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
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3answers
337 views

Find the exact value of the trigonometric Function csc(-630 °) Answer is csc( 90°) but why is it positive 1?

I evaluated csc(-630 °) = csc(90°) and understand it is a quadrantal, but I do not understand why it is 1 nor why it is +1 and not -1. Can someone explain?
0
votes
1answer
24 views

Graphing functions

I am having problems understanding how to graph the product $fg$ when $f(x) = x$ and $g(x) = |x|$. Any help would be much appreciated!
1
vote
2answers
59 views

Proving double derivatives with the chain rule (I think?)

Hey StackExchange I'm having trouble understating where to start with this problem, I'm supposed to prove something about double derivatives and the chain rule but I'm having trouble understanding ...
0
votes
1answer
21 views

Discovering the derivatives of functions combined with trig values.

Hey StackExchange I have a problem that I don't really understand and I could use some hints for starting it. Suppose $m(\frac{\pi}{3}) = 4$ and $ m'(\frac{\pi}{3}) = -2$, and let $g(x) = m(x)\sin x$ ...
1
vote
2answers
53 views

How to express a trigonometic equation in $\sin 2\theta $ and $\cos 2\theta $?

How do I express the given equation in $\sin 2\theta $ and $\cos 2\theta $ in terms of x? $x + 3 = 7\sin \theta $ with $\frac{\pi }{2}{\text{ < }}\theta {\text{ < }}\pi $ for $\sin 2\theta ...
4
votes
1answer
78 views

Simplify continued fraction with $\pi$

I'm not even sure where to start on this: Simplify: $$\pi+\dfrac{2}{\pi+\dfrac{2}{\pi +\dfrac{2}{\dots}}}$$ The second term is a rational expression with 2 in the numerator and the denominator is ...
0
votes
2answers
41 views

Solving equations with powers without logarithms

Im taking an introduction to logarithms. Of course a short review of exponentiation is inherent for a clear understanding of logarithms. I was asked to find, for example, $27^x = 3$. (without the use ...
1
vote
3answers
65 views

Express the length a, b, c, and d in the figure in terms of the trigonometric ratios of θ.

I have memorized this chart and know that a= Sine, b= Tangent, c= Secant, and d= cosine. However, while a and d make sense intuitively, I do not understand how Secant is c and b is tangent. Can ...
4
votes
2answers
60 views

If $x,y \in (0,\frac{\pi}{2})$ then expression $\sin x +\cos y +\tan^2y+\cot^2x+5>\ldots?$

Problem : If $x,y \in (0,\frac{\pi}{2})$ then expression $\sin x +\cos y +\tan^2y+\cot^2x+5$ is always greater than : (a) $\ 7 $ (b) $\ 8 $ (c) $\ 9 $ (d) $\ $none of these Solution : We ...
0
votes
2answers
39 views

solving $|(x-3)(x-1)| $$\le$ $|\frac{1-x}{x-3}|$ graphicly [closed]

how to solve $|(x-3)(x-1)| $$\le$ $|\frac{1-x}{x-3}| $ in the graphic method?
1
vote
1answer
40 views

Formula alteration

is there any way to transform the formula$ \frac {1-x}{x-3}$ into something that can be easily sketched, or which will help eliminate $x$ from the denominator?
2
votes
4answers
210 views

Solve The Triangle

I am having a tough time trying to solve this problem. I have utilized the 30, 60, 90 triangle measures for the length of sides. However, I am stuck since the side that would be √3 has 100 as its ...
2
votes
1answer
41 views

Greatest value of f

If $f'(x)=6-x$ then which of the following has the greatest value? $f(2.01)-f(2)$ $f(3.01)-f(3)$ $f(4.01)-f(4)$ $f(5.01)-f(5)$ $f(6.01)-f(6)$ I know the answer is $f(2.01)-f(2)$ but how to prove?
1
vote
1answer
74 views

How to solve $\frac{2x+1}{2x-3}+\frac{7x\:}{9-4x^2}=1+\frac{x-4}{2x+3}$ for $x$?

Can somebody explain me this one! $\frac{2x+1}{2x-3}+\frac{7x\:}{9-4x^2}=1+\frac{x-4}{2x+3}$ My book says the answer is $x_1 = 0$; $x_2 = 6$. I tried to solve it and got stuck somewhere in: ...
1
vote
1answer
46 views

Completing the following equation by the suitable method

i got this linear equation two variable problems for my school. I understand the basics of the normal linear equation but this seems different instead having a pure number after the "=" they got a ...
2
votes
2answers
34 views

Help requested with integer means

Do there exist two distinct positive integers whose arithmetic mean, geometric mean, and harmonic mean are all integers? In illustrating these three means to students, it would be nice if a non-messy ...
2
votes
0answers
82 views

Help calculating Combinations

A boy has n objects to paint, ordered in a row and numbered form left to right starting from 1. There are totally c colors, numbered from 0 to c-1. At the beginning all objects are colored in color ...
0
votes
2answers
45 views

Is $3(2k+1)(2^{2k+1}-1)>(2^{k+3}-1)(2^{k+1}-1)$?

Let $k$ be an integer. I need to prove that: $$3(2k+1)(2^{2k+1}-1)>(2^{k+3}-1)(2^{k+2}-1)$$ where $k>a$ for a suitable $a$. thanks in advance.
0
votes
2answers
55 views

Solve $\dfrac{|x|}{|x+2|}<2$

Got it. $$\dfrac{|x|}{|x+2|}<2$$ $$|x|<2|x+2|$$ $$|x|<2x+4$$ $$-x<2x+4<x$$ $$-2x-x<4<x-2x$$ $$-3x<4<-x$$ $$x>\dfrac{-4}{3},x<-4$$ ...
0
votes
0answers
41 views

Prove (non)differentiability in piecewise functions

I'm looking for some help on proving that this function is not differentiable at a specific value. My first instinct is to approach the limit of the value from positive and negative, but that doesn't ...
0
votes
3answers
50 views

Factoring when differentiating expressions

I'm having trouble with differentiating a expression. I do it one way, wolfram alpha does it another. Let me show you what I mean. The original expression is this: $$\frac{1}{2u^3}$$ I start by ...
2
votes
5answers
154 views

deriving $y=\sqrt{x+\sqrt{x+\sqrt{x}}\cdots} $ [closed]

How to derive $y=\sqrt{x+\sqrt{x+\sqrt{x}}\cdots}$ at $x=6$ ?
1
vote
2answers
71 views

Finding the derivative of sinus and cosinus. Trigonometric identities

How can we see that $$\sin(x+h)-\sin(x)=2\sin\left(\frac h2\right)\cos\left(x+\frac h2\right)$$ How can we see that $$\cos(x+h)-\cos(x)=-2\sin\left(\frac h2\right)\sin\left(x+\frac h2\right)$$ Do ...
-1
votes
2answers
436 views

What is the efficient way to calculate number of divisors of N that are divisible by 2?. [closed]

For example if a number is given let say 8 then its factors are 1,2,4,8 hence total numbers of divisors which are divisible by 2 are (2,4,8) that is 3.
1
vote
1answer
55 views

If $F(x)=4x^3 +2x^2 -2ax - 4a^2$ and $F(a) = 0$, find all values of $a$

I used synthetic division with $x=a$ and since $F(a)=0$, I knew that the last term of synthetic division should be $= 0$. So, I set the last term $(-4a^2 + 4a^3) = 0$ and solved for $a$. This gave me ...
2
votes
0answers
117 views

Evaluate this product $n \times \frac{n-1}{2} \times \dots \times \frac{n-(2^k-1)}{2^k}$

For $k = \lfloor \log_{2}(n+1) \rfloor - 1$ evaluate $n \times \frac{n-1}{2} \times\frac{n-3}{4} \times \frac{n-7}{8} \times \dots \times \frac{n-(2^{k}-1)}{2^k}$ So the product goes up to $k$ and I ...
0
votes
3answers
34 views

Describe to me what is happening here (Pre-algebra)

I get confused on the second step. Can someone describe to me what is happening? If I want to solve this, what should the first thing I look at in my head such that I get the correct answer in the ...
3
votes
2answers
78 views

Trigonometry sum of solutions question

Problem: For which $a$ will the sum of solutions be equal to $100$, in $\sin(\sqrt{ax-x^2})=0$. The attempt at a solution: For $\sin(x)=0$, $x$ must be equal to $0$, so we get ...
0
votes
1answer
34 views

solving the number of boys who are left handed

In a group of 45 children, 60 percent of the children are boys, and 60 percent of the children are left-handed. If I want to solve the number of boys who are left handed then will it not be ...
6
votes
2answers
114 views

Fermat's Last Theorem for Negative $n$

While studying Fermat's Last Theorem and Pythagorean triples, the following question occurred to me: For the equation $a^n+b^n=c^n$, where $n$ is a negative integer, a) does a solution exist, and b) ...
0
votes
1answer
226 views

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept.

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. f(x) = (1/5)x^4(x^2 - 3) the choice 1- 0, ...
0
votes
3answers
69 views

The real cubic root expression

$2x^3-2x^2-3x+2=0$ has 3 real root, but they are all express in such way: $x=\dfrac{1}{3}\left(1+\dfrac{\sqrt[3]{-23+3i\sqrt{237}}}{\sqrt[3]{2^2}}+\dfrac{11}{\sqrt[3]{2(-23+3i\sqrt{237}})}\right)$ ...
0
votes
1answer
36 views

shortest distance between the point $(0,-3)$ and the curve $y=1+a_{1}x^2 + a_{2}x^4 + …+a_{n}x^{2n}$

If each $a_{i}>0,$ Then the shortest distance between the point $(0,-3)$ and the curve $$y=1+a_{1}x^2 + a_{2}x^4 + \cdots +a_{n}x^{2n}$$ is $\bf{My\; Try::}$ Let $P(x,y)$ be ant point on the ...
-1
votes
4answers
94 views

using trial and error in math problems

Suppose one bacterium is in a jar at 12AM, and then suppose there are two bacteria at 12:30AM, and then there are four bacteria at 1AM, etc. (a) How many bacteria will be in the jar at 12PM that ...
0
votes
2answers
72 views

Angle Measure: Angles in Standard Position, is it a useless term?

I have a pre calculus test next week, and I have been going over the chapters to gain a deeper understanding; however, I find it difficult, or at least I find the concept of "Angles in Standard ...
0
votes
1answer
22 views

How can I determine the row index of a parking space?

This represents a parking lot with 7 rows, containing a total of 392 spaces: In the example above, the row indexes from left to right would be: ...
0
votes
1answer
33 views

Rational Inequality Question

Is my process correct? Solve: $\dfrac{x-3}{x+5}\leq3$ $$\dfrac{x-3}{x+5}-\dfrac{3(x+5)}{x+5}\leq0$$ $$\dfrac{x-3-3x-15}{x+5}\leq0$$ $$\dfrac{-2(x+9)}{x+5}\leq0$$ $$\dfrac{x+9}{x+5}\geq0$$ ...
2
votes
1answer
37 views

Trigonometric identity proof problem

My problem is from Israel Gelfand's Trigonometry textbook. Page 48. Exercise 5: d) $\frac{\sin\alpha}{1+\cos\alpha}=\frac{1-\cos\alpha}{\sin\alpha}$ I would appreciate some hints on how to ...
0
votes
2answers
55 views

The best way to factorize?

$f(t) = t^3 -t^2 +t + 7$. Just made it up, but looking through previous tests, they come up a lot when trying to find eigenvalues. How would I easily factorize this or make it=0? Wow, thanks for the ...
1
vote
2answers
104 views

Omitting $i$ in calculations

Is it possible in various calculations related to the complex plane which also include analytic geometry , calculating distances etc, to omit $i$ and treat the imaginary axis as simply the cartesian ...
1
vote
2answers
55 views

How to find out number of real solutions to this without graphing?

$$ 3^x = 4x^2 $$ My prof's notes just says to look at the graph and you'll see that they intersect in 3 points. But is there another way of doing this if you can't graph it?
6
votes
1answer
70 views

What am I doing wrong in this algebra excercise?

This is my first question here, so please forgive me if the format etc. are not quite right. I've been attacking an algebra question, and my workings are below. There's a mistake somewhere (I don't ...
2
votes
3answers
92 views

Solve exponential equation $3^x= 2^x+2$

How do we solve this? I can't think of an easy way.. Is there any way to solve it without using newton's method or other approximations? $3^x=2^x+2$
0
votes
0answers
17 views

bounds of solution to the system of nonlinear equations

I have a system of nonlinear equations: \begin{eqnarray*} F_1(x,y) &=& 0,\\ F_2(x,y) &=& 0, \end{eqnarray*} where $F_i(x,y)$ with $i=1,2$ are continuosly differentiable in $(x,y)$. ...
3
votes
3answers
59 views

Solve $\dfrac{x}{x-2}>2$ by first rewriting it in the form $\dfrac{P(x)}{Q(x)}>0$

Edit: So then is this the correct final solution? $x<4,(\infty,4), x\ne2$ I am asked to do this: Solve $\dfrac{x}{x-2}>2$ by first rewriting it in the form $\dfrac{P(x)}{Q(x)}>0$ ...
2
votes
3answers
69 views

Square root of a squared number changes sign, which to apply first?

Heres something Ive always found interesting. Supose we have a variable $x$, and $x$ equals a negative number: Say: $$x=-17$$ Now, I can apply a square to both sides of the equation and preserve ...
0
votes
1answer
29 views

simplify following 4y+6x divided by xy

simplify: 4y+6x/xy I did 4y*xy = 4xy^2 and 6x^2y / xy I then /xy = 4y+6x = (2) 2y+3x. But this was not correct. Help in steps please.
0
votes
2answers
55 views

SAT Algebra and elementary number theory

Question 1. If $a>5$ and $b>4$, then which of the following must be true? i. $a>b$ ii. $a+b>9$ iii.$a+b>11$ I think all 3 are true, but the answer is ii. only. Question 2. ...
0
votes
3answers
57 views

Find the range of (x+1)/(2x+1)

I recently encountered with this question in an exam, but wasn't able to solve it. At first I equated the og eqn to y and created an eqn in y, but that didn't help. any hint?
1
vote
1answer
36 views

consider a square of side length $x$, find the area of the region which contains the points which are closer to its centre than the sides.

Any ideas how to start. I am having trouble figuring out the region itself All ideas are appreciated thanks