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)

0
votes
1answer
106 views

derive a parabola from two tangent lines

I have two tangents of a parabola, one representing initial speed (y=mx) and the other the maximum permissible extent (y=e), and I want to find the gentlest deceleration required to stop in time. It ...
1
vote
1answer
254 views

Solve $x - 2\arctan(x)= 0$

$x - 2\arctan(x)= 0$. I can find one root (0) from the equation $\tan(x/2) = x$ but there are two others, namely ($-2.3312, 2.3312$) that I don't know how to find. Looking for help! Thanks :) ...
1
vote
2answers
141 views

Confusion regarding the Logarithmic function change of base formula

My textbook seems to be making a big leap when trying to prove the change of base formula for logarithms. If someone could help clear this up it would be very appreciated. It starts with: $b^{x ...
1
vote
1answer
348 views

How to prove this equals something else.

I'm trying to prove that $$\dfrac{l}{c+u}+\dfrac{l}{c-u}=\dfrac{2l}{\sqrt{c^2-u^2}}$$ My workings so far: $$\dfrac{l}{c+u}+\dfrac{l}{c-u},$$ put over common denominator, ...
0
votes
3answers
88 views

Finding a formula for the sequence $10, \,110,\,1110,\,11110,\dotsc$

How can I find a formula for the sequence $$10,\,110,\,1110,\, 11110,\dotsc$$ to make it ready for summation?
0
votes
2answers
141 views

Solve discrete Math Problem using abstract algebra, postage problem?

The question I am looking at is not very hard: Determine which amounts of postage can be written with $5$ and $6$ cent stamps. To determine the amount, use a brute force way to solve it. Counting ...
2
votes
2answers
1k views

How to prove that $\lim_{x \to 0} \sin(x) = 0$ using the epsilon-delta definition?

How do I prove that $\lim_{x \to 0} \sin(x) = 0$ using the episilon-delta definition of a limit? Do I have to divide the domain of $x$ into 4 cases for each quadrant? Update: Based on the ...
2
votes
1answer
68 views

Proof by induction that $(a^n-1)$ is divisible by $(a-1)$

I have to proof that $(a^n-1)$ is divisible by $(a-1)$ where $a \in \mathbb {N_{>1}}$ I think that I have the proof but I am not sure if that is the correct format. Induction hypothesis: ...
1
vote
1answer
298 views

Let p and q be distinct odd primes. Define $n=pq$ and$ \phi(n)=(p−1)(q−1)$

(a) Show that $p+q = n−\phi(n)+1$ and $p−q = \sqrt{(p+q)^2−4n}$. (b) Suppose you are given that $n = 675683$ and are told that $p−q = 2$. Explain how this information can help us factor $n$ ...
0
votes
1answer
60 views

Equality of corresponding variables

This question might be silly, but I was wondering for $x_1 y_1 + y_1 = x_2 y_2 + y_2$ , $y_1 = y_2$ is true. And likewise, $x_1 y_1 = x_2 y_2$ ?
1
vote
2answers
69 views

Solving for $y$ in $y= 14x + 1000 = y= 16x + 800$

Given $y= 14x + 1000 = y= 16x + 800$, solve for $y$. I think I have it: $x= 100$. Subbing that in would leave me with: $y = 14x + 1000$ $y = 14 \cdot 100 + 1000$ $y = 2400$ $y = 16x + 800$ ...
4
votes
1answer
40 views

Sequence of nonzero digits with sum dividing decimal representation

Is there an infinite sequence of nonzero digits $a_1,a_2,\ldots$ such that $$a_1+a_2+\ldots+a_n\mid\overline{a_1a_2\ldots a_n}$$ for all $n\geq 1$, where $\overline{a_1a_2\ldots a_n}$ denotes the ...
1
vote
4answers
91 views

Inequalities and absolute values

My book asks that if $$-5\leq x\leq 1$$ then find the boundaries of absolute value of $x$. Can you please help me in finding that?
0
votes
2answers
39 views

Change in square root?

I'm looking at an example problem where $\sqrt{1-(x^2/36)}$ is changed to $\sqrt{36-x^2}$ with no explanation. How does that work?
4
votes
2answers
57 views

Rational sequence with $a_{n+1}=2a_n^2-1$

Suppose we start with a rational number $a_0$, and define $a_{n+1}=2a_n^2-1$ for $n\geq 0$. For what $a_0$ will it be the case that $a_i=a_j$ for some $i\neq j$? We can start with something like ...
-1
votes
5answers
106 views

Solving a Quadratic Equation

$f(x) = 9x^2 - 48x + 14$ I need help in solving this equation. I cannot simply factorise it, so do I need to use the 'quadratic' formula to solve it?
0
votes
1answer
63 views

solution of $y^2 - x = 15$ and $x^2 -xy = 2009$

Find all the integer solutions to the equations: \begin{eqnarray} y^2 - x &=& 15 \\ x^2 -xy &=& 2009 \end{eqnarray} Not sure how to solve this :/, tried the usual algebra way ...
2
votes
2answers
78 views

Use De Moivre's Theorem to determine $(-1 +i)^{184}$ in the form $x + iy$

Use De Moivre's Theorem to determine $(-1 +i)^{184}$ in the form $x + iy$ I first rewrite the equation in polar form. To do this I first determine $z$ $z = -1 + i$ I then solve $|z| = \sqrt{-1^2 + ...
0
votes
3answers
49 views

Removing logs from equation

I have a simple question that I need clarification on: If $$\log(a) = \log(b) + c$$ is it true that $$a = b + \exp(c)$$ Is this correct or am I missing something really basic that I cant ...
0
votes
2answers
54 views

System of equations with multiplication

I want to find all $x,y,z\in\mathbb{R}$ such that $(x+1)yz=12, (y+1)zx=4, (z+1)xy=4$. I can multiply all three equations to get $(x+1)(y+1)(z+1)x^2y^2z^2=192$. I can divide the first equation by the ...
1
vote
1answer
248 views

Prove using Jensen's Inequality

Let $\alpha_1, \alpha_2, . . . , \alpha_n$ be the interior angles of a convex (but not necessarily regular) n-gon. Prove, that for all integers $n\geq3$: $$\cos \alpha_1 + \cos \alpha_2 + \cdots + ...
0
votes
1answer
64 views

how can exponential of two terms be expressed as sum of the two terms?

How can exp(x+y) be expressed (or separated) as a sum (and NOT PRODUCTS) of exp(x) and exp(y) only? exp(x+y)=exp(x)exp(y), is there a way of having this a sum?
1
vote
3answers
2k views

Express $cos2\theta$ in terms of $cos$ and $sin$ (De Moivre's Theorem)

Use De Moivre's to express $cos2\theta$ in terms of powers of $sin$ and $cos$ What I have is: $cos2\theta + isin2\theta\\ = (cos\theta + i sin\theta)^2\\ = cos^2\theta + 2 cos\theta i sin\theta + (i ...
0
votes
3answers
717 views

How can I calculate large powers manually?

I was just wondering if there was nice, pencil and paper approach to calculating large powers with small base values. For example, can someone calculate $$(1.05)^{15}$$ or numbers of that sort on ...
0
votes
1answer
61 views

Given the index of an element in a triangular array, how do I find its row?

Consider a triangular array(numbers laid out in rows, where the r-th row contains r elements). Given the index i of an element in this array (assuming the numbers are laid out at indices 1, 2, 3, etc. ...
1
vote
1answer
41 views

Question on Polynomial function

Suppose $f(x)$ is a polynomial in $x$ having integer coefficients and $13 < a < b < c$ are integers such that $f(13) = f(a) = 13$ and $f(b) = f(c) = 19$: Determine the possible values of ...
2
votes
2answers
169 views

How do you find the value of an equation with nested fractions?

Am stuck on this problem in electronics as I have ran into a bit of algebra. Am generally not too bad with algebra but I cant for the life of me solve this equation: $V=IR$ $P=IV$ So $V = ...
1
vote
1answer
26 views

How can I know that the graph is symmetric with respect to the x axis or y axis?

How can I know that the graph is symmetric with respect to the x axis or y axis? $y=x^4-2x^2-21$ $9x^2+4y^2=36$ $y^2=x-4$
14
votes
2answers
207 views

Prove $\log_5{30}<\log_8{81}$

It's easy to prove this by calculator or computer, and I wonder can we prove that $$\log_5{30}<\log_8{81}\tag 1$$ by pencil and paper ? Thanks in advance ! Edit: $(1)$ can be written as ...
2
votes
1answer
44 views

Polynomial equation should be easy to solve for $c$ given $c \ne 1$?

So I am reviewing questions for a probability exam tomorrow, and have paused on the following Let $X$ be such that $\mathbf{P}(X=1) = p = 1 - \mathbf{P}(X = -1)$. Find $c \ne 1$ such that ...
1
vote
2answers
57 views

How do I find the range of this function?

$$f(x) = \frac{3x^2}{x^2 - 25}$$ How do I find the range of the function? I found its inverse $$ x = \frac{3y^2}{y^2 - 25}$$ but don't know how to isolate $y$. Are there any other ways to find the ...
0
votes
2answers
75 views

Prove completing the square

Prove that: $x + y + xy - x^2 - y^2 \leq 1$ If I use $-x^2 + xy - y^2$ to start completing the square, I get: $x + y -((x+y)(x-y)) - xy$ I am confused on how to keep going.
1
vote
3answers
85 views

Simplify the equation, containing radicals

I'm having trouble simplifying this equation, while keeping it as an exact expression. $x=\sqrt{28800-14400\sqrt{3}}$ I'm looking for the steps to change that above equation into ...
1
vote
1answer
135 views

Algebra Math Contest Question

A novel has 6 chapters. As usual, starting from the first chapter begins on a new page. The last chapter is the longest and the page numbers of its pages add up to 2010: How many pages are there in ...
2
votes
3answers
2k views

What is the value of sin(arcsin(4))?

In this case arcsin() is the unrestricted sin inverse function. I know that it is either undefined or has the value of 4. It could be undefined because arcsin() has only a doman of -1...1 and 4 is out ...
1
vote
1answer
59 views

For how many seconds do I need to turn the pedal?

For how many seconds do I need to turn the pedal of a bike, so that the number of turns is equal to the value of my velocity in the given moment measured in km/h.
1
vote
3answers
54 views

Algebraic simplification

I have never learned this in school, I only learned algebra when you have $x$ and numbers, in equations, like this: $$2x = 5(-2 + 5x)^2$$ I can solve that, but I cannot solve this one: $$-3(7 ...
1
vote
1answer
2k views

Rationalizing the denominator with 3 roots

Well, I can't find the example on how to solve this. If I multiply $$ \dfrac{2}{\sqrt[3]{9}+\sqrt[3]{15}+\sqrt[3]{25}} $$ with $$ ...
0
votes
4answers
112 views

How to simplify this?

How would I go by simplifying this: $$\frac{a-b}{2}+\frac{a+b}{3}-\frac{b-a}{4}$$ Also, this: $$\frac{a^2-16^2}{2a+8b}$$ Tried looking around, but letters in equations just fuzzles me.
0
votes
1answer
53 views

What is the solution to this problem

I'm sorry I can't be more specific in the title. I have this exercise I can't solve by myself, I've tried many times and I never get the right answer: $[(\frac{1}{a^2}-b^2):(\frac{1}{a}+b)]^{-1}$ This ...
0
votes
1answer
40 views

$2^{2x}-3 \cdot 2^{x+1}=16 \implies 2^x=-2$ or $2^x=8$

How did they get from $$2^{2x}-3 \cdot 2^{x+1}=16$$ to $2^x=-2$ or $2^x=8$? I know that you could also write it as $2^{2x}-3 \cdot 2^{x+1}$ and $2^{2x}-6\cdot 2^x=16$ But I got stuck there...
1
vote
1answer
462 views

Determining the minimum angle of a baseball trajectory to get a home run

This is a homework question: The skydome in Toronto has a center field fence that is 10 feet high and 400 feet from home plate. A ball is hit 3 feet above the ground with an initial velocity of ...
0
votes
1answer
35 views

If I have the equation, $\frac{p}{K-p}=Ce^{rt}$, how can I solve for p?

There is where I'm at: $$ \frac{p}{(K-p)}=Ce^{rt} $$ $$ p=(K-p)(Ce^{rt}) $$ $$ p=Ce^{rt}K-Ce^{rt}p $$ $$ p+Ce^{rt}p=Ce^{rt}K $$ $$ p(1+Ce^{rt})=Ce^{rt}K $$ $$ p=\frac{Ce^{rt}K}{1+Ce^{rt}} $$ Is it ...
1
vote
1answer
49 views

Mistake in simplification of large polynomial inequality?

We are to solve for $p$, and the inequality to simplify is $$10p^3(1-p)^2 + 5p^4(1-p) + p^5 - 3p^2(1-p) - p^3 > 0$$ On the next line of the textbook, the author simplifies this expression to ...
1
vote
0answers
40 views

Solving for $y$ from multiple $y$ terms

If you were given a problem like, $y^3+7y=16x^2-3x+2$, where there are multiple terms with $y$ of different powers in them, how would you solve for $y$? Also, are there many situations where you ...
0
votes
2answers
47 views

Prove that $X$ is a finite set

Base case: $7 \in X$ Recursive case: If $x \in X$, either $\dfrac{x}{2} \in X$ (if $x$ is even) or $3 \times x + 1 \in X$ (if $x$ is odd) Prove that $X$ is a finite set by explicitly listing all of ...
0
votes
3answers
391 views

Basic Math equation Question

I am finding it difficult to solve the following question, tried solving it by simultaneous equation method but the problem is the value of third equation is unknown. 2 oranges, 3 bananas and 4 ...
0
votes
1answer
44 views

Arithmetic difficulty

The sum of two people's age is 52. 8 years ago, one person was eight times as old as the other person. Provide two equations showing this and solve for both people's age.
1
vote
1answer
49 views

Need help solving an equation that includes percentages.

Three groups, Group A, Group B, Group C, of Sam's friends decided to buy a watch as a memorable gift for his birthday. They contributed 1200 AED in the ratio of 3:4:5. How much did each group ...
2
votes
3answers
70 views

Proving that a $f:\mathbb{R} \to (-1,1); x \mapsto \frac {x}{\sqrt{1+x^2}}$ is bijective

I am trying to prove that a function is bijective and I really am not sure how to go about it. I know that I must show that the function is both injective and surjective for it to be bijective. The ...