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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2answers
73 views

Proof of a summation of $k^4$

I am trying to prove $$\sum_{k=1}^n k^4$$ I am supposed to use the method where $$(n+1)^5 = \sum_{k=1}^n(k+1)^5 - \sum_{k=1}^nk^5$$ So I have done that and and after reindexing and a little algebra, ...
2
votes
6answers
122 views

Show that ${n \choose 1} + {n \choose 3} +\cdots = {n \choose 0} + {n \choose 2}+\cdots$ [duplicate]

Show $${n \choose 1} + {n \choose 3} +\cdots = {n \choose 0} + {n \choose 2}+\cdots$$ A hint is given to consider the expansion $(x-y)^n$ However, when I plug in a number for $n$, I don't get an ...
2
votes
4answers
238 views

Are these proofs logically equivalent?

Here are two proofs, firstly: x = 0.999... 10x = 9.999... = 9 + 0.999... = 9 + x 9x = 9 x = 1 And secondly: ...
2
votes
2answers
139 views

How many different proofs are there that $a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $?

How many different proofs are there that $a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $ for positive integer $n$ and real $a, b$? You can use any techniques you want. My proof just uses algebra, ...
2
votes
3answers
120 views

Mathematical way to solve integer numbers $217 = (20x+3)r+x$

Is there any mathematical way to find the integer numbers that solve the following equation: $$217 = (20x+3)r+x$$
2
votes
0answers
84 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 ...
2
votes
1answer
36 views

How do you prove a function is defined for a a certain set?

Spin-off from here. Context: Highschool textbooks often ask students to find the domain of functions. Let's say $f(x) = x+2$. The domain is $\mathbb{R}$...suppose a student (highschool or o/w) asks ...
2
votes
2answers
111 views

How do you prove the domain of a function?

Suppose we have a function, say, $f(x) = x+2$. Its domain is $\mathbb{R}$. How do you prove this? Or is this something not needed to be proven since it is "defined" $\forall$ x $\in \mathbb{R}$? If ...
2
votes
3answers
271 views

Translating text to functions

I am having problems understanding how to extract this information into a formula. ...
2
votes
2answers
63 views

Prove that $\binom {n}{k} = \frac {n!} {(n-k)!k!}$, viewed as a function of $k$, has maximum at $k=\lfloor n/2 \rfloor, \lceil n/2 \rceil$. [duplicate]

Prove that the binomial coefficient $\binom {n}{k} = \frac {n!} {(n-k)!k!}$, viewed as a function of $k$, has maximum at $k=\lfloor n/2 \rfloor, \lceil n/2 \rceil$ if $n$ is odd and maximum at $k=n/2$ ...
2
votes
3answers
179 views

Proof that $\sqrt{x}=-\sqrt{x}$ [duplicate]

$\sqrt{x}=\sqrt{1\cdot x}=\sqrt{(-1)^2\cdot x} = \sqrt{(-1)^2} \cdot \sqrt{x} = (-1) \cdot \sqrt{x}=-\sqrt{x}$ The idea popped into my head while I was evaluating an integral. I have a feeling that I ...
2
votes
3answers
2k views

Finding surface area of part of a plane that lies inside a cylinder???

I have a question:: Let $S$ be the part of plane $x+2y+3z=1$ that lies inside cylinder $x^2 + y^2 = 3$ They want me to find the surface area of S?? This is a way harder question than all my ...
2
votes
2answers
32 views

no. of Digit in $x^y\;,$ where $x,y\in \mathbb{N}$

$(1)$:: Calculation of no. of Digits in $2^{100}$ .$(2)$:: Calculation of no. of Digits in $3^{100}$. If it is given that $\log_{10}(2)=0.3010$ and $\log_{10}(3) = 0.4771$ $\bf{My\; Try::}$ I have ...
2
votes
4answers
117 views

Notation: is it correct to state $3a=a3$?

If $a$ is a real constant, do you regard $3a$ and $a3$ as equal or different?
2
votes
2answers
688 views

Derivation of slope of line formula

The formula for slope of a line as we know: $y_2 - y_1/x_2 - x_1$ or just rise / run What is the derivation for this formula? E.g. Why is it not rise times run for example?
2
votes
2answers
10k views

How to calculate the percentage of increase/decrease with negative numbers?

I feel like an idiot for asking this but i can't get my formula to work with negative numbers assume you want to know the percentage of an increase/decrease between numbers ...
2
votes
5answers
245 views

Derivation of factorization of $a^n-b^n$

How does one prove that: $$a^n-b^n=(a-b)\left(a^{n-1}+a^{n-2}b+a^{n-3}b^2+\dots+a^2b^{n-3}+ab^{n-2}+b^{n-1}\right)$$ Better yet, why is $a^n-b^n$ divisible by $a-b$? I would very much appreciate some ...
2
votes
6answers
140 views

for $n$ an integer, why is $n^0=1$ ??

This is so going to cost me.... I was wondering why for any integer $n$: $n^0 =1$. Perhaps It's because $n$ is a round number and if $m$ is a non negative integer, also round then: $$n^m = 1 \cdot ...
2
votes
2answers
283 views

Finding the “triangular root” of a number.

A triangular number is a number that is the sum of the natural numbers up to some $n$. The closed form is $x = \frac{n(n+1)}{2}$. How do I get $n$ on one side? I've been looking at it from every ...
2
votes
2answers
829 views

Exponent rule and square roots?

For some $x$, $\sqrt{x^2} = |x|$ However, for $x= -1$. $\sqrt{(-1)^2} = (-1^2)^{1/2} = (-1)^{2/2} = (-1)^1 = -1$ Isn't this paradoxical?
2
votes
3answers
107 views

Combinatorial proofs of the identity $(a+b)^2 = a^2 +b^2 +2ab$

The question I have is to give a combinatorial proof of the identity $(a+b)^2 = a^2 +b^2 +2ab$. I understand the concept of combinatorial proofs but am having some trouble getting started with this ...
2
votes
1answer
46 views

Simplify the following indices

$3^{x+4} * 5^{x+7} * 15^{2x-1}$ I tried it in this way: $3^{x+4}*5^{x+1}*(3*5)^{2x-1}$ Then: $3^{x+4}*5^{x+1}*3^{2x-1}*5^{2x-1}$ And what about next?
2
votes
4answers
325 views

Simplify the surd expression.

Simplify the surd. $(2\sqrt 3 + 3\sqrt 2)^2$ I know I should us this formula: $(a^2+2ab+b^2)$ But this gets complicated later. Please explain. :(
2
votes
2answers
75 views

Determining the domain of a function

What is the domain of: $$\left(\frac{5x+4}{x^2+9x+8}\right)^{1/3}$$ I got $(-\infty, -8) \cup (-8,-1) \cup (-1, \infty).$ But according to Wolfram Alpha it is $(-8, -1) \cap [-4/5, \infty)$. Could ...
2
votes
3answers
231 views

Quadratic formula - math error

I'm attempting a past paper and I have been asked to compute the derivative for $(x^2-2x+2)$ and from this I calculated $2x-2$. Once I completed this, I was then asked to find and classify the ...
2
votes
2answers
32 views

What can be said about a function that is odd (or even) with respect to two distinct points?

This question is a little open-ended, but suppose $f : \mathbb R \to \mathbb R$ is odd with respect to two points; i.e. there exist $x_0$ and $x_1$ (and for simplicity, let's take $x_0 = 0$) such that ...
2
votes
1answer
77 views

Showing $(a+b+c)(x+y+z)=ax+by+cz$ given other facts

$$x^2-yz/a=y^2-zx/b=z^2-xy/c$$ None of these fractions are equal to 0.We need to show that, $(a+b+c)(x+y+z)=ax+by+cz$ This question comes from a chapter that wholly deals with factoring ...
2
votes
1answer
2k views

Determine the cube roots of -8 in polar form

Exam time tomorrow and I am not entirely sure if I am doing this right. I first write -8 as a complex number $z^3 = -8 = -8 \times 0i$ Calculate the modulus of z $|z| = \sqrt{-8^2} = 8$ Get the arg ...
2
votes
2answers
74 views

How to prove: $-\frac{1}{\sec2x}=\frac{\cos^3x-\sin^3x}{\cos x +\sin x}+\frac{\cos2x}{(\cos x +\sin x)^2}$

How do you do it? I'm really stuck on this proof. Can someone please explain? Thanks
2
votes
3answers
113 views

Prove that for real numbers $x$, if $x^2 - 5x + 4 \ge 0$, then either $x \le 1$ or $x \ge 4$.

Its another homework question that I'm having trouble understanding. The full question is write a detailed structured proof that uses a proof by cases to prove that for real numbers $x$, if $x^2 - 5x ...
2
votes
1answer
115 views

Help with understanding a proof that $f$ is bounded on $[a,b]$ (Spivak)

I need help on the proof of Theorem 7-2 in Spivak: If $f$ is continous on $[a,b]$, then $f$ is bounded above on $[a,b]$. So, the proof starts with this: Let $$A= \{x:a\le x \le b \text{ ...
2
votes
3answers
151 views

Inequalities - Absolute Value $|2x-1| \leq |x-3|$

$$|2x-1| \leq |x-3|$$ Answer is $$-2 \leq x \leq \frac43$$ My Question is HOW?
2
votes
2answers
98 views

Questions on powers of a bijection $f\colon\{1,2,\dots,n\}\to\{1,2,\dots,n\}$ [closed]

Let $f$ be a one-to-one function from $X=\{1,2,\dots,n\}$ onto $X$. Let $f^k=f\circ f\circ \cdots \circ f$ denote the $k$-fold composition of $f$ with itself. Show that there are distinct positive ...
2
votes
2answers
125 views

Partial fractions of $\frac{-5x+19}{(x-1/2)(x+1/3)}$

Alright, I need to find the partial fractions for the expression above. I have tried writing this as $$\frac{a}{x-1/2}+\frac{b}{x+1/3}$$ but the results give me $a=25.8$ and $b=-20.8$, which are ...
2
votes
1answer
38 views

Stuck on rearranging of this equation

I need to get from $[(1-p)f+p(1-f)](1+v)-[(1-p)(1-f)+pf] = x$ to $(2+v)(f+p-2pf)-1 = x$ but I'm stuck. I'd appreciate any tips on what I should I do after the following. $(f+p-2pf)(1+v) + (f + p ...
2
votes
1answer
2k views

Quadratic equation which has rational roots

If the following quadratic equation $$qx^2+(p+q)x+bp=0$$ always has rational roots for any non-zero integers $p$ and $q$ what will be the value of $b$? My book's solution says the value of ...
2
votes
3answers
313 views

How do you know when you can substitute certain limits into others?

I know there are some limits where you can't to certain substitutions such as $\sin(x)=x$ as $x$ approaches $0$. How do you know when you can or can't do that? I wish I could give you an example ...
2
votes
3answers
138 views

A tricky logarithms problem?

$ \log_{4n} 40 \sqrt{3} \ = \ \log_{3n} 45$. Find $n^3$. Any hints? Thanks!
2
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4answers
2k views

Solve the equation $\sqrt{3x-2} +2-x=0$

Solve the equation: $$\sqrt{3x-2} +2-x=0$$ I squared both equations $$(\sqrt{3x-2})^2 (+2-x)^2= 0$$ I got $$3x-2 + 4 -4x + x^2$$ I then combined like terms $x^2 -1x +2$ However, that can not ...
2
votes
3answers
298 views

Spivak problem on Schwarz inequality

I have a question regarding problem 19 in the 3rd Ed. of Spivak's Calculus. Specifically, part (a). The question concerns the Schwarz inequality: $$ x_1y_1 + x_2y_2 \leq ...
2
votes
2answers
71 views

Slope of a straight line

Why is this so that a higher value of slope indicates a steeper incline? I can't take it into my head. What could be the reason behind that? I know that it is a fact because I've also noticed it but ...
2
votes
1answer
147 views

License plate consisting of 4 letters and 4 numbers

While doing homework today, the following question popped into my head: Can you easily calculate the amount of unique license plates consisting of 4 letters and 4 numbers in any order? It doesn't ...
2
votes
1answer
213 views

Given that $\sec \theta = k$, $|k| \geq 1$, and that $\theta$ is obtuse, express in terms of $k$: $\cos \theta$, and $\csc \theta$

Given that $\sec \theta = k$, $|k| \geq 1$, and that $\theta$ is obtuse, express in terms of $k$: $\cos \theta$, and $\csc \theta$ For $\cos \theta$ I get: $\frac{-1}{k}, (as \theta$ is obtuse, ...
2
votes
1answer
4k views

What are the most famous (common used) precalculus books and its differences?

I'm trying to decide which one to pick up to begin a self study of mathematics. One of the factors is how much content is covered and the amount of associated solved problems the book has. EDIT: ...
2
votes
1answer
3k views

Perpendicular line passing through the midpoint of another line

I have several $2d$ line segments. for example, if I take a one line segment having end points $(x_1, y_1)$ and $(x_2, y_2)$. Then, I want to make a perpendicular line which passes through the ...
2
votes
3answers
218 views

Can $n(n+1)2^{n-2} = \sum_{i=1}^{n} i^2 \binom{n}{i}$ be derived from the binomial theorem?

Can this identity be derived from the binomial theorem? $$n(n+1)2^{n-2} = \sum_{i=1}^{n} i^2 \binom{n}{i}$$ I tried starting from $2^n = \displaystyle\sum_{i=0}^{n} \binom{n}{i}$ and dividing it ...
2
votes
4answers
2k views

Factoring Cubic Equations

I’ve been trying to figure out how to factor cubic equations by studying a few worksheets online such as the one here and was wondering is there any generalized way of factoring these types of ...
2
votes
2answers
203 views

The Roots of Unity and the diagonals of the n-gon inscribed in the unit circle

I want to prove that the sum of the squares of the diagonals of a regular $n$-gon inscribed in the unit circle is equal to $2n$. So what I've done is I considered the $n$th roots of unity and said ...
2
votes
4answers
169 views

Fast square roots

I need to compute the square roots of lots of numbers. The numbers increase monotonically by fixed step. For example, 1, 2, 3, ..., 1 000 000. What is the fastest way to do so? Is it possible somehow ...
2
votes
8answers
242 views

Why does $\frac{a}{\frac{b}{x}} = x \times \frac{a}{b}$?

As much as it embarasses me to say it, but I always had a hard time understanding the following equality: $$ \frac{a}{\frac{b}{x}} = x \times \frac{a}{b} $$ I always thought that the left-hand side ...