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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1
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6answers
57 views

How to determine the derivative of $ f $ at $ x=2$ by looking at the graph only?

How to determine the derivative of $ f $ at $ x=2$ (i.e., $ f^\prime(2) $) by looking at the graph only ? I am well aware of the theory of the derivative and how to compute it. But how can I ...
1
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4answers
808 views

Solved to be 7 after arithmetic

I recently made a blunder while trying to explain a question asked to me in an interview, The question was Think of $X$ Add $X$ to itself ($X+X = y$) Times the result by $3$ ($y\times 3 = z$) ...
2
votes
3answers
71 views

Finding all natural $n$ such that $2^n+2^{2n} +2^{3n}$ has only $2$ prime factors.

Find all natural $n$ such that $2^n+2^{2n} +2^{3n}$ has only $2$ prime factors. I've tried checking the first 6-7 $n$'s on wolframalpha, but I don't see any patterns for even nor odd $n$'s. At first ...
1
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2answers
34 views

Proof by induction of sum

My question is from Apostol's Vol. 1 One-variable calculus with introduction to linear algebra textbook. Page 40. Exercise 10. Prove by induction, that for $n\ge1$ we have ...
3
votes
2answers
62 views

If $f(x) = \sqrt{x}$, what is the domain of $f^4(x)$?

I am unclear if I should consider the function's domain before or after raising it to the power. My textbook gives the following definition of raising a function to a power: By $f^n$, we mean the ...
0
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2answers
34 views

How many milliliters to fill cone

A right circular cone has a depth of 103 mm and a top diameter of 82.4 mm. The cone contains water to a depth of 30.0 mm. How many more milliliters of liquid need to be added in order to fill the ...
0
votes
2answers
24 views

Completely factor a polynomial using the rational root theorem and synthetic division

I am currently seriously confused. My problem, as stated above, is about completely factoring a polynomial. My question is, once you get your possible factors, how do you then simplify it down? Ill ...
0
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1answer
25 views

Linear equation with 3 equal signs

Here is my linear equation: Solve for p: $\frac{5}{6} = \frac{n}{72} = \frac{m+n}{84} = \frac{p-m}{120}$ How am I supposed to solve for this 1 variable when there are multiple equal signs and 3 ...
-2
votes
1answer
15 views

linear word problem 2 [closed]

Two ships steam towards each other from positions 60 miles apart, one at 12 knots and one at 9 knots. After how long will they be 18 miles apart? Please show the working out I have the answer.
1
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6answers
73 views

Solve $\sqrt{3x}+\sqrt{2x}=17$

This is what I did: $$\sqrt{3x}+\sqrt{2x}=17$$ $$\implies\sqrt{3x}+\sqrt{2x}=17$$ $$\implies\sqrt{3}\sqrt{x}+\sqrt{2}\sqrt{x}=17$$ $$\implies\sqrt{x}(\sqrt{3}+\sqrt{2})=17$$ $$\implies ...
0
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4answers
62 views

How to solve the inequality $x^4<4x^2$? [closed]

How do you solve the below inequality? $x^4<4x^2$ My answer is (-2, 2)
0
votes
2answers
16 views

Two ships leaving a port at different times and different speeds. When do they meet?

Can someone please show me the working out to this word problem I have the answer but have no clue how to do the working out. At noon ship A leaves port steaming at 8 knots 2 hours later ship B ...
-1
votes
1answer
16 views

divide clock into halfs

John has special clocks one hands do 1 turn per minute, second do 1 turn per 3 minutes and third do 1 turn per 15 minutes. how many times and when the first divide clockface into three equal parts in ...
0
votes
5answers
56 views

Prove that for all positive integers $x$, $\left\lfloor \frac{x^2 +2x + 2}{4}\right\rfloor =\left\lfloor \frac{x^2 + 2x + 1}{4}\right\rfloor$.

Title says it all, basically. I believe it to be true that $$\left\lfloor \dfrac{x^2 + 2x + 2}{4} \right\rfloor=\left\lfloor \dfrac{x^2 + 2x + 1}{4} \right\rfloor$$ for all positive integers $x$. I ...
1
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1answer
25 views

Trouble understanding factorial algebra

I am having trouble understanding some of the algebraic concepts used here. In fact, the entire thing to me makes sense, except for the second red line. I don't understand how the diagonal swap ...
0
votes
0answers
22 views

related rates of a sphere

So this is my problem and I can't figure out for the life of me how to get the two answers that have X's next to them. If someone can just give me an explanation I would really appreciate it. Thanks.
0
votes
1answer
33 views

Solving Polynomial Equations and Inequalities

The distance, in km, of a ship from its harbour is modeled by the function $d(t)= -3t^3 + 3t^2 + 18t$ where $t$ is the time elapsed in hours since departure from the harbour. a) When does ...
0
votes
3answers
51 views

Show that $\, 0 \leq \left \lfloor{\frac{2a}{b}}\right \rfloor - 2 \left \lfloor{\frac{a}{b}}\right \rfloor \leq 1 $

How can I prove that, for $a,b \in \mathbb{Z}$ we have $$ 0 \leq \left \lfloor{\frac{2a}{b}}\right \rfloor - 2 \left \lfloor{\frac{a}{b}}\right \rfloor \leq 1 \, ? $$ Here, $\left \lfloor\,\right ...
0
votes
0answers
20 views

Coordinates of the center of the sphere with given equation

I´m trying to solve this question, which asks for the coordinates of the center of the sphere $$ 4(x^2+(l-y)^2+z^2) = x^2+y^2+z^2 $$ I know the answer should be $(0, 4l/3,0)$. There's a picture of ...
6
votes
3answers
93 views

Why is Division harder than Multiplication?

Both conceptually and computationally it feels easier to see that: $ 6 \cdot 3.7 = 22.2$ than it is to see that $ 22.2 \div 6 = 3.7 $. Thoughts about the roots of this asymmetry? An analogous ...
2
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3answers
49 views

Positive values of $x$ that satisfy the inequality $\frac{1}{x}-\frac{1}{x-1}>\frac{1}{x-2}$

Determine the set of positive values of $x$ that satisfy the inequality $$\frac{1}{x}-\frac{1}{x-1}>\frac{1}{x-2}.$$ My attempt: \begin{align} \frac{-1}{x(x-1)} & >\frac{1}{(x-2)} \\[0.1in] ...
1
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1answer
21 views

Mortgage payment calculation without annuty.

I have been asked the following problem by a student of mine and there is a specific method that he requested. A mortgage of $\$450,000$ is loaned for a monthly payment for $30$ years with nominal ...
0
votes
2answers
29 views

Sum of roots of an equation $\sqrt{x-1}+\sqrt{2x-1}=x$

Find the sum of the roots of the equation $\sqrt{x-1}+\sqrt{2x-1}=x$ My attempt: Squaring the equation: $(x-1)+(2x-1) +2\sqrt{(x-1)(2x-1)}=x^2$ $\implies x^2-3x+2=2\sqrt{(x-1)(2x-1)} $ $\implies ...
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votes
5answers
85 views

Sigma Notation Involving Even and Odd Numbers [closed]

How would you write this sequence in sigma notation? $ \frac {1}{2} + \frac {3}{4} + \frac {5}{6} + \frac {7}{8} + ... \frac {49}{50}$
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0answers
18 views

Algebraic problem

Let $T:R\to R$ be given as $Tx=x^3-1$ and $d(x,y)=\left|x-y\right|$. Then is it possible to find $\lambda\in (0,1)$ and $L\geq0$ such that $$d(Tx,Ty)\leq \lambda M(x,y)+LN(x,y)$$ for all $x,y\in R$, ...
2
votes
0answers
25 views

Solve $x/(4x^2+1) = \tan(6x)$ for $x$

$$ \frac{x}{4x^2+1} = \tan(6x) $$ Can this equation be solved for $x$ algebraically and can I get exact answer for this question? Or do I have to approximate it?
0
votes
5answers
58 views

How do I solve this inequality $\frac{c-1}{\sqrt{c}}<2$?

How do I solve this inequality $\frac{c-1}{\sqrt{c}}<2$? Wolfram Alpha says that it is $0<c<3+2\sqrt{2}$ but my brain is just not letting me see how to get there.
1
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3answers
38 views

Needing the answer and working out to $11(y+2)=55$ [closed]

what is the answer (showing working out) to $11(y+2)= 55$?. it is for a student who is really stuck on this topic and needs a helping hand on this question in particular.
0
votes
3answers
31 views

Remainder Theorem, solve for K

For what values of $k$ does the function $f(x) = x^3 + 6x^2 + kx – 4$ give the same remainder when divided by $(x-1)$ and $(x + 2)$?
0
votes
2answers
56 views

Solving a system of two cubic equations

I'm trying to solve a system of two cubic equations with two variables x and y. The original problem was to solve the equation $z^3=-4i \overline{z}$. I know how to solve it using polar form. Now I ...
0
votes
3answers
43 views

Find numerical value without using a calculator [closed]

4^n+4^n+1/4^n-2 Should I leave it as 4 or change it to 2^2. I'm unable to get rid off all variables. Please help.
0
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2answers
37 views

Gelfand trigonometry question

If we start with a lemma that states that when $ a^2+b^2=1$ there exists an angle $ \theta $ such that $ a=\cos\theta $ and $ b=\sin\theta$ Suppose that $\alpha$ is some angle if ...
1
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3answers
35 views

Using IVT prove that a polynomial of even degree has atleast two real roots if $a_n a_0 \lt 0$

$P(x) = a_n x^{n} + a_{n-1}x^{n-1} + \cdots +a_1x+ a_0$ $n = 2k$ Show that $P(x)$ has at least two real roots if $a_na_0 \lt 0$ I think I need to find some interval of length $|N|$ in which the ...
6
votes
2answers
161 views

how find all the zeroes of the polynomial

Find all the zeroes of the polynomial $f(x) = 2x^7 - 17x^6 -45x^5 +390x^4 + 28x^3 + 1832x^2 +960x$ this is my try $f(x)= 2x^7 - 16x^6-x^6-42x^5-3x^5+390x^4 + 28x^3 + 1820x^2+12x^2 +960x$ ...
1
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1answer
24 views

Proving these are equal

We have the following equality (in physics) $$c^2t^2-x^2 = c^2t'^2-x'^2$$ where: $t' = \gamma (t- \dfrac{vx}{c^2})$ $x' = \gamma(x-vt)$ $\gamma = \dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}$ My ...
0
votes
4answers
36 views

Can someone walk me through how this expression simplifies to y/x?

I am just wondering how this equation comes to be: it is from an economics problem involving marginal utilities. I have my two variables, $x$ and $y$. Intuitively, how does $$\frac{0.5\times ...
3
votes
6answers
96 views

Proving AM-GM for the special case $n=3$

I know the AM-GM inequality and its proof which is relatively complex, though the case for $n=2$ is quite simple. However, I don't know of any special easier proof for the case $n=3$, specifically: ...
0
votes
1answer
14 views

Switching a parametric form of a plane to a non-parametric form

So I have a parameterized representation of a plane: $$ (1,0,2) + t(-1,0,1) + u(0,1,3) : t ϵ R, u ϵ R $$ I want to swicth it to the equation $ax + by + cz = d$ in order to find the intersection line ...
0
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0answers
25 views

writing sum as a product and vice versa.

$\Pi = k$ from k = 1 to n Can you write this in form of sigma? So that you can evaluate it as a sum? Also, are there any shorthand formula to evaluate a product like there are for summations? ...
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0answers
39 views

Proper use of implication and equivalence

I think I have a pretty good understanding of implication and equivalence (I also found this question), but there are some things I am unsure about. First of all, in maths class in high school, when ...
1
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1answer
34 views

How to obtain this geometric progression

How do I obtain this from the formula of the geometric progression (which I 'only' know as $1+q+q^2+...+q^{n-1} = \frac{1-q^n}{1-q}$)? $$\frac{x_1^p-x^p}{x_1^q-x^q} = ...
1
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1answer
25 views

Finding the intersection points of 2 circles with a given equation, center and radius

Determine the points of intersection of the circle with equation $x^2 + y^2 - 12x - 4y + 30 = 0$ and the circle with center $(3,5)$ and radius $4$. My attempt : With given center and radius of the ...
0
votes
3answers
25 views

What is the ratio between the speed of the boat and speed of the water current?

A boat running upstream takes 8 hours 48 minutes to cover a certain distance, while it takes 4 hours to cover the same distance running downstream. What is the ratio between the speed of the boat and ...
2
votes
2answers
39 views

Roots of simultaneous power sum equations (numerically or otherwise)

I'm a physicist, and I've come across a problem in my research where I need to solve a set of equations looking like (e.g. in 3D) $$r_1 + r_2 + r_3 = k_1$$ $$r_1^2 + r_2^2 + r_3^2 = k_2$$ $$r_1^3 + ...
0
votes
1answer
67 views

How do I solve two equations in two unknowns?

In my free time, I've been challenging my mind with IQ problems. I found this question: Jim has as many sisters as he has brothers, but his sister has twice as many brothers as she has ...
1
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1answer
33 views

Simplify this fraction with square roots; application to arctangent equation.

I need your help. I don't know how to simplify: $\frac{-1+\sqrt{3}+\sqrt{4+2\sqrt{3}}}{2\sqrt{3}} $ and $\frac{-1+\sqrt{3}-\sqrt{4+2\sqrt{3}}}{2\sqrt{3}}$ Thank you in advance. I found $1$ and ...
0
votes
1answer
32 views

Simplifying with wxMaxima

I have this expression in wxMaxima: ...
2
votes
4answers
41 views

Show that $2\cos(x)$ is equal to $2\cos(2x)\sec(x)+\sec(x)\tan(x)\sin(2x)$

This is from the derivative of $\dfrac{\sin(2x)}{\cos x}$ I tried to solve it and arrived with factoring the $\sec(x)$ but I still can't get it to $2\cos(x)$. Could you help me out, please? Thanks
1
vote
1answer
39 views

can I apply Cauchy-Schwarz to this problem?

The question says, if $n \geq 3, k \in \mathbb{R}$, what is the smallest $k$ such that $\forall a_i\in \mathbb{R}, \sum_{i=1}^n a_i^3\leq k\sqrt{\sum_{i=1}^na_i^6}$. Tried: apply Cauchy-Schwarz ...
2
votes
0answers
60 views

Is it possible to solve $k = \frac{x}{\ln(x)}$ for $x$?

Is it possible to solve $k = \frac{x}{\ln(x)}$ for $x$? My suspicion after a fruitless hour of manipulation is that it is not.