim\limits_{n\to\infty}(1+\sqrt[2]{2+\sqrt[3]{3+\dotsb+\sqrt[n]n}})$? Edit : I really tried ...
0
votes
0answers
26 views

Mensuration problems [closed]

The total cost of painting the four walls of a square hall at Rs. 40 per square metre is Rs. 12000. If the hall contains 600 m^3 of air, find the height of the hall. My solution is as follows; Area ...
0
votes
1answer
30 views

Real values of $x$ in $(1+2^x)\cdot (1+8^x)\cdot (1+9^x)^2 = (1+6^x)^4$

$(1)$ Real values of $x$ in $2^x+3^{-x}+4^{-x}+6^x = 4$ $(2)$ Real values of $x$ in $(1+2^x)\cdot (1+8^x)\cdot (1+9^x)^2 = (1+6^x)^4$ My Try: For First one:: Here ...
0
votes
1answer
49 views

Binominal expression simplification

I need to simplify the expression $$\sum_{k = 1}^{10} k\binom{10}{k}\binom{20}{10 - k}$$ Thank you.
3
votes
2answers
35 views

Calculations with an exponentially-weighted moving average

I need help figuring out the following formula: Where: CTLy = yesterdays CTL TSS = current Training Stress Score TC_c = your CTL Time Constant Now I have TSS, thats a number between 20-500 ...
0
votes
1answer
28 views

If $2^{x + 1} < y$, then what is the “smallest” function of $x$ that cannot be an upper bound for $y$?

(This is a follow-up question to this MSE post.) The title says it all. Let $x$ be a positive integer. If $2^{x + 1} < y$, then does there exist a minimum function $f(x)$ that cannot be an ...
0
votes
1answer
36 views

If $2^{x + 1} < y$, then what is the largest polynomial in $x$ that cannot be an upper bound for $y$?

Update: I have posted a follow-up question here. The title says it all. If $2^{x + 1} < y$, then what is the largest polynomial in $x$ (of maximum possible degree) that cannot be an upper ...
1
vote
3answers
68 views

How to go upon proving $\frac{x+y}2 \ge \sqrt{xy}$? [duplicate]

I'm trying to prove this but am having some difficulty. For any $x,y\in\mathbb R$ such that $x\ge 0$ and $y\ge 0$ we have $$\frac{x+y}2 \ge \sqrt{xy}.$$ So far what I have gotten to is ...
0
votes
1answer
17 views

Is there an analytic definition of reflection?

Consider some real function $f$. Then its reflection about the y-axis is defined as the function $g$ satisfying $g(x) = f(-x)$; its reflection about the x-axis is defined similarly as the function $h$ ...
2
votes
0answers
72 views

Having a hard time getting an expression into the desired form

My goal is to obtain the following: $$ g(0)=\frac{B}{1-2p}-\frac{A+B}{1-2p}\frac{r^B-1}{r^{A+B}-1}. $$ This is what I know: $$ g(A)=g(B)=0, \ g(k)=a+br^k+\frac{k}{1-2p}, \ a \text{ and } b ...
3
votes
2answers
117 views

Does $\overline{ \sqrt{1 + i}} = \sqrt{1-i} \ $?

I am having trouble with complex conjugates today. Can someone help me? $$\overline{ \sqrt{1 + i}} \stackrel{\color{#2222FF}{?}}{=} \sqrt{1-i} \tag{$\ast$} $$ In this case, since $\cos ...
-1
votes
0answers
61 views

Solve for $n$: $12n^3 + 40n \log n = 5n^4 -100n^2$

Need to solve for $n$: $12n^3 + 40n \log n = 5n^4 -100n^2$ tried it out and my answer isn't making sense
4
votes
1answer
114 views

How do you find the probability of A winning if the probability of getting a favourable outcome in the $r^{th}$ turn is a function of $r$?

Problem: Two players A and B are playing snake and ladder. A is at 99 and he needs 1 to win in rolling of a dice. However, he is always allowed to re-throw the dice if 6 appears. What is the ...
0
votes
1answer
28 views

A Binomial coefficient sequence

If 'n' is a positive integer and $C_k=^nC_k$, then find the value of: $[\sum\limits_{k=1}^n\frac{k^3}{n(n+1)^2.(n+2)}(\frac{C_k}{C_{k-1}})^2]^{-1}$ [![enter image description here][1]][1] I have ...
2
votes
2answers
35 views

Polynomial with no integer roots

This is an excercise given to a kid I am tutoring, as part of a set of problems regarding polynomials. He is currently at the last class before graduation year. Let $p$ be a polynomial in $ℤ[x]$ such ...
1
vote
2answers
72 views

Evaluating limit $\lim_{n\to\infty}({\sqrt{4^n + 3^n} - 2^n})$

I have to find: $$\lim_{n\to\infty}\left({\sqrt{4^n + 3^n} - 2^n}\right)$$ I plugged in some numbers and it seems as if this sequence were approaching infinity, but I do not know how to begin ...
0
votes
3answers
39 views

How $n\pi +(-1)^n \pi/6 = n\pi + \pi/6$ and $n\pi - (-1)^n \pi/6 = n\pi - \pi/6$?

I'm learning Trigonometry right now with myself and at current about general expression of the angles. I am confused in a problem from sometime . I don't know how $n\pi +(-1)^n \pi/6 = n\pi + \pi/6$ ...
0
votes
2answers
38 views

When does $P(x,y)$ is a function of $x+2y$?

Suppose $P$ is a polynomial of two real variables $x$ and $y.$ How can I prove that $P(x,y)$ is a function of $x+2y$ if and only if $P_y=2P_x$ ? Here $P_x=\dfrac{\partial P}{\partial x}.$ Is ...
0
votes
2answers
19 views

Polynomial with bounded coefficients and real root

A polynomial with degree $2n$ has all coefficients in the range $[100,101]$ and has a real root. What is the minimum possible $n$? Degree $0$ is clearly not possible. For degree $2$, the discriminant ...
0
votes
3answers
51 views

What comes first here? pemdas doesnt really tell me what to do here

So I have this equation: $2x(x+3)(x+3)$ Do I FOIL the $(x+3)$ first or multiply the $2x$ to the first $(x+3)$? Would there be a difference? Isn't multiplication commutative?
2
votes
1answer
19 views

Given that there is at least one error in the bit, what is the probability that it will be retransmitted?

A communication channel can increase the probability of successful transmission by using error-correcting codes. One of the simplest of these is called a "parity scheme". In such a scheme, the message ...
0
votes
1answer
39 views

A Combination of decreasing functions

I have a strictly decreasing convex function $f$ (at least over $\Bbb R^+$ ), and the non negative numbers $a_1 , a_2$ and $b_1 , b_2$. Is the following a decreasing function ( at least on $t \in \Bbb ...
3
votes
1answer
33 views

Probability of having at least one error in block of three bits?

A communication channel can increase the probability of successful transmission by using error-correcting codes. One of the simplest of these is called a "parity scheme". In such a scheme, the message ...
0
votes
1answer
24 views

Working out expression values

What is the value of this expression? $$\frac{1}{\dfrac{1}{\frac{1}{2} + \frac{1}{3}} + \dfrac{1}{\frac{1}{4} + \frac{1}{5}}}$$ I thought I'd start by working out 1/2 + 1/3,which is 5/6, and then ...
-1
votes
1answer
19 views

Roots of polynomial [closed]

If the roots of the equation $x^3-5x^2+8x-6=0$ are alpha,beta and gamma. How to find the $\sum \alpha^2 \beta^2$ ? Can't anyone give me some hints?
0
votes
2answers
17 views

Prove minimum of $\sum_{i=1}^n=S_i$ where all $S_i$ are limited by $x \le S_i \le y $

Sorry if this has already been asked or answered somewhere on the net. I have a set of values $S=\{S_1,S_2,S_3,... S_n\} $ where $x \le S_i \le y $. S has an unknown number of discrete members, ...