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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0
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1answer
23 views

“$\frac{dy}{dx}= -0.11$. It is easy to see that except at one particular point, dy will be of a different size from dx”.What point?

From "Calculus made easy" by Thompson. A ladder of fixed length is against a horizontal wall.Height of ladder is y,distance of base of ladder from wall is x.For positive increment to x, there will be ...
1
vote
2answers
41 views

How to solve this 3rd degree polynomial?

I looked up the factoring method, but I think this one is calculated using a calculator. With a basic calculator, how do I set this up? $16x^3 - 18x^2 - 2x - 1 = 0$ I factored it to become... ...
0
votes
0answers
63 views

Numbers can be written as $A^B =B^A$ [duplicate]

Of course 16 is the only integer that can be written as $A^B = B^A$ $(4^2 = 2^4)$, where $A$ and $B$ are two distinct integers. My question is: Are there infinitely many (real) numbers that can be ...
0
votes
1answer
16 views

How many different groups of 12 are being in an average of 1.7 groups?

If I have $10$ million people, how do I calculate how many different groups of $12$ with each person being in an average of $1.7$ groups?
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votes
0answers
33 views

Find the range of the function : $\frac{1}{\pi}(\sin^{-1}x+\tan^{-1}x) + \frac{x+1}{x^2+2x+5}$ [duplicate]

Problem : Find the range of the function : $\frac{1}{\pi}(\sin^{-1}x+\tan^{-1}x) + \frac{x+1}{x^2+2x+5}$ My approach : Let $g(x) = (\sin^{-1}x+\tan^{-1}x)$ and $h(x)=\frac{x+1}{x^2+2x+5}$ and ...
0
votes
2answers
24 views

Prove that $-x^2 \leq x^n \leq x^2$ for $-1<x<1, n\in \mathbb N, n \geq3$

Prove that $-x^2 \leq x^n \leq x^2$ for $-1<x<1, n\in \mathbb N, n \geq3$ I have no idea how to do this, I don't even know how to begin. Please help!
11
votes
3answers
59 views

Prove that $4x^2-8xy+5y^2\geq0$ - is this a valid proof?

I need to prove that $4x^2-8xy+5y^2\geq0$ holds for every real numbers $x, y$. First I start with another inequality, i.e. $4x^2-8xy+4y^2\geq0$, which clearly holds as it can be factorized into ...
-1
votes
1answer
16 views

Algebraic isolation of integer with power function attached

$$\frac{T_3}{T_2}=\left(\frac{P_3}{P_2}\right)^{\frac{R}{C_p}}$$ I need to find $P_3$, I'm not sure how to deal with the power function? Any help appreciated!
0
votes
1answer
54 views

Prove the inequality, $\root3\of4\sin^2(x/2)<3(\sin x+1-x)^{2/3}$

Prove that $$\left(\sin^2{\frac{x}{2}}\right) \cdot \frac{\sqrt[3]{4}}{3} \cdot \frac{1}{{(\sin x + 1 - x})^{\frac{2}{3}}} <1$$
3
votes
2answers
48 views

Is $\sin^4 x-\cos^4 x = \cos2x$ or is it $-\cos2x=\cos2x$?

A test question I received and got wrong stated that $$\sin^4x-\cos^4x = \cos2x$$ After solving the equation from lower powers of tragicomic functions it came out ...
3
votes
2answers
40 views

Cyclic Equation. Prove that: $\small\frac { a^2(b-c)^3 + b^2(c-a)^3 + c^2(a-b)^3 }{ (a-b)(b-c)(c-a) } = ab + bc + ca$?

This is how far I got without using polynomial division: \begin{align} \tiny \frac { a^{ 2 }(b-c)^{ 3 }+b^{ 2 }(c-a)^{ 3 }+c^{ 2 }(a-b)^{ 3 } }{ (a-b)(b-c)(c-a) } &\tiny=\frac { { a }^{ 2 }\{ { b ...
0
votes
1answer
34 views

How to express a fraction into product of two terms

I want to express the following fraction: $$x^2\left(\frac{y-1}{y+1}\right)^2+x\left(\frac{y-1}{y+1}\right) + 1$$ into $$K(y-c)(y-d),$$ where $c$ and $d$ are zeroes of the above function. How can I ...
-3
votes
3answers
66 views

not sure how to do this [on hold]

Julie is required to pay a 2 percent tax on all income over 3,000. She also has to pay 2.5 percent on all income over 20,000. She earned more than 20,000 and paid 992.50 what was her total income
0
votes
1answer
34 views

how to find area of segment of only given the radius? [on hold]

I have a thought question it will be tough to explain but here we go. My teacher gave us three circles which each have a radius of 5in. Inside the circle there's a triangle with no angles? Inside the ...
5
votes
2answers
91 views

What wolfram does to factor $x^6+x^2+2$?

I am learning polynomials and I am trying to understand what wolfram did to obtain $$(x^2+1)(x^4-x^2+2)$$ from $$x^6+x^2+2$$ It does not show me the step-by-step option in this case and I got ...
1
vote
1answer
26 views

Precalculus algebra: Greatest term in polynomial

What is the term with the highest power in $$(x^3-2)^{16}-(x^4 + 3)^{12}$$ I am not sure what the question entails. Is it the highest power or the highest coefficient? The textbook answer is ...
-1
votes
1answer
27 views

Determine the three digits of a number.

A three-digit number in base 7 is expressed by the same numerals, but in reverse order on the basis 9. Determine the three digits.
0
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3answers
45 views

Solving an equation algebraicly

Just wondering, how can ALL the answers be found to the equation: $2^\pi+\pi=2^n + n$ Obviously $\pi$ is a real solution, but how can I get this result; and can I obtain other solutions?
2
votes
1answer
32 views

5 pears and 1 apple cost as much as 2 pears and 2 apples. If each apple costs $0.75,find the total cost of 100 pears and 450 apples.

I know it's in algebra based on the context and the next question is also algebra. This is what I could do: $y + $0.75 = $x $d + $1.50 = $x I don't know what to do next.
0
votes
0answers
40 views

A nice set of squares. [duplicate]

Are there integers $a, b, c, d$ such that $$a^2+b^2=c^2$$ $$a^2-b^2 = d^2?$$ I have tried by showing that $a^2 = b^2 + d^2$ and thus $a^2+ b^2 = 2b^2+d^2 = c^2$ But how do I show that there are no ...
0
votes
1answer
30 views

Scaling a grading system

I wasnt quite sure what to call this question. But here is my issue (it might be super simple) I have the following grading system: Some pseudocode ...
2
votes
2answers
59 views

A problem in definite integral.

What will be the value of $a$ for which the integral $$\int \limits^{\infty }_{0}\frac{dx}{a^{2}+(x-\frac{1}{x})^{2}} =\frac{\pi}{5050}$$ where $a^{2}\geq0$ It seems like a standard integral but ...
-3
votes
0answers
34 views

Provide a logarithm with a base $10$ for $1.6 \times 10^{-19}$ [on hold]

Provide logarithms with a base $10$ for: a. $1.6 * 10 ^{-19}$ b. $3 * 108$
0
votes
1answer
16 views

Tensor Product of Extension of Scalars

Let $M$ and $N$ be modules over commutative ring $A$. Let $\varphi:A\to B$ be a morphism of rings. We use the notation, $M_B = M\otimes_A B$, this is a module over $A$, but we will rather consider ...
1
vote
4answers
99 views

Can some explain very quickly what $ |5 x + 20| = 5 $ actually means?

I don’t mean to bother the community with something so easy, but for the life of me, I can’t remember how to do these. I even forgot what they were called, and I’m referring to the use of the “$ | $” ...
0
votes
0answers
14 views

Sine and cosine graph transformation

I'm having some difficulties with this question A bike is on a stand such that the highest point of the back wheel is 47 inches above the ground. If the pedal is turned counter clockwise, the back ...
1
vote
3answers
32 views

Find the interval on which $ x^{2} - \lfloor x \rfloor - 3 < 0 $ holds.

On what interval does the equation $ x^{2} - \lfloor x \rfloor - 3 < 0 $ hold? My attempt: I tried sketching the graph, but it’s a bit complicated. Is there any other approach?
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vote
4answers
20 views

Finding the exact values of trig functions in a quadrant

I need some help solving some questions because I have no idea how to solve them, and some explanation would be appreciated. The questions says: Given $\cot\alpha=\frac{\sqrt{13}}{6}$ and $\alpha$ ...
-1
votes
1answer
24 views

Algebra. Relationships of x and y [on hold]

What is the relationship between 1 yard of fabric and 8 napkins? Is it make x=fabric and y=napkins. So 1 yd/8 napkins or 36 inches/ 8 napkins= 4.5 inches per napkin?
1
vote
2answers
29 views

Help Finding the Equation of a Line with a Given a Point and a Slope

I have been receiving a lot of help today and I really appreciate it, I am nearing the end of my class and need help studying for my final. In my review I came across a problem that i am having ...
-1
votes
1answer
37 views

Triangles and law of sine, cosine question [on hold]

Im having problems with this question and I've tried lots of approaches yet keep getting the similar or a close answer to what im getting an its always wrong
3
votes
0answers
33 views

Triangles, sine and cosine problem

Hi everyone I tried solving this countless times but I always get the wrong answer! what I did first is 600/tan(46) - 600/tan(40) and that sounded reasonable to find the answer! but I keep getting it ...
1
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2answers
23 views

Simplify the Complex Fraction

I am having trouble with the following complex fraction. I have simplified everything for the most part, but I am stuck on the last part and need to know what I have to do next. ...
1
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2answers
36 views

I am given two points : A(2,4,-8) and B(0,-2,-6) I am supposed to find a point three times as far from point A as point B.

I am given two points : A(2,4,-8) and B(0,-2,-6) I am supposed to find a point three times as far from point A as point B. How am i supposed to approach this problem?
1
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0answers
25 views

How do I find the domain and range of this piecewise defined function?

Are both conditions true when $x>3$? If so, how do I graph it? $$ f(x)=\left\{\begin{aligned} &x^2-4&&:x>3\\ &2x-1&&:x\geq 3 \end{aligned} \right. $$
0
votes
2answers
36 views

Why does $1-\frac{1+n}{4+2n}=\frac{3+n}{4+2n}$

I have an understanding of proofs and I came to the point where $1-\frac{1+n}{4+2n}=\frac{3+n}{4+2n}$. I couldn't figure out why the math worked out though. Could someone explain to my why this math ...
0
votes
5answers
73 views

Help understanding how to factor completely $x^3-x^2-x+1$

I need someone to help explain the steps to completely factor the problem $x^3-x^2-x+1$. Here is what I have done so far: $x^3-x^2-x+1$ to $x^3-x^2+-1(x+1)$ Since there is a ...
0
votes
3answers
53 views

Let $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ be unit vectors, with $\mathbf{a+b+c=0}$. The angle between any two of these vectors is $120^\circ$.

Let $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ be unit vectors, such that $\mathbf{a}+\mathbf{b}+\mathbf{c} = \mathbf{0}$. Show that the angle between any two of these vectors is $120^\circ$. Hi, ...
0
votes
2answers
54 views

Show that $f(a)$ converges after some point

There is a row of 1000 integers. There is a second row below, which is constructed as follows. Under each number $a$ of the first row, there is a positive integer $f(a)$ such that $f (a)$ equals ...
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0answers
19 views

Algebra inequality problem

Suppose: $ x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1$ , and $x_1x_3x_5 + x_2x_4x_6 \ge \dfrac {1}{540} $ and $\dfrac{p}{q}$ is the maximum possible value of $x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 ...
-2
votes
1answer
30 views

Anna, Bill and Carl entered a race [on hold]

Anna, Bill and Carl entered a race. Bill's speed was 4/5 Anna's, and Carl's speed was 3/4 Bill's. How many times the mean of the two boys' speeds was Anna's?
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votes
0answers
21 views

Polynomial division of $x^3-y^3$ using lexical ordering $x > y$

I am trying to divide $f = x^3 - y^3$ with $f_1 = x - y$ as the divisor using lexical ordering $x > y$. I know that $LT(f) = x^3$ and $LT(f_1) = x$, so $f_2 = f - x^2 f_1 = x^2 y -y^3$. So, ...
-1
votes
4answers
64 views

Solving $\frac{x+y}{xy}=2$, $\frac{x-y}{xy}=6$

$$\frac{x+y}{xy}=2,\ \ \frac{x-y}{xy}=6$$ I am not understanding how to solve the equation. I tried dividing the whole equation by $xy$, but, that didn't work too. Any hint or help would be much ...
1
vote
1answer
20 views

Finding out the logarithmic function for the situation below

The situation reads as follows: There are 3000 barbs in a pond and every year 20% of the barbs die and then 1000 new barbs come to the pond. A logarithmic function needs to be plotted to graph ...
0
votes
0answers
18 views

how to find principal when rate and sum of compound interest is given? [on hold]

Find the money, invested at 10% compounded annually, on which the sum of interest for first year and third year is 1768
-1
votes
1answer
41 views

tricky inside/outside of brackets in algebra [on hold]

Oh hi guys I'm hoping to get some help with these two questions. I know the basic rule about brackets, but I'm stumped here: a) $8P - (3 + P) = 12$ b) $28 - (1 - P) = 42$ Many thanks :d
1
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0answers
33 views
+50

How to determine when this two variable transformation is invertible?

I am given: $$ X= U \cos(V) \tag{1}\\ $$ $$ Y = U \sin V \tag{2}$$ Now, I need to: a) Give the respective ranges for $U$ and $V$ in order that the transformation defined is one to one. and ...
10
votes
3answers
125 views

What is the connection between the discriminant of a quadratic and the distance formula?

The $x$-coordinate of the center of a parabola $ax^2 + bx + c$ is $$-\frac{b}{2a}$$ If we look at the quadratic formula $$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ we can see that it specifies two ...
1
vote
1answer
12 views

When can you not do a mapping composition?

Suppose I have $\alpha:\mathbb R^3 \to \mathbb R$ and $\beta:\mathbb R \to \mathbb R^+$. Looking over my notes, it says $\alpha \circ \beta$ can not be done but $\beta \circ \alpha$ can. What is the ...
3
votes
2answers
55 views

Prove that $f(x)=x$ can have at most one solution if $f'(x)\ne1$

Prove that $f(x)=x$ can have at most one solution if $f'(x)\ne1$ What I did : Use $g(x) = f(x)-x$, then $g'(x) = f'(x)-1\ne0$ I suspect I have to use Rolle's theorem now, But I am having difficulty ...