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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1answer
30 views

For which points ($P$) on the $x$-axis, $\angle APB= 90^\circ $?

Let $A =(-2, 3, -2)$ and $B =(-6, -1, 1)$. For which points ($P$) on the $x$-axis, $\angle APB= 90^\circ $? I figured, since $P$ is supposed to be on the $x$-axis, the $y$ and $z$ coordinates ...
5
votes
4answers
528 views

Evaluate $\lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^m}{n^{m+1}}$

By considering: $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^1}{n^{2}} = \frac 1 2$$ $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^2}{n^{3}} = \frac 1 3$$ $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^3}{n^{4}} = ...
2
votes
1answer
104 views

Logarithms - Find the solution of $\ln(x^2+1) = \ln(x) + 2$, how to isolate $x$ in a meaningful way?

When solving $\ln(x^2+1) = \ln(x) + 2$ I'm getting stuck at $e^2 = \dfrac{x^2+1}{x}$ How do I isolate $x$?
1
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0answers
153 views

Solve $x^x=2x$ for $x$, s.t. $x\in\mathbb{C}$

Solve $x^x=2x$ for $x$, such that $x\in\mathbb{C}$. I'm not sure if the question has a closed form solution.
1
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1answer
59 views

Quadratic Equation - What am I doing wrong?

For which value of $c$ does the quadratic equation $5x^2 - 6x + c = 0$ have exactly one solution in terms of $x$? The solution is supposed to be $c = 1.8$, but I only ever get $c = 1$ ...
1
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2answers
422 views

Vectors - Find the vector $\vec{c}$ which is orthogonal to $\vec{a}$ and $\vec{b}$ and whose first component is 1

Find the vector $\vec{c}$ which is orthogonal to $\vec{a}$ and $\vec{b}$ and whose first component is 1 $\vec{a}$ = (0 / 4 / -2), start point: $P$ (2 / -3 / 5), end point: $A$ (2 / 1 / 3) ...
2
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2answers
46 views

How to determine $\mathbb{E}$ for which a maximum defined function is a bijection?

Assume that $f\colon\mathbb{E}\subset\mathbb{R}\rightarrow\mathbb{R}$, $f(x)=\max\lbrace2x-5,x-2\rbrace$. Determine $\mathbb{E}$ for which $f$ is a bijection. I was thinking it is ...
1
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3answers
93 views

How to determine m for which $f(x)$ is surjective?

Assume that $m \in \mathbb{R}$, $f: \mathbb{R} \rightarrow \mathbb{R}$, and \[ f(x) = \begin{cases} x+m,&\qquad x \le 3 \\ mx+2,&\qquad x > 3 \end{cases} \] I need to find $m$ for which ...
0
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2answers
39 views

Determine $x$ for which the function has the maximum value?

We have $f:R\rightarrow R$ $f(x)=\frac{3x-1}{3x^2+1}$ Determine $x$ for which the function has the maximum value. How can I determine the maximum of this function? Thank you very much in advance! ...
1
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2answers
68 views

How to find the minimum of the expression $E$?

Find the minimum of the expression $E=a^2+2b^2-3a+3b $ $ a,b\in R$ Is there a formula I can apply? How do I find the minumum? Thank you very much in advance!
0
votes
1answer
142 views

Teaspoons of Vinegar to Gallon.

I am trying to regulate my ponds PH level, but I'm having trouble on the teaspoon conversion. Vinegar is supposed to lower my PH level so I need to put vinegar in my water, but how much? 1/4 Vinegar ...
3
votes
2answers
160 views

How to solve this equation, when the unknown variable just disappears?

This problem $\sqrt{1-x^2} + \sqrt{3+x^2} = 2$ has the solution $x = 1$ and $x = -1$. However, I always get stuck like this: $1-x^2 + 3+x^2 = 4$ $4 = 4$ How do I isolate that darn ...
1
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2answers
203 views

Equality with absolute values - Is this a valid solution?

For this problem $|2 - |x-2|| = 2$ I've found the values $x = -2$ and $x = 2$. However, an third solution was presented to me, which I can't seem to find by myself: $x = -6$. Is this solution ...
1
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4answers
1k views

How to factor the quadratic polynomial $2x^2-5xy-y^2$?

How do I factor this polynomial: $2x^2-5xy-y^2$ ?
2
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2answers
65 views

What to do with these e's to solve for x?

I started here: $({14x-21})e^{x^{^{2}-3x+6}}$ took the derivative to end up here: $0=14xe^{x^{2}-3x+6}-21e^{x^{2}-3x+6}$ and now I must solve for x. Or, find where the tangent line's slope is ...
3
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2answers
2k views

Find a solution for f(x)=g(x) having 2 given values.

If $f,g\colon\mathbb{R}\to\mathbb{R}$ have the following properties: i) $f(g(x))=x^2-3x+4 $ ii) $g(f(2))=2 $ Determine at least a Real solution for the equation $f(x)=g(x)$ Choose the right ...
1
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2answers
230 views

Vectors - Define a vector of length 1 orthogonal to $\vec{v} = (-4 \qquad 3)^t$

Define a vector of length 1 orthogonal to $\vec{v} = (-4 \qquad 3)^t$ I'm looking for the solution in terms of $\vec{a} = \binom{x}{y}$. How do I go about it? I'm familiar with addition, ...
1
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2answers
483 views

Rudin's “Principles of Mathematical Analysis” Example 1.1

This probably involves some very easy algebra, but I am stuck and would appreciate some help. Walter Rudin's Example 1.1 on page 2 of Principles of Mathematical Analysis includes the following ...
4
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3answers
7k views

Finding out whether two line (segments) intersect

I need to know whether or not two line segments intersect. I thought the formula for that is y = mx + b but I don't think that will work for what I need, at least I think I need to first know whether ...
3
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3answers
73 views

Help needed with whole numbers

I need help on this. I tried simplifying it, putting some numbers on it but there has to be a simple way to figure this one out. Thanks in advance. a, b and c are whole numbers. $$a > b > ...
0
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1answer
344 views

Finding set of possible values of a constant in a function definition.

It is given that the function $f(x) = \sqrt{(mx + 7)} - 4$ $x \geq -\frac{7}{m}$ and its inverse do not intersect, and that neither intersect the line $y = x$. How could one determine the set of ...
1
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1answer
88 views

The Law of Sines and a second forces magnitude

How do you solve for the second forces magnitude? The 17' and 30' threw me off.
0
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2answers
6k views

How do you find a vector in the form <a,b> when only the angle and magnitude are given?

How do you find a vector in the form when only the angle and magnitude are given? Here is an example where an angle of 80 degrees is given along with a magnitude of 3.
1
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3answers
223 views

Finding any point on a line if you know the slope and $y$-intercept.

I am wondering if there is a way to determine where a point is if I only know the slope and $y$-intercept. For example, say I am told that the line has a slope of $3$ and a $y$-intercept of $-3$. ...
10
votes
3answers
263 views

Does the fact that $\sum_{n=1}^\infty 1/2^n$ converges to $1$ mean that it equals $1$?

I have a clueless friend who believes that $$ \sum_{n=1}^\infty \frac{1}{2^n} $$ doesn't equal $1$ in the 'normal arithmetical sense'. He doesn't believe that this series flat out equals $$1$$ Is ...
2
votes
2answers
277 views

Parabola Question: How to derive an appropriate equation based on specific criteria?

I've seen similar questions, but none worded quite like what I'm asking here. I'm not a math major, so I may have follow-up questions if answers befuddle me. Scenario: I'm trying to emulate a ...
6
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5answers
726 views

How to solve an equation involving the floor of a number?

I'm looking for a solving procedure for this type of exercises. If $[x]$ represents the floor of $x$, solve the equation: $$\left[\frac{6x+5}8\right]=\frac{15x-7}5$$ Choose the correct ...
6
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3answers
519 views

Puzzle: A coin rolls without slipping around another coin

If a coin rolls without slipping around another coin of the same or different size, how many times will it rotate while making one revolution? The proof given is like this: Cut the curve ...
2
votes
2answers
136 views

How to solve an equation involving the whole part of a number?

If $[x]$ represents the floor of $x$, solve the equation: $$\left[\frac{x+1}3\right]=\frac{x-1}2$$ Choose the correct answer: a) $(2,7)\cup(9,15)$ b) ...
0
votes
1answer
180 views

How long does it take to reach deimos?

If a planet is 93,341,952 km away and you use a space shuttle, which travels at a speed of 28,000 km/hr to get there, how many days would it take to reach your destination?
0
votes
1answer
573 views

How to find values of constants in the definition of a function given that function's stationary points?

The function is defined as $f : x \mapsto a + b \csc x$ $x \in \mathbb{R}$ $0 < x < 2\pi$ $x \neq \pi$ The stationary points of the graph $y = f(x)$ are $(\frac{\pi}{2}, ...
0
votes
2answers
69 views

Equation $x = y + k$, where $x$ is known.

In an equation x = y + k Where: "$x$" is a known variable, "$y$" is an unknown variable and "$k$" is an unknown constant. How could "$y$" and "$k$" values be calculated? Is it possible to get exact ...
1
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3answers
70 views

How to Calculate the function of an interval?

I need to calculate $f((1,4])$ for the function $$f(x)=x^2-4x+3.$$ The answers I can choose from are: a) [0,3] b) [-1,0) c) (0,3] d) [-1,3] e) (-1,0) f) (0,3) Can someone guide me? It may be ...
0
votes
1answer
638 views

How to write an equivalent statement?

What is a good equivalent statement to : If the garden is not watered every day, the flowers wilt. Using the Morgan's Laws
2
votes
3answers
151 views

Question on word Problem, Algebra Question.

I'm not sure if these types of questions are accepted here or not (I'm very sorry if it's not), but it would be great if anyone could explain me this. Question: Using his bike, Daniel can ...
11
votes
5answers
1k views

Why aren't the graphs of $\sin(\arcsin x)$ and $\arcsin(\sin x)$ the same?

(source for above graph) (source for above graph) Both functions simplify to x, but why aren't the graphs the same?
1
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2answers
54 views

Constructing an average equation - Standardized Test

A question in the standardized test is: The average of $n$ numbers is $a$. If $x$ is subtracted from each number the average will be a) $(ax)/n\quad$ b) $(an)/x\quad$ c) $an-x\quad$ d) ...
4
votes
2answers
479 views

Russian Peasant Method for multiplication

What exactly happens with the remainder in this algorithm? I don't understand why it is "dropped". Example: $$\begin{array}{c} \text{Half}&&\text{Double}&\text{Remainder}\\ \hline ...
0
votes
3answers
144 views

How to get $x$ by itself in this equation?

How do I get $x$ by itself in this equation? This is work related not school related. Thanks! $$x^2 +(tx)^2 = 1.333^2$$
0
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1answer
49 views

Which of the following cannot possibly be the average - Standardized Test

I came across a question while preparing for a standardized test that is kind of confusing. Hope you guys could help me out here. The question is: Three men have weights that range from 110 to 135 ...
0
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1answer
189 views

Calculating average of word problem

I was preparing for a standardized test and came across the following problem: During a closeout sale 30 suits are sold at \$60 each.The price is the reduced to \$50 each and 20 suits are sold. At ...
1
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1answer
777 views

whole number equation

2x + 5y = 90 x and y are whole numbers. how many (x,y) can this equation get? I thought it like this; 2x = 90-5y x = (90-5y)/2 x is a whole number so must be divisible by 2. So I thought--- by ...
30
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6answers
2k views

What is the larger of the two numbers?

What is the larger of the two numbers? $$\sqrt{2}^{\sqrt{3}} \mbox{ or } \sqrt{3}^{\sqrt{2}}\, \, \; ?$$ I solved this, and I think that is an interesting elementary problem. I want different points ...
2
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2answers
261 views

Average of all numbers from $1$ to $100$ that end in $2$

Can anyone please tell me what would be the fastest way to find the average of all numbers from $1$ to $100$ that end in $2$?
0
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2answers
53 views

Algebra system of equation?

How would I solve this system of equation? $$\begin{align*}&.05(w+2000)=.03(y+3000)\\ &4w=\frac{y}2+500 \end{align*}$$ I end up setting them up like this but I am not sure if it is correct. ...
0
votes
1answer
90 views

Weighted Values

I hope to be able to get some assistance here, I tried reading through some of the posts, however, they are far above my math abilities..so I apologize if this is a re-post or this is a basic ...
3
votes
5answers
1k views

Why are zeros/roots (real) solutions to an equation of an n-degree polynomial?

I can't really put a proper title on this one, but I seem to be missing one crucial point. Why do roots of a function like $f(x) = ax^2 + bx + c$ provide the solutions when $f(x) = 0$. What does that ...
3
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3answers
144 views

What equation intersects only once with $f(x)=\sqrt{1-(x-2)^2}$

Being $f(x)=\sqrt{1-(x-2)^2}$ I have to know what linear equation only touches the circle once(only one intersection), and passes by $P(0,0)$. So the linear equation must be $y=mx$ because $n=0$. I ...
0
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1answer
108 views

What would be the prize value in today's terms

If a $1 million dollar competitions came out roughly in 1985. Would would the equivalent prize money be today? Many thanks!
0
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5answers
110 views

Solving a system of equations

Could anyone help me to solve the following system of equations: $$\begin{align*} x-y&=17 \\ \frac{4}{3} x+ \frac {3}{2} y &= 0 \end{align*}$$ How should I go about solving this, I am stuck. ...