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.

learn more… | top users | synonyms (2)

6
votes
2answers
114 views

Changing $|1-x|$ to $|x-1|$

I'm trying to get the limit $$\displaystyle\lim_{x \rightarrow 1} \frac{(x-1)(x-1)}{|1-x|}.$$ I think what I need to do is change $|1-x|$ to $|x-1|$ so I can cancel out one of the terms... but how do ...
0
votes
1answer
190 views

Prove if $a$ & $b$ & $d$ $\in \mathbb{N}^*$ and $gcd(a,b)=d$ and $a=d\cdot k$ and $b=d\cdot n$ then $gcd(k,n)=1$.

Let's define $\mathbb{N}^*$ first: $\mathbb{N}^* = \mathbb{N} - \{0\}$ Prove if $a$ & $b$ & $d$ $\in \mathbb{N}^*$ and $gcd(a,b)=d$ and $a=d\cdot k$ and $b=d\cdot n$ then $gcd(k,n)=1$. ...
0
votes
3answers
106 views

How do you factor this using complete the square? $6+12y-36y^2$

I'm so embarrassed that I'm stuck on this simple algebra problem that is embedded in an integral, but I honestly don't understand how this is factored into $a^2-u^2$ Here are my exact steps: ...
0
votes
1answer
153 views

Using table of integrals to solve $\int y \sqrt{6+12y-36y^2}dy$

I'm supposed to use a table of integrals to solve the below equation: $$\int y \sqrt{6+12y-36y^2}dy$$ I'm having trouble identifying the form to use because I guess my weakness in algebra shows in ...
5
votes
4answers
216 views

How to solve equations of this form: $x^x = n$?

How would I go about solving equations of this form: $$ x^x = n $$ for values of n that do not have obvious solutions through factoring, such as $27$ ($3^3$) or $256$ ($4^4$). For instance, how ...
0
votes
3answers
67 views

Why is $t=\frac 3 2$ in $1+t=\sqrt{4+t^2}$?

I am confused about solving $1+t=\sqrt{4+t^2}$. When I solve it per hand I come to the conclusion that $t$ can be everything. $$\begin{align*} 1+t =\sqrt{4+t^2}& \qquad | \cdot^2 \tag{1} \\ ...
0
votes
2answers
88 views

Linear Approximation 2

Use a linear approximation of $$f(x) = \sqrt[3]{x}$$ at $$x=8$$ to approximate $$\sqrt[3]{7}$$. Express your answer as an exact fraction.
2
votes
4answers
153 views

Find number of solutions of $2^x$+$3^x$+$4^x$=$5^x$

Find number of solutions of $$2^x+3^x+4^x=5^x$$ I tried using graphs but don't know how to draw graph of L.H.S.
0
votes
3answers
103 views

Least Value Of $x+y+z$

If $x$, $y$ and $z$ are positive integer and $3x=4y=7z$, then What is the least possible value of $x+y+z$?
1
vote
1answer
55 views

How to evaluate this using algebra?

We have $$1.001^6 - 1.001^5$$ How do I evaluate this? Normally I would use algebra to rewrite it, but I don't know how to cleverly rewrite $a^6 - a^5$ to a simpler form.
1
vote
1answer
42 views

Interval of p in the quadratic equation

In what interval does p lie for which both the roots of this equation are less than 2? Since both the roots are less than 2, their sum should be less than 4; giving 20p/4<4, that is, 5p<4, or, ...
0
votes
2answers
52 views

Factoring with 5 terms

I'm doing some factoring and have arrived at the point where the book says that: $$(k+1)/30(6k^4 + 39k^3 + 91k^2 + 89k + 30)$$ factors to: $$(k+1)/30(k+2)(2k+3)(3(k+1)^2+3(k+1)-1)$$ I cannot ...
1
vote
1answer
50 views

Number of steps to rotate for convergence in a circle

I am trying to come up with a formula to calculate how many rotations of discrete steps I need to converge on the exact same point on a circle. For instance I start at $0$ degrees and rotate $45$ ...
1
vote
2answers
70 views

Solving this logarithm equation?

How do I solve this equation using common logarithms? $\log x = 1-\log(x-3)$
0
votes
2answers
51 views

How does this factor to 6(1-ln6)

-6(ln6)+6 I got.... 6(-1(ln6) shouldn't the 1 be negative not the ln6
2
votes
3answers
167 views

nitpicking the definition of a polynomial function

A textbook I'm reading says that $f(x)=0$ is NOT a polynomial function, yet $g(x)=8$ IS a polynomial function since $g(x)=8=8x^0$ which satisfies the non-negative integer degree requirement. Yet, ...
3
votes
1answer
451 views

$f(x,y) = \sqrt{x^2+(y-1)^2}+\sqrt{(x-3)^2+(y-4)^2}-\sqrt{x^2+y^2}-\sqrt{(x-1)^2+y^2}\;\;,x,y\in \mathbb{R}$.

Let $f(x,y) = \sqrt{x^2+(y-1)^2}+\sqrt{(x-3)^2+(y-4)^2}-\sqrt{x^2+y^2}-\sqrt{(x-1)^2+y^2}\;\;,x,y\in \mathbb{R}$. Then Max. of $f(x,y)$. $\underline{\bf{My\;Try}}::$ We can convert into Complex no. ...
4
votes
2answers
927 views

system of equations $\sqrt{x}+y = 11$ and $x+\sqrt{y} = 7$. [duplicate]

If $x,y\in \mathbb{R}$ and $\sqrt{x}+y = 11\;$ and $x+\sqrt{y} = 7$. Then $(x,y) = $ $\underline{\bf{My\;\; Try::}}$ Let $x=a^2$ and $y=b^2$, Then equation is $a+b^2 = 11$ and $a^2+b = 7$. ...
1
vote
2answers
56 views

Why do the conditions $x_1+x_2=b$ and $x_1\cdot x_2=ac$ hold for any quadractic equation?

Consider the equation $$ax^2+bx+c=0.$$ The factorization of the left hand side is of the form $(x+x_1)(x+x_2)$, where the solutions $x_1$ and $x_2$ must satisfy $$(1)\quad x_1+x_2=-\tfrac ...
0
votes
2answers
157 views

polynomial maximum / minimum problem [precalculus]

What would be the minimum dimensions of a rectangular box with the smallest surface area that has a volume of 288 and has a width 3 times its length. $$V = lwh$$ $$l = 3w$$ so $V = 3w^2 \cdot h$ ...
0
votes
1answer
51 views

Make up $a$, $b$, and $c$ such that $f(x)=\frac{1}{ax^2+bx+c}$ is even and positive

Make up numbers for $a$, $b$ and $c$ to make this true: the function $f(x)=\dfrac{1}{ax^2 + bx +c}$ has even symmetry and is entirely above the x-axis.
1
vote
3answers
57 views

Elementary Algebra Inequality question

$$ \frac{1}{x-1} < -\frac{1}{x+2} $$ (see this page in wolframalpha) Ok, so I think the main problem is that I don't really know how to do these questions. What I tried to do was move $-1/(x + ...
1
vote
2answers
50 views

None exact first order ODE

i have to solve the following $1^{st}$ order differential equation $(xy+1)dx+(2y-x)dy=0$ i am in the elementary differential class,and have not learned multivariate functions, the equation below is ...
5
votes
3answers
619 views

Minimizing the length of wire between two poles?

There are two poles (lets say poles A and B) $50$ feet apart and the poles are $15$ and $30$ feet tall. There is a wire which runs from the top of pole A to the ground, and then to the top of pole B ...
1
vote
1answer
417 views

Polynomial word problems

Find all points on the circle $x^2 + y^2 = 9$ that are 3.5 units from $(4,5)$ and Find the point on the circle $x^2 + y^2 = 9$ that is closest to $(4,5)$. I only need to set both x and y to a ...
2
votes
1answer
42 views

Maximum of the product $\displaystyle P(N)=\prod_{k=1}^N\frac{k^a}{a^k}$

Given the following product: $$\displaystyle P(N)=\prod_{k=1}^N\frac{k^a}{a^k}$$ it can be expressed as: $$P(N)=\Gamma(N+1)^a\frac{1}{a^{\frac{1}{2}(N+1)^2-\frac{1}{2}N-\frac{1}{2}}}$$ I have to find ...
0
votes
3answers
48 views

A question about the solution of $x^2+4x-32 \lt 0$

I was given the following problem $$ x^2+4x-32 \lt 0 $$ and I came up with the following solution $x \lt -8 , x \gt -4$ and then I, unsucessfully, put it in interval notation $(-\infty, -8)\cup ...
1
vote
0answers
52 views

Equations Match the Columns

$f(x)=2x^3-9x^2-24x+p,\quad p\in\mathbb R$ Match the columns $f(x)$ has three distinct real roots, then p belongs to $f(x)$ has two positive roots and one negative root, then p belongs to $f(x)$ ...
4
votes
4answers
126 views

Comment upon nature of the roots

How many roots are there of the following polynomial? How many are real, and how many are complex? ...
2
votes
3answers
368 views

Find number of positive integral solutions of

Find number of positive integral solutions of $x^4-y^4=30108012$ How to do it?
0
votes
2answers
146 views

Calculus question with optimization homework

A piece of wire 30 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. (a) How much of the wire should go to the square to maximize the total area ...
0
votes
1answer
346 views

Quadratic equations tricky question

Solve for x : $x^4-6x^3+12x^2-12x+4=0$ Tried to factorize and used substitution but no result.
-1
votes
1answer
39 views

Linear transformation / Polynomial Question

$T:P_{3}\rightarrow P_{3}$ defined by $T(p(t))=tp'(t)+p(0)$ is a linear transformation. Determine whether $T$ is invertible. If yes, find $T^{-1}(q(t))$, where $q(t)$ is a polynomial of degree at ...
1
vote
2answers
23 views

What is the length of the bar needed to represent 75 kilometers( in centimeters)?

In a bar graph, 1 centimeter represents 30 kilometers. What is the length of the bar needed to represent 75 kilometers( in centimeters)?
0
votes
1answer
125 views

Need help in determining the volume of styrofoam used with dimensions $2.50ft + 1.50ft + 1.00ft$

So to elaborate on the title, the question is this: The average density of Styrofoam is $1.00 \frac{kg}{m^3}$. If a Styrofoam cooler is made with outside dimensions of $3.00ft$ $x$ $2.00ft$ $x$ ...
1
vote
3answers
953 views

If I weigh 250 lbs on earth, how much do I weigh on the moon?

One of my homework questions is to determine how much a 250 lb person weighs on the moon. I first googled a calculator for this and found that the weight is 41.5 lbs. So I tried to derive it myself ...
3
votes
1answer
254 views

Help find the derivative of $e^{2^x}$ using the definition of the derivative

Let $f(x) = e^{2^x}$, where $e$ is the exponential function. So the $f'(x)$ is: $\begin{align}f'(x) &=& \lim_{h \to 0} \frac{e^{2^{x+h}}-e^{2^x}}{h}\\ &=& ...
1
vote
2answers
624 views

Suppose $g$ is even and let $h=f \circ g$. Is $h$ always an even function?

I came across one of the following problems in my homework set: $$ \text{Suppose} \, g \, \text{is even and let} \, h=f \circ g. \text{Is} \, h \, \text{always an even function?}$$ I came to the ...
1
vote
3answers
222 views

To what extent can I square both sides of an absolute equation?

I am working on some absolute equation problems like the following: $$\begin{align} & {|x-4|} \lt 1 \\ & 1 \le |x| \le 4 \\ & |x+3| = |2x+1| \end{align}$$ Now, for both of these ...
2
votes
0answers
43 views

Making a quantity dimensionless

OK, here's another that might be simpler than it looks but I am about ready to give up. Given the following: $$U_\mathrm{eff}(r) = - \frac{k}{r}+ \frac{L^2}{2mr^2}$$ $$\frac{U_\mathrm{eff}}{U_0} = ...
3
votes
2answers
320 views

Sum of derangements and binomial coefficients

I'm trying to find the closed form for the following formula $$\sum_{i=0}^n {n \choose i} D(i)$$ where $D(i)$ is the number of derangement for $i$ elements. A derangement is a permutation in which ...
4
votes
1answer
60 views

Simple upper bound for $\binom{n}{k}$

I remember seeing an upper bound for the binomial $\binom{n}{k}$ with an exponential function, something like $\binom{n}{k}\leq \left(ne/k\right)^k$. What exactly is it, and are there other similar ...
0
votes
1answer
183 views

Does $e^{i\theta}$ Relate To Hyperbolic Sine/Cosine?

I would like to understand the relationship betwene $e^{i\cdot \theta}$ and hyperbolic sine and cosine. Here is what I have done so far: Given: $$\sinh(x)+\cosh(x)=e^x $$ ...
0
votes
1answer
37 views

Can you help solve this cubic in root x?

Here's the original equation: $$\frac{1}{\beta}arctan\left(\frac{\sqrt{2rx+x^{2}}}{r}\right)+\left(r\sqrt{2rx+x^{2}}-r^{2}arctan\left(\frac{\sqrt{2rx+x^{2}}}{r}\right)\right)=\frac{\pi}{2\beta}$$ ...
4
votes
2answers
66 views

For $x>0$, $x + \frac1x \ge 2$ and equality holds if and only if $x=1$ [duplicate]

Prove that for $x>0$, $x + \frac1x \ge 2$ and equality holds if and only if $x=1$. I have proven that $x+ \frac1x \ge 2$ by re-writing it as $x^2 -2x +1 \ge0$ and factoring to $(x-1)^2\ge0$ ...
0
votes
1answer
37 views

Find an Inverse function

I need to find the inverse of those functions: $x \mapsto \sin e^{x}$ $x \mapsto e^{\sin x}$ I know that the way is to solve the equation $y = f(x)$ for $x$, and I did it with functions like ($x ...
0
votes
2answers
27 views

How many rackets did he throw away?

Siva bought $\$945$ worth of rackets at $\$7$ each. He had to throw some away as they were damaged. He sold the rest at $\$9$ each and collected $\$1116$. How many rackets did he throw away?
0
votes
1answer
67 views

Example ill-conditioned nonlinear system of equations

I need ill-conditioned multivariate nonlinear systems of equations for testing purposes of a computer program. Do u know any specific problems from science? I read about chebyschev polynoms, but they ...
0
votes
1answer
65 views

Unknown terms of the proportion

please help me solving this problem. The question is, find the unknown terms of the proportion $$\frac 23 = \frac x{12} = \frac y{15}.$$
1
vote
1answer
50 views

Plot implicit equation

I'm working with a frequency-response curve of a nonlinear oscillator and came across the following equation (Kovacic & Brennan 2011, p. 179): $$ A^2 = \frac{f^2}{4 \xi^2 \omega^2 + (\omega^2 - ...