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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2
votes
1answer
62 views

$\cos ^2(x)=\frac{1}{2} \cos (2 x)+\frac{1}{2}$

I am wondering how this is rewritten? $$\cos ^2(x)=\frac{1}{2} \cos (2 x)+\frac{1}{2}$$, I think it has something to do with double-angle/half angle? but I am not sure and I do not see the ...
5
votes
1answer
28 views

Newton's way of getting a Taylor expansion

I don't understand how Newton find the Taylor expansion of $\frac{a^2}{b+x}$ by the following method : **This screenshot is from : The method of fluxions and infinite series Any idea ?
-1
votes
2answers
28 views

For which values of $m$, $f(x)=mx$ intersect the function $g(x)=\log x$?

For which values of $m$, the function, $f(x)=mx$ intersect the function, $g(x)=\log x$ I suppose that this problems reduce to the next form. Find for which values of m, exist solution for the ...
0
votes
1answer
30 views

Simultaneous function with three variables using subsititution method

Use any substitution method and solve the following equations: $$2x+5y+7z=86$$ $$3x+y+5z=60$$ $$x+4y+3z=54 $$ I used $x+4y+3z=54$ to make $x$ the subject $x=54-4y-3z$.
-2
votes
0answers
20 views

Nursing Home Question

A 1-year old 120-bed for-profit facility with a \$2,500,000 mortgage at 7.4% reports a net operating margin before depreciation of \$25,000. The answer is that in today's financial market the owners ...
0
votes
1answer
33 views

Find the range of function $f(x) =\cos(\sin(\ln(\frac{x^2+e}{x^2+1})))+\sin(\cos(\ln(\frac{x^2+e}{x^2+1})))$…

Problem : Find the range of function $f(x) =\cos(\sin(\ln(\frac{x^2+e}{x^2+1})))+\sin(\cos(\ln(\frac{x^2+e}{x^2+1})))$ My approach : maximum value of the function is when denominator term is ...
2
votes
1answer
47 views

Beautiful evaluation inequality

a,b,c>0 are such that $a^2+b^2+c^2=4$ and $4(a^2+2)=(a^2+b+c)^2$. What is the biggest possible value for a+b+c? I tried a lot of stuff like $a^2=4-b^2-c^2$. And i think it's somehow connected to ...
0
votes
0answers
15 views

Real world situation/model using (factored) higher degree polynomial function?

(Form A) $$y=3x^5+23x^4-7x^3+x^2-4x+9$$ (Form B) something in factored form $$y=(x-1)^2(x+2)^5(x-5)^4(x+7)$$ 2 part question: 1) Anyone know of examples where these types of functions arise? 2) Is ...
2
votes
1answer
36 views

simple SAT practice question

I tried to solve this and I am getting 11n/6. Am I seriously doing something really wrong or is this question wrong?
1
vote
5answers
97 views

Is 1^2^3 = $1^{2^3}$ or $(1^2)^3$ [duplicate]

Caret ^ signs can be used to describe the power of numbers. Is $1$^$2$^$3 = 1^{(2^3)}$ or $(1^2)^3$ How do you calculate it? Do you start with $2^3$ and then do $1^8$ or do you start with $1^2$ ...
0
votes
2answers
22 views

Trigonometry: How to determine the Period

I'm still kinda confused with solving the period on the diagram above. Amplitude= $3$ Max = $3$ Min = $-3$ Period = ? $y=a\cos(bx+c)$ Value of $a$ = $3$ Value of $b$ = ? Value of $c$ = ?
1
vote
1answer
35 views

To show $(x+y)^p\leq x^p+y^p$, where $0\leq p\leq1, x>0,y>0$?

How to show that, $(x+y)^p\leq x^p+y^p$, where for $0\leq p\leq 1,x\geq 0, y\geq0?$ Any suggestion how to prove it? Thanks in advance.
1
vote
2answers
63 views

Remainder of $3^7/8$

I read here that the remainder of $\frac{ab}{c}$ is equal to the remainder of $\frac{a}{c}\frac{b}{c}$ implying that the remainder of $\frac{a^b}{c}$ is equal to the remainder of $[\frac{a}{c}]^b$. ...
1
vote
4answers
40 views

quadratic equation: $5x^2 + 9x - 170 = 0$

I have a problem, my textbook says the solution of $5x^2 + 9x - 170 = 0$ is $5$ but the book didn't describe how it solved the equation. How can I solve this?
2
votes
4answers
62 views

Find the sum of an infinite series of Fibonacci numbers divided by doubling numbers. [duplicate]

How would I find the sum of an infinite number of fractions, where there are Fibonacci numbers as the numerators (increasing by one term each time) and numbers (starting at one) which double each time ...
2
votes
2answers
36 views

Is there a way to get the closed form of $x$ considering $x^2 + 3x = \sqrt{x + 2}$, not using calculator/computer?

How would one go about finding the exact answer to $$x^2 + 3x = \sqrt{x + 2}$$ Solving for $x$? Using paper and pencil to plot a graph, I've found the solution lies at $\approx 0.453$, but I am ...
-4
votes
3answers
51 views

Solving $x^2-6=\sqrt{x+6}$ [on hold]

How can I solve: $$ x^2-6=\sqrt{x+6}$$ Thanks for any help.
0
votes
0answers
47 views

Solving equations, Math olympiad, using vieta relation?

So the question asks to solve for real valued $a$ such that $b,c,d\in\mathbb{R}$ $$abcd=-1$$ $$(a+c)(b+d)=-1$$ $$ac+bd+a+b+c+d=-1$$ $$ab+cd=ac+a+c$$ So assuming the four numbers are roots of a quartic ...
2
votes
2answers
21 views

Find one set of solutions for the following system:

Find one set of solutions for the following system: \begin{cases} 1+a^2+d^2=3+b^2+e^2=3+c^2+f^2 \\ 1+ab+de=0 \\ ac+df=0 \\ bc+ef=0 \\ \end{cases}
2
votes
2answers
39 views

Probability of 8 or 9 digit sequence colliding in the same place in two 65 digit numbers

I have two numbers: 3032643431333337636238613038343231383364303731376566303037663231 3861663464383131656131653461343961343364303737663565356561653361 36430373 ...
7
votes
1answer
44 views

How do I find a constant for a polynomial so its roots are reflective around a linear function?

How can I find all complex numbers $w$ so that the roots of the following polynomial are reflected around a linear function $f(x)$ $$p(q) = q^2-4q+w = 0$$ If I want to find all the complex numbers ...
-2
votes
0answers
24 views

Explain why every natural number (other than one) is divisible by at least one prime number? [on hold]

Explain why every natural number (other than one) is divisible by at least one prime number? This is through Euclid's idea. A prime number is a natural number with exactly two distinct positive ...
3
votes
4answers
172 views

Solve $\frac{x^2+2xy+y^2}{x^2-y^2} >x+y$

Find the set of integer solutions $(x,y)$ to $$\frac{x^2+2xy+y^2}{x^2-y^2} >x+y$$ I can't seem to multiply both sides by the expression in the denominator. Nor can I simplify and cancel any ...
-1
votes
1answer
13 views

Trapezoids and Bases [on hold]

A trapezoid has bases of length $x$ and $4$. Let $P$ and $H$ be points on opposite legs of the trapezoid. $PH$ is parallel to the bases and divides the trapezoid into two quadrilaterals of the same ...
0
votes
2answers
28 views

Finding relationship between two numbers directly that were changed cumulatively

Bear with me. I'm not sure how to express this question let alone answer it. Here goes... I have a program that can calculate change from a single rate, we'll call 'A'. Known: $$C = A + 400$$ ...
1
vote
1answer
19 views

Third degree polynomial with unknown coefficients $q^3-3aq^2+b^2q+c = 0$

For an equation $q^3-3aq^2+b^2q+c = 0$ we know the roots $c, (a+b), (a-b)$. What is a good place to start with such equations? I've tried setting up a system of equations, but this is supposed to be ...
-1
votes
1answer
56 views

Mathematically prove that a bench which 2 chidlren fit in can't fit 3. [on hold]

You have a bench( Only 2 children can sit on it), 3 children and you have to prove logically that 3 children don't fit on the bench.
1
vote
2answers
23 views

How do I solve for $\delta$ in this question

$316.45 = 100e^{\delta(10)} + 100e^{\delta(5)}$ I don't know why I can't do this. I thought of using $\ln$ but I don't think $\ln(A+B) = \ln(A) + \ln(B)$ or does it?
0
votes
4answers
81 views

How do I solve this one? It's irritating me!

How do I solve this question? I can't think of anything to do! As $(x,y)$ ranges over all pairs of real values, what is the smallest value of: $(2x-3y-4)^{ 2 }+(2x-3y+10)^{ 2 }$
0
votes
2answers
32 views

Expand $-(x-2y)^2$

I'm having a hard time solving this. this is a part of a bigger "count sequence" and I need to expand: $$-(x-2y)^2$$ Bigger "part": $$(x+2y)^2-(x-2y)^2-4x(2y-1)$$ Doesn't matter how much I twist ...
0
votes
0answers
23 views

Silly technical question about polynomials in Lagrange's “résolution algébrique”

I decided that I'd go through Lagrange's "Sur la Résolution Algébrique des Équations" (see, if you have the time, a previous, unanswered question of mine: Value in retracing mathematicians' steps ...
2
votes
0answers
25 views

Interchanging Powers

Let $f(x)=(x^b)^{1/a}$ and $g(x)=(x^{1/a})^b$, where $a,b$ are even numbers. The domains are clearly different. $\operatorname{dom}\{ f\}=\mathbb{R}$, while $\operatorname{dom}\{ g\}=\mathbb{R}^+_0$. ...
1
vote
1answer
21 views

Calculating Covariance missing understanding for one step

I am following this once I get to this point I don't understand the transition/calculation to get -0.01 I mean to me that equals ...
0
votes
1answer
19 views

How do I get an Archimedean spiral that decreases from an initial radius?

So, where the equation of an archimedean spiral is: $$r = a + b\theta$$ I want to be able to use the equation in this form to then have a function where r decreases in exactly the same way by an ...
2
votes
4answers
65 views

Solving $y^2 - yx - y + x = 0$ for $y$?

I solved this equation for $y$ by inspection and confirmed it with Wolfram Alpha - $y^2 - yx - y + x = 0$ I got the values $y = 1$ and $y = x$ However I was wondering is there a formal method for ...
0
votes
1answer
31 views

Work out percentages and commisions

I have a price 265.69 top price. Of this 265.69, 180.83 is cost. 45 is profit 13.29 is a commission at 5% of price and 26.57 is another commission at 10% of the price 180.83 + 45 + 13.29 + 26.57 = ...
1
vote
1answer
19 views

Trigonometry graphs sinusoidal waves

i need help on this questions. I couldn't figure how to determine for both question A and B. But i have the answers for them, i just don't understand how the amplitude is 3 and so on.
2
votes
4answers
127 views

If $\omega + 1 = \omega$, find $\omega$ ($\omega \not= - \infty$ or $\infty$)

If $\omega + 1 = \omega$, find $\omega$ ($\omega \not= - \infty$ or $\infty$). It does not have to be a real number. My teacher gave us this question just to play around with, and my first ...
-3
votes
1answer
40 views

Pipe A is a inlet pipe filling the tank at 8000 litre/hour .Pipe B is the outlet pipe which empties the tank in 3 hours. [on hold]

Pipe A is a inlet pipe filling the tank at 8000 litre/hour . Pipe B is the outlet pipe which empties the tank in 3 hours.The capacity of the tank is?
1
vote
2answers
32 views

Interesting Functional Equations Problem?

Find all functions $f:\mathbb R \to \mathbb R$ that satisfy $f(x) + 3 f\left( \frac {x-1}{x} \right) = 7x$. How would we solve this? I noticed that if you plug in $\frac{x-1}{x}$ in for $x$, and ...
8
votes
4answers
84 views

Finding the sum of $\sin(0^\circ) + \sin(1^\circ) + \sin(2^\circ) + \cdots +\sin(180^\circ)$

I need help understanding the sum of $\sin(0^\circ) + \sin(1^\circ) + \sin(2^\circ) + \cdots +\sin(180^\circ)$ or $\displaystyle \sum_{i=0}^{180} \sin(i)$ This might be related to a formula to find ...
0
votes
1answer
24 views

Linear Algebra Analytical Exercise

This one has me stumped... $$H=C(sI-A)^{-1}B$$ and $$H_{CL} = C(sI-A+BK)^{-1}BG$$ Show that $$H_{CL} = H[I+K(sI-A))^{-1}B]^{-1}G$$ Any hints would be greatly appreciated!
0
votes
1answer
16 views

Geometric progression in annuity

I am working on the following problem that involves annuity which deposits form a geometric progression. Stan elects to receive his retirement benefit over $20$ years at the rate of $2,000$ per ...
-5
votes
4answers
61 views

Tea with grandmother [on hold]

My mother still makes tea with the old saying: one spoon per person and one for the pot. We used to buy a packet of tea every week but since grandmother came to live with us we have to buy two packets ...
0
votes
2answers
36 views

Mathematics and logs

For this indirect utility function: v(y) = ln(1/3 y) + 2 ln (2/3 y) = 3 ln(y) + ln (4/27) How did they simplify to 3 ln(y) + ln (4/27)? Im a bit confused, if there is a site helping with this ...
-1
votes
3answers
34 views

Why examples of the order of algebraic computations do not agree with calculator results?

This my first lesson in Algebra I replaced the dot signs with x. I've never seen times done with a dot before Lesson taken from MathPlanet Operations in the correct order When you are faced with a ...
7
votes
2answers
59 views

Function composition: $f^{653}(56)=?$

Let $f(x) = \frac1{(1-x)}$. Define the function $f^r$ to be $f^r(x) = f(f(f(...f(f(x)))))$. Find $f^{653}(56)$. What I've done: I started with r=1,2,3 and noticed the following pattern: $$f^r(x)= ...
0
votes
0answers
9 views

Calculating the interest rate for an annuity (Exam FM)

I have been searching for a way to solve for the interest rate given the monthly payments of a loan. I would like to set up a problem as the following. $X$=monthly payment , $i$=effective ...
-1
votes
1answer
30 views

Factoring/Expansion explantion

Sorry if I call something by the wrong name since I didnt learn math in english. ok so for example this: (a+b)(a-b) if you break it down to the second "()" you will end up with this: a+-b could ...
0
votes
2answers
48 views

Why is $y{(\log_a(x))} = \log_a{(x^y)}$?

Why is $y{(\log_a(x))} = \log_a{(x^y)}$? I feel like I'm missing something here. Sorry if I put the title wrong..