Articles of algebra precalculus

Bending a paper sheet to a cone

EDIT1: Changing description on how to bend paper of $A4$ size for example to make portion of a cone The long side $L$ of a rectangular paper sheet $L \times W $ is bisected and without any cutting the two half length $L/2$ edges are rolled out of plane/ bent/glued to form a pointed vertex […]

Determining whether there are solutions to the cubic polynomial equation $x^3 – x = k – k^3$ other than $x = -k$ for a given parameter $k$

Let $k$ be a real parameter, and consider the equation $$x^3 – x = k – k^3 .$$ Obviously, $x=-k$ is a solution. Is it the only one? How to prove it?

using the same symbol for dependent variable and function?

Is it wrong to represent a dependent variable and a function using the same symbol? For example, can we write the parametric equations of a curve in xy-plane as $x=x(t)$, $y=y(t)$ where $t$ is the parameter? Also, if someone write the following equation $y=y(t) = t^2$ where $y$ represents the dependent variable and t represents […]

Alternative ways to write $ k \binom{n}{k} $

I am looking for a way to simplify $ k \binom{n}{k} $. I don’t understand what effect the factor $k$ has on the formula. So can anyone please explain what $\ k\binom{n}{k} $ would equate to?

How to find the sine of an angle

How to find the sine/cos/tangent/cotangent/cossec/sec of an angle: In degrees $\sin(23^{\circ}) =$ ? In radians $\sin(0.53) =$ ?

Showing if $n \ge 2c\log(c)$ then $n\ge c\log(n)$

Is this true that if $n \ge 2c\log(c)$ then $n\ge c\log(n)$, for any constant $c>0$? Here $n$ is a positive integer.

Find minimal value $ \sqrt {{x}^{2}-5\,x+25}+\sqrt {{x}^{2}-12\,\sqrt {3}x+144}$ without derivatives.

Find minimal value of $ \sqrt {{x}^{2}-5\,x+25}+\sqrt {{x}^{2}-12\,\sqrt {3}x+144}$ without using the derivatives and without the formula for the distance between two points. By using the derivatives I have found that the minimal value is $13$ at $$ x=\frac{40}{23}(12-5\sqrt{3}).$$

Writing without negative exponent

can someone please describe to me the method behind writing numbers without negative exponents such as: Maybe just show me the logic / process? Especially for number (c) because fractions really confuse me. Thanks in advance

Prove that $\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3} +…+ \frac{n}{2^n} = 2 – \frac{n+2}{2^n} $

I need help with this exercise from the book What is mathematics? An Elementary Approach to Ideas and Methods. Basically I need to proove: $$\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3} +…+ \frac{n}{2^n} = 2 – \frac{n+2}{2^n} $$ $i)$ Particular cases $ Q(1) = \frac{1}{2} ✓ $ ———- $ P(1) = 2 – \frac{1+2}{2^1} = \frac{1}{2} ✓ $ $+ = 1 […]

Show that $d \geq b+f$.

This question already has an answer here: How prove that $q \geq b+d$ for $ad-bc = 1$ and $\frac{a}{b} > \frac{p}{q} > \frac{c}{d}$? 1 answer