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
1
vote
3answers
23 views
How should I go about this approaching-infinite limit problem?
I'm doing some exercises of limits approaching infinite, most are simple polynomials where only the highest degree term will matter in the end but for this one I couldn't find a solution (not correct ...
3
votes
3answers
120 views
How many functions $f:\{1,2,3,4\}→\{1,2,3,4\}$ satisfy $f(1)=f(4)$?
I just need a hint or a way to think a about this problem: $f(1)$ can be $1, 2, 3, 4$ and $f(4)$ can be $1,2,3,4.$
1
vote
1answer
25 views
Linear Approximation
I have an exercise, giving this question.
Find the linear approximation $Y$ to $f(x)$ near $x=a$.
$$
f(x) = x + x^4,\quad a=0
$$
I can see in my result list that it says $Y=x$, however, after ...
0
votes
3answers
50 views
Finding the number of integer solutions, why is this wrong?
The question is to find the number of solutions such that $(x, y)$ are integers: $(x-8)(x-10)=2^y$. Here's what I did: $u(u-2)=2^y$. From the quadratic formula, $u=1+\sqrt{1+2^y}$. This is where I ...
5
votes
3answers
46 views
Algebraic expression in its most simplified form
I am trying to simplify the algebraic expression:
$$\bigg(x-\dfrac{4}{(x-3)}\bigg)\div \bigg(x+\dfrac{2+6x}{(x-3)}\bigg)$$
I am having trouble though. My current thoughts are:
...
0
votes
1answer
24 views
Tell me the ideal selling price to get back a specified number
If I am buying something at xxx,
what is the price to sell it if I want a profit of $2.50 after minus-ing 0.63% (broker fee) from the selling price?
I need to make this into an excel formula, but ...
0
votes
1answer
35 views
Solve this system of equations
Solve this system of equations
$$a+b = 3 -c$$
$$\frac{1}{a}+\frac{1}{b}= \frac{5}{12}-\frac{1}{c}$$
$$ a^3+b^3 = 45 -c^3$$
0
votes
2answers
25 views
How to find the minimum value of the expression? $\sqrt{(x-0,5)^2+0,75}+\sqrt{(x-\frac{\sqrt{3}}{2})^2+0,25}$?
What is the minimum value of this expression $\sqrt{(x-0,5)^2+0,75}+\sqrt{(x-\frac{\sqrt{3}}{2})^2+0,25}$ ?
-1
votes
1answer
34 views
How find this $(8\sqrt{6}-12\sqrt{2})\left(t_{2}+160e^{-\frac{1}{160}t_{2}}-1\right)=15$ [closed]
someone can help me?
use MATLAB find $$(8\sqrt{6}-12\sqrt{2})\left(t_{1}+120e^{-\frac{1}{120}t_{1}}-1\right)=20$$
and
$$(8\sqrt{6}-12\sqrt{2})\left(t_{2}+160e^{-\frac{1}{160}t_{2}}-1\right)=15$$
I ...
4
votes
3answers
170 views
how to solve this multivariate quadratic equation?
Any hope to acquire an analytic solution to such equations:
Solve:
$$\sum_{j=1}^n a_{ij} x_i x_j = b_i$$
for $i=1,\ldots,n$, where $a_{ij}$'s and $b_i$'s are known constants and $x_i$'s are unknowns ...
12
votes
6answers
601 views
If $2^x=0$, find $x$.
If $2^x=0$, find $x$.
Solution: I know range of $2^x$ function is $(0,\infty)$.
So $2^x=0$ is not possible for any real value of $x$
Hence, equation is wrong. We can't find value of $x$. Am I ...
5
votes
3answers
88 views
How to solve this simultaneous equation of $3$ variables.
I've stuck in this equation system.No clue how to start ?
$x+y+z=a+b+c\,\cdots(1)$
$ax+by+cz=a^2+b^2+c^2\,\cdots(2)$
$ax^2+by^2+cz^2=a^3+b^3+c^3\,\cdots(3)$
Find the value of x,y,z is in the form ...
1
vote
2answers
96 views
What's wrong with this conversion?
I need to calculate the following limes:
$$
\lim_{n\rightarrow\infty} \sqrt{\frac{1}{n^2}+x^2}
$$
My first intuition was that the answer is $x$, but after a bit of fiddling with the root I got ...
2
votes
1answer
27 views
f(n) for rows with $2^x$ bits on
This is similar to a question I asked here but now I need to alter it so I have brought across the relevant parts.
Given a binary table with n bits as follows:
$$\begin{array}{cccc|l}
...
5
votes
4answers
65 views
Where does the function $f(x) = \frac{2x}{x - 7}$ have an increasing slope?
Where does the function $f(x) = \frac{2x}{x - 7}$ have an increasing slope?
$a. x \le 0, x > 7$
$b. x<7$
$c. x > 7$
$d. x \in \Bbb R, x \neq 7$
This question is from a test of mine in a ...
1
vote
2answers
49 views
How to solve this system of equation.
$x^2-yz=a^2$
$y^2-zx=b^2$
$z^2-xy=c^2$
How to solve this equation for $x,y,z$. Use elementary methods to solve (elimination, substitution etc.).
Given answer is:$x=\pm\dfrac{a^4-b^2c^2}{\sqrt ...
2
votes
3answers
50 views
Algebra Equation with equals on left hand side
$6 = 4 - 2x =$
Show the answer with the mechanics of working out.
6
votes
3answers
62 views
How do I factor this?
How do I factor $p^2+8pq+16q^2-9r^2$? I know how to group the first two terms, but I dont know what to do with the other half. Can someone help me with this problem?
2
votes
1answer
38 views
If $y<x$ is there a way to phrase it in terms of $y>$something?
For instance I know $y<x$ implies $-y > -x$ but is there a way to phrase it in terms of $y>$ something (that does not itself contain $y$)?
1
vote
0answers
27 views
Inverse function of product of exponential matrices
I am looking for the value of $\mathbf{X}$ in a function of the type
\begin{align}
(\mathbf{X}-\mathbf{A})e^{\mathbf{X}}e^{-\mathbf{A}} = \mathbf{B}
\end{align}
where ...
1
vote
3answers
47 views
Which one is the correct series expansion?
Is
$$p^{n+1} = p^0+p^1+ \dots + p^n$$
or
$$p^{n+1} = p^0\times p^1\times \dots \times p^n\text{ ?}$$
I am confused.
please explain the correct one.
2
votes
3answers
136 views
Order of precedence: in $ab^{c}$, which operation goes first?
If you have
$$x^3(x^2 + 1)^{-\frac{1}{2}},$$
os the power or the product calculated first? I'm assuming the power comes first but I don't like to just assume.
5
votes
1answer
291 views
Looking for a trick to solve $2\sqrt {2x}+\sqrt {2x+3}=\sqrt {3x+2}+\sqrt {6x+20}$
Consider the equation:
$$2\sqrt {2x}+\sqrt {2x+3}=\sqrt {3x+2}+\sqrt {6x+20}.$$
Find a trick ( if exists ) which allows to solve it elegantly i.e. with avoiding the systematic squaring.
(The ...
8
votes
5answers
346 views
solving $\sqrt{3-\sqrt{3+x}}=x$.
Can we solve the following equation in $\mathbb{R}$ without expanding it into a fourth degree equation :
$$ \sqrt{3-\sqrt{3+x}} = x.$$
squaring both sides and squaring again is the only thing I ...
4
votes
6answers
117 views
Solving $\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$
Where do I start to solve a equation for x like the one below?
$$\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$$
After squaring it, it's too complicated; but there's nothing to factor or to ...
5
votes
6answers
5k views
How to solve this equation with two square root terms?
So guys, my girlfriend is taking a college algebra class this summer and I figured I would help her study for her upcoming final because I am an engineering major and this kind of math would be easy ...
18
votes
7answers
440 views
Solving $\sqrt{x+5} = x - 1$
I'm currently learning about radicals and simplifying them, and I came across this problem on the internet and tried to solve it:
$$\sqrt{x+5} = x - 1$$
So I used this logic:
$$
\begin{align}
...
0
votes
1answer
22 views
I need help with a word problem .
The length of a rectangle is 1 meter less than twice its width. If the area of the rectangle is 120 square meters, find the dimensions of the rectangle
17
votes
17answers
2k views
Interesting calculus problems of medium difficulty?
I would like to know sources, and examples of good "challenge" problems for students who have studied pre-calculus and some calculus. (differentiation and the very basics of integration.) Topics could ...
0
votes
1answer
37 views
How do I solve this solution-mixing problem?
A chemist has a 55% acid solution and a 40% acid solution. How many liters of each should be
mixed in order to produce 100 liters of a 46% acid solution?
1
vote
2answers
36 views
Equation of a line passing through a point and forming a triangle with the axes
How can I find the equation of a line that;
is passing through the point (8, 6) and
is forming a triangle of area 12 with the axes
?
So I tried to start using $A = |{\frac{mn}{2}}|$ and ...
1
vote
2answers
40 views
System of equations word problem
Andrea's Sequoia gets 12 miles per gallon while driving in the city, and 20 miles per gallon while on the highway. The last time she filled up the car it took 16 gallons of gas, and she had driven 280 ...
6
votes
2answers
190 views
Equating sums of square roots
I solved the following equation the hard way:
$$\sqrt{x+1} +\sqrt{x+33}=\sqrt{x+6} +\sqrt{x+22}$$
The only solution is $x=3$.
I am wondering if there is some easy observation that solves the equation ...
3
votes
1answer
141 views
Solving $\sqrt[3]{x^2} + \sqrt[3]{x} = 2$
My sister asked me for some help on her algebra homework the other day, and I was stumped by her question. The problem is to find the root of $\sqrt[3]{x^2} + \sqrt[3]{x} = 2$.
The internet tells me ...
22
votes
9answers
2k views
Is this way of teaching how to solve equations dangerous somehow?
Two years ago, I bought the book Mathematics for the Nonmathematican, by Morris Kline.
There I learned a new way of solving equations, which is related to the principle that states that any ...
1
vote
1answer
87 views
For which numbers $c$ is there a number $x$ such that $f(cx)=f(x)$?
This is one exercise in Spivak's book that is bugging me for a while, first I thought that $c=1$, but there's a hint:
There are a lot more than you might think at first glance.
And here I'm ...
3
votes
3answers
82 views
is $(x+1)^4-x^4$ non-prime for all natural positive integers $x$
Looking at difference between two neighbouring positive integers raised to the power 4, I found that all differences for integer neighbours up to $(999,1000)$ are non-prime.
Does this goes for all ...
4
votes
1answer
45 views
Is it possible to rationalize a denominator containing two cube roots?
The fraction in question is
$$-\frac{12}{\sqrt[3]{12\sqrt{849} + 108} - \sqrt[3]{12\sqrt{849} - 108}}$$
And was reached in calculating the solution to $x^4 - x - 1 = 0$. I've tried all the standard ...
1
vote
1answer
26 views
Like term reduction
In finding the derivative of $f(x) = 4x - x^2$ we first find the difference of the numerator $f(x + h) - f(x)$. Therefore we have $f(x + h) = 4(x + h) - (x + h)^2 = 4x + 4h - x^2 - 2xh - h^2$ minus ...
19
votes
11answers
870 views
Comparing $\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$
Without the use of a calculator, how can we tell which of these are larger (higher in numerical value)?
$$\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$$
Using the calculator I can see that the first one ...
2
votes
2answers
59 views
How do I solve $x^2 + y^2 + xy = z$ for $y$
How do I solve the following equation for $x$ or $y$ (does not matter because you can swap them):
$$
x^2+y^2+xy=z
$$
3
votes
2answers
160 views
Does this polynomial factorize further?
I just did a national exam and this question was in it; I am convinced this does not work:
Given that $(x - 1)$ is a factor of $x^3 + 3x^2 + x - 5$, factorize this cubic fully.
My attempt
1 | ...
1
vote
1answer
38 views
How do you solve for Y?
I know there has got to be a way to solve for $Y$ but I just can't seem to figure it out. Does anyone know how to solve this? Please help :)
$$5(Y(8))=C$$
$$C(Y(4))=B$$
$$B(Y(2))=A$$
...
3
votes
4answers
68 views
Solve equation $\sqrt{s+13} - \sqrt{7-s} = 2$
Solve the equation
$$\sqrt{s+13}-\sqrt{7-s} = 2$$
I moved the $-\sqrt{7-s}$ to the right side
Thus, I had
$$\sqrt{s+ 13} = 2 +\sqrt{7-s}$$
I then squared both sides
$$\sqrt{s+ 13}^2 = \left(2 ...
3
votes
3answers
72 views
Determining $\sin(15)$, $\sin(32)$, $\cos(49)$, etc.
How do you in general find the trigonometric function values? I know how to find them for 30 45, and 60 using the 60-60-60 and 45-45-90 triangle but don't know for, say $\sin(15)$ or $\tan(75)$ or ...
0
votes
0answers
15 views
Find the terminal point when the distance is not in terms of $\pi$
From Stewart Precalculus 5th edi, P407
I am not sure what to do here, in the textbook, Steward didn't provide any example as to finding the terminal point when the distance $t$ is an integer. I ...
1
vote
1answer
50 views
For which values is $x^3$ less than or equal to $3x$?
The title says it all. The answers say:
$x\le -\sqrt{3}$ and $0\le x\le \sqrt{3}$
(can someone edit this so all the $<$ have an 'or equal to' sign. Edit the roots as well please.
I'm not sure ...
1
vote
1answer
21 views
Number of equivalent rectangular paths between two points
I am trying to determine the number of paths between two points.
I am representing the paths as a list of steps "ruru" = right -> up -> right -> up
For my purposes, we can assume that there will ...
0
votes
0answers
40 views
Quaternion exponential map, rotations and interpolation
A code snippet I need to optimize is performing something peculiar. It seems that it's somehow related to transforming from a frame of reference to another. This is what it does, in mathematical ...
12
votes
3answers
117 views
$\sum_i x_i^n = 0$ for all $n$ implies $x_i = 0$
Here is a statement that seems prima facie obvious, but when I try to prove it, I am lost.
Let $x_1 , x_2 \dots x_k$ be complex numbers satisfying:
$$x_1 + x_2 \dots + x_k = 0$$
$$x_1^2 + x_2^2 ...
