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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2answers
38 views

$(5 + (24)^{\frac{1}{2}})^x + (5 - (24)^{\frac{1}{2}})^x = 10$ , solve for $x$

I have been stuck to this question lately $(5 + \sqrt{24})^x + (5 - \sqrt{24})^x = 10$ , solve for $x$
2
votes
2answers
61 views

Solving the roots of Redlich Kwong Equation

Good evening everyone. After studying some equations of state, I've read about the mathematical steps formulated to model some. In the particular case of Van der Waals, where ...
0
votes
1answer
14 views

forming log equation from graph points

Okay so I need to form a logarithmic equation from the points (1960,4.7) (1964,5.1) (1968,5.4). I have 'guess and checked' to get the equation 2.7421 log(x-1950)+1.9579, and was wondering if there was ...
-3
votes
0answers
28 views

How to generate RBG color from an arbitrary range of values? [closed]

Given some vector like values = [23,123,6,5 .. ] How can I define a list of RBG values for each number, such that The max element is RBG{0, 122, 255}, or blue ...
0
votes
0answers
59 views

What are the classic or great books for Algebra, Geometry, and Trigonometry that are similar to what Spivak, Courant, and Apostol are for Calculus?

There are classic textbooks for Calculus like Spivak, Courant, Apostol, etc that do a fantastic job at explaining the fundamental concepts and theory along with great problem sets. My question is the ...
1
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1answer
36 views

If $f(x)=2x^2+2x-4$ and $g(x)=x^2-x+2$, Find the number of integral values of $x\in[1,10]$ such that $\sqrt{f(x)}+\sqrt{g(x)}\ge \sqrt{2}$

If $f(x)=2x^2+2x-4$ and $g(x)=x^2-x+2$, Find the number of integral values of $x\in[1,10]$ such that $\sqrt{f(x)}+\sqrt{g(x)}\ge \sqrt{2}$ I tried squaring two times to remove the square root and ...
6
votes
7answers
339 views

Is $202^{303}$ greater or $303^{202}$?

Find without use of calculator which of the two numbers is greater $202^{303}$ or $303^{202}$. I think we have to do this with calculus because I got this question from my calculus book. I tried ...
1
vote
1answer
23 views

Optimization Problem - Trigonometric Derivatives Application

You need to get from point A to point B as fast as possible. But there is a circular lake between A and B. You can run twice as fast as you can swim. At what angle (theta) should you swim? ...
3
votes
1answer
41 views

Solution to a simple system of quadratic equations

I am hoping to find a closed-form solution to the following system of $n$ quadratic equations: $$ x_j^2 = \sum_{i=1}^n B_{ij}x_i $$ for $j\in\{1,\dots,n\}$, where $B_{ij}\geq 0$. There is a trivial ...
0
votes
2answers
40 views

Prove minimum of piecewise function

$$F(x)= \begin{cases} \dfrac{\tan x}{x}, x \neq 0\\ 1, x = 0 \end{cases} $$ Prove that there is a minimum at $x=0$
0
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0answers
17 views

Graphing with transformation: translations or reflections.

$${y = (x - 4)^2}$$ I come up with 4 units to the right, but my problem is, how do I come up with other points on the parabola? The second question is ${y = -x^2 - 3}$. The vertex is ${-3}$. With ...
1
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1answer
34 views

Polynomial having all integral coefficients $P_n(a)=b$ and $P_n(b)=c$ and $P_n(c)=a$

Let $a,b,$ and $c$ denote three distinct integers, and let $P_n$ a polynomial having all integral coefficients. Show that it is impossible that $P_n(a)=b$ and $P_n(b)=c$ and $P_n(c)=a$. I started ...
-1
votes
1answer
39 views

Exercise: Evaluate a polynomial function such as $P(x)=2x^3-3x^2+7x-2$ at a surd such as $x=1+2\sqrt{3}$.

Exercise: Given polynomial function $P(x)=2x^3-3x^2+7x-2$ evaluate $P(x)$ at the surd $x=1+2\sqrt{3}$.
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4answers
49 views

Simplifying Square Roots of a Negative Number

I've been thinking about proving exponent laws: By induction, we have $(ab)^n=(ab)(ab)\cdots (ab)=a^nb^n$, for any natural $n\in \mathbb{N},a,b\in \mathbb{R}$ We can extend this to any $n\in ...
-1
votes
1answer
19 views

fill in a table with a 7% yearly growth [closed]

There is $1000$ put in a account and it grows $7\%$ each year starting at $1000$. I'm confused on how to get the calculation and I have several problems just like this.
0
votes
0answers
13 views

How do I show I've rounded to a certain decimal?

I noticed that we could say $\sqrt{0.9}\approx0.9$ if we round to one decimal, or $\approx0.95$ if we round to two decimals. But what is the correct way to show how many decimals we rounded to? ...
3
votes
4answers
128 views

The Definition of the Absolute Value

The Absolute Value can be defined in many ways, but these are the two most common : 1. As a Piecewise Function $$ |x|= \begin{cases} -x&\text{if } x < 0\\ x&\text{if } x\geq 0 \end{cases} ...
0
votes
0answers
9 views

Relationships between values increasing in chunks

I know this must be a common problem, but I can't find the name for it. Suppose I have two quantities of thing $a$ and thing $b$. $a$ can only increase in chunks of $A$, and $b$ can only increase in ...
1
vote
2answers
26 views

Algebra. Multiplying binomials [duplicate]

could you please explain, share link about multiplication binomials issue. For example $(2a + 7)(a - 5)$ As i know, need to do next $2a \times a; 2a \times -5$ $7 \times a; 7 \times -5$ But ...
0
votes
1answer
18 views

Deriving the logarithmic form of inverse hyperbolic cosecant

I am having trouble finding my mistake in deriving the logarithmic form of inverse hyperbolic cosecant function. Here is my work: $$ y= \mathrm{csch} ^{-1} x \implies \mathrm {csch} \ y= x $$ $$ ...
3
votes
1answer
42 views

How to minimize a distance (more than one problem)

Minimizing a Distance When we seek a minimum or maximum value of a function, it is sometimes easier to work with a simpler function instead. (a) Suppose $g(x) = \sqrt{f(x)}$ where $f(x) \geq 0$ ...
0
votes
2answers
58 views

What is the purpose of approximating solutions to equations?

Many times one wants to approximate solutions to equations, particularly when the equation in question has no closed form. For example: $x^5 - x + 1 = 0$. The approximate solution to this, which one ...
8
votes
3answers
143 views

solve for $x$: $\frac{\sin(x)}{x}=\frac{5}{6}$

Is it possible to solve for x the following equation without root finding: $$\frac{\sin(x)}{x}=\frac{5}{6}$$
0
votes
2answers
21 views

Coefficient of product of polynomials.

Suppose we have the polynomial $f(x)$ and another polynomial $g(x)$. How can I find the coefficient of say $x^n$ in the product of the polynomials without actually multiplying. I am not that ...
1
vote
1answer
35 views

Where i am going wrong in solving the inequality?

If $\cos x \left(\cos x+\frac12\right) >0$ then where should $x$ lie in the interval $(0,\pi)$ What I tried When i made two cases i got correct answer but when i used wavy-curve method. I am not ...
0
votes
3answers
22 views

Find the Capacity of the Water Tank?

A water tank has three taps attached, $A,B$ and $D$. $A$ and $B$ fill the water tank completely in $\displaystyle\frac{25}{3}$ minutes and $\displaystyle\frac{25}{2}$ minutes, respectively. ...
0
votes
3answers
44 views

Write $f(x) = x \cdot |x|$ as a piecewise function

$$f(x) = x\cdot|x|$$ I was wondering how this function should look if I expanded it to have the format of a piecewise defined function? I know how to write a piecewise defined function, but the ...
-3
votes
4answers
62 views

Prove that, if $a>c$ and $b>d$, thus $ab>cd$ [closed]

I would like to ask you a question: how could I prove that, if $a>c$ and $b>d$, thus $ab>cd$? Thank you for help. P.s. I forgot to tell you that $a>0, b>0, c>0, d>0.$
1
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5answers
121 views

Show that $n^2+11n+2$ is not divisible by $113^2$ ( n is integer)

Show that $n^2+11n+2$ is not divisible by $113^2$ ( n is integer) It's obvious that if we show $113$ doesn't divide $n^2+11n+2$ we are done...
-1
votes
3answers
71 views

Let $(\sqrt{3} + \sqrt{2})^5 = a\sqrt{3} + b\sqrt{2}, a,b \in \mathbb Z$ Find $a+b$.

Let $$(\sqrt{3} + \sqrt{2})^{\color{red}{5}} = a\sqrt{3} + b\sqrt{2}, a,b \in \mathbb Z$$ Find $a+b$. I don't know if that's supposed to be $\color{red}{5}$ or $\color{red}{3}$. By binomial ...
-2
votes
2answers
38 views

Why does $-(3e^{-x})(1-x)-(3e^{-x}) = (-3e^{-x})(2-x)$?

I am looking at an old exam. The first part of the task wants you to differentiate $$ f(x) = 3xe^{-x}, $$ which is $$ f'(x) = 3e^{-x}(1-x) $$ but then, it wants you to differentitate $f'(x)$. While ...
0
votes
1answer
50 views

Is it possible that $2-2\cos^2x$ is equivalent to $1-(2\cos^2x-1)$

There is this exercise and for the first time in my life, I don't want to go to see the solution. Instead, I'm more asking of a tiny help to see if I'm right in my conclusion Kids are getting ...
0
votes
1answer
85 views

About the solution to “Finding the range of $y= \sqrt x + \sqrt{3-x}”$

I was reading the solution of "Find the range of $y = \sqrt{x} + \sqrt{3 -x}$" and I had some points of confusion about the solution posted in the OP. I wrote here my interpretation of the solution. ...
1
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0answers
28 views

About the identity $(\vec{R}+\vec{a})\cdot(\vec{R}-\vec{a})=R^2-a^2$

From basic vector algebra, we know that $(\vec{R}+\vec{a})\cdot(\vec{R}-\vec{a})= R^2-\vec{R}\cdot\vec{a}+\vec{a}\cdot\vec{R}-a^2=R^2-a^2$, where the result is independent of the angle between ...
3
votes
3answers
78 views

Solution to $\sqrt{x-2} = 3- 2\sqrt{ x}$

The above question is from Serge Lang's basic mathematics. The question asks if there are any values of x which satisfy the above equation. Serge Lang's answer key states that there is no solution. ...
0
votes
1answer
67 views

can someone explain this simplification for me?? [closed]

Can someone tell me how $$−56−173\,\ln(11)+366\,\ln(13)−\left(\frac{105}2+20\,\ln(2)+366\,\ln(3)\right)$$ simplifies to $$\frac{-217}2−20\,\ln(2)−173\,\ln(11)+732\,{\rm arctanh}\left(\frac58\right)?$$ ...
0
votes
2answers
61 views

Hello i am confused with this quadratic question.

Why is it that when we look at equation for example $(x-2)^2+5$ and the question states "state the minimum point" the minimum point is $5$. I get that the coordinates of minimum point is $(2,5)$ but ...
0
votes
1answer
26 views

solve system inequalities derived from a function

I have this system of inequalities $$ \begin{cases} y^2-3 \geq 0\\ 16y^4-96y^2 \geq 0 \end{cases} $$ the solution for the first inequality is $y\leq -\sqrt{3}$ or $y\geq \sqrt{3}$ and the solution ...
1
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2answers
26 views

find the result of $16y^4-96y^2 \leq 0$

$16y^4-96y^2 \leq 0$ I have not clear the last step of this inequality to get the result $-\sqrt{6} \leq y \leq \sqrt{6}$. Change $t=y^2$ and $t^2 = y^4$ $16t^2-96t \leq 0$ I compute the ...
2
votes
1answer
36 views

When you divide the polynomial $A(x)$ by $(x-1)(x+2)$, what remainder will you end up with?

When you divide the polynomial $A(x)$ by $x-1$, you get a remainder of $10$. When you divide $A(x)$ by $x+2$ you get remainder $0$. When you divide $A(x)$ by $(x-1)(x+2)$ what remainder will you end ...
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votes
2answers
39 views

How to rearrange this equation [closed]

Can you please help me to solve the following equation for $s$? $$[w-s]^{-0.5}=[(1+r)s]^{0.5}$$
0
votes
2answers
37 views

Ascertaining a from logarithmic equations

I've just been accepted on to a PHD program at Melbourne, studying chemical engineering. I'm working my way through some standard pure and further mathematics books just to get the concepts into my ...
1
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1answer
80 views

Solve the following equation for $x$

$(1)$ Solve for $x: \sum_{i=1}^p\frac{1}{(x-x_i)^2}=\sum_{i=p+1}^n\frac{1}{(x-x_i)^2}$ where $x_i$'s are fixed real numbers. Note: It's a generalization of the usual problems like solving ...
2
votes
3answers
46 views

For real numbers $a,b,c$ calculate the value of: $\frac{c}{a+b}+\frac{a}{b+c}+\frac{b}{c+a}$ if we have…

For real numbers $a,b,c$ we have: $a+b+c=11$ and $\frac1{a+b}+\frac1{b+c}+\frac1{c+a}=\frac{13}{17}$, calculate the value of: $\frac{c}{a+b}+\frac{a}{b+c}+\frac{b}{c+a}$ I think we should use a trick ...
2
votes
1answer
35 views

$\lfloor\frac{18}{35}\rfloor+\lfloor \frac{18(2)}{35}\rfloor+\lfloor \frac{18(3)}{35}\rfloor+…+\lfloor \frac{18(34)}{35}\rfloor$

Value of the expression $$\bigg\lfloor \frac{18}{35}\bigg\rfloor+\bigg\lfloor \frac{18(2)}{35}\bigg\rfloor+\bigg\lfloor \frac{18(3)}{35}\bigg\rfloor+....+\bigg\lfloor ...
-1
votes
2answers
35 views

$100 + [110/(1+r)] = [1/ (1+r)] + [(232 /(1+r)^2 ]$

Need to learn how to solve this: $100 + \frac{110}{1 + r} = \frac{1}{1 + r} + \frac{232}{(1 + r)^{2}}$. Checked this site got to the 3rd line and am completely lost. Can someone help me solve for r ...
0
votes
2answers
36 views

Negative Ratio----Math

I have always studied the ratios of the type $a:b$, where a and b are natural numbers and I can also understand the ratio where both a and b are negative. But what I don't get is that when a is ...
2
votes
2answers
79 views

Number of polynomials which are divisible by $x+1$

Let $a,b,c,d$ be four integers (not necessarily distinct) in the set ${1,2,3,4,5}$ . The number of polynomials $f(x)=x^4+ax^3+bx^2+cx+d$ which are divisible by $x+1$ are: $(A)$ Between 55 and 65 ...
0
votes
0answers
40 views

Suppose ($ a_{1} ,…, a_{n} $) is an arithmetic sequence.

Suppose ($ a_{1} ,..., a_{n} $) is an arithmetic sequence. Then: $$ \frac{1}{ a_{1} } +....+ \frac{1}{a_{n} }=؟ $$ Is it possible to obtain high regard.
1
vote
1answer
62 views

Calculating $\sqrt[3]{\sqrt 5 +2}-\sqrt[3]{\sqrt 5 -2}$

We want only the real 3rd root. By calculation, $[\sqrt[3]{\sqrt 5 +2}-\sqrt[3]{\sqrt 5 -2}]^3= 4-3[\sqrt[3]{\sqrt 5 +2}-\sqrt[3]{\sqrt 5 -2}]$ Therefore, the answer is a root of $t^3=4-3t$ , which ...