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
votes
0answers
55 views

Solve this equation for x

I've come up with an equation whilst solving a problem but I need to rearrange it for $x$. Putting it in Wolfram Alpha doesn't give me anything. This is the equation $$(1+x)^c - (1-x)^c = d.$$ $c,d$ ...
-1
votes
0answers
32 views

Interesting and challenging problem [on hold]

I've been given this problem to solve, but didn't succeed until now. Can you help me? A city has 5 billion paper money (bills) in circulation. Thirty million of them are taken daily to the bank ...
0
votes
1answer
24 views

Summation operation for precalculus

Studying Spivak's Calculus I came across a relation I find hard to grasp. In particular, I want to understand it without using proofs by induction. So please prove or explain the following ...
-5
votes
1answer
23 views

Find the domain and range of $f$ and $f^{-1}$ [on hold]

Find the domain and range of $f$ and $f^{−1}$ $f(x) = x^2 − 9, \ \ \ x \le 0$ $f^{−1}(x) = -\sqrt{x+9}$
-8
votes
1answer
43 views

Simplify $2^3-3^{\frac{5}{8}}+2^2+3^{\frac{5}{8}}+2^1$ [on hold]

How can I simplify this expression? I really need to know how. $2^3-3^{\frac{5}{8}}+2^2+3^{\frac{5}{8}}+2^1$
-1
votes
1answer
38 views

Is this Factored out fully? (Exponents) [on hold]

$2x^2 + 32$ $\Rightarrow$ $2(x + 4)^2$ Is this correct?
4
votes
7answers
71 views

Calculate $\lim_{x \to 0} \frac{e^{3x} - 1}{e^{4x} - 1}$

Question: Calculate $$\lim_{x \to 0} \frac{e^{3x} - 1}{e^{4x} - 1}$$ using substitution, cancellation, factoring etc. and common standard limits (i.e. not by L'Hôpital's rule). Attempted ...
0
votes
0answers
16 views

transforming an equation into a difference equation

I know how to rewrite a differential equation into a difference equation using Euler's forward difference. However, I'm at a loss as how to convert a given equation into a difference equation. For ...
3
votes
3answers
259 views

Solving a Radical Equation $5(\sqrt{1-x} + \sqrt{1+x}) = 6x + 8\sqrt{1-x^2}$ (squaring doesn't help)

How should I approach this problem: $$ 5(\sqrt{1-x} + \sqrt{1+x}) = 6x + 8\sqrt{1-x^2} $$ I've tried squaring both sides but to get rid of all the radicals requires turning it into a quartic equation, ...
2
votes
2answers
17 views

How do I use the $A= Pe^{rt}$ formula in this question?

So the question is in $2000$ the deer population in a certain area was $800$. The number of deer increases exponentially at a rate of $7%$ per year. Predict the population in $2009$. a) $1408$ b) ...
0
votes
1answer
33 views

Working out “break even” point

Please bear with me, my brain is hopeless at math. My colleague has a Jeep Grand Cherokee. He had a chip installed which cost him $\textrm{ZAR}3500$. He wants to know when his purchase of the chip ...
0
votes
1answer
43 views

Confusion with Summations

I am having a little bit of confusion regarding summations. I know that $$\sum_{i=m}^n a_i = a_{m}+a_{m+1}+\cdots +a_{n-1}+a_n$$ Here is my confusion. How do we interpret/decompose the following: ...
-3
votes
0answers
27 views

Synthetic division [on hold]

This is about changing fractions into a mixed expression. So I have to do divide them. But I don't know why this problem tells me to leave spaces! Here is the problem: $\dfrac{k^3 - 1}{k - 1}$ ...
2
votes
4answers
345 views

How to solve certain types of integrals

I'm asking for a walk through of integrals in the form: $$\int \frac{a(x)}{b(x)}\,dx$$ where both $a(x)$ and $b(x)$ are polynomials in their lowest terms. For instance $$\int ...
0
votes
1answer
86 views

Why is $\sqrt{X}\times\sqrt{X}=X$?

Today I was solving the limit $(\ln(x))/(2*(x^{1/2})$ but then faced the step after applying the derivation that ended up with $(1/x)/(1/x^{1/2})$ and the result of that was $1/x^{1/2}$. When I asked ...
0
votes
2answers
41 views

Proof of $xyz+1= 2yz$, Given $x=\log_{2a}a$, $y=\log_{3a} 2a$, $z=\log_{4a} 3a$ [on hold]

Proof of $xyz+1= 2yz$, Given $x=\log_{2a}a$, $y=\log_{3a} 2a$, $z=\log_{4a} 3a$
1
vote
3answers
19 views

Complex plane (Show that triangle is right-angled)

The points $O$,$P$ and $Q$ in the complex plane represent the complex numbers $0+0i$, $4+2i$ and $3-i$ respectively. Find the exact length of $PQ$ and hence, or otherwise, show that triangle $OPQ$ is ...
1
vote
1answer
32 views

Complex Number (Angle)

The complex number $z$ is given by $z=-2+2i$ Find the modulus and argument of $z$ Write down the modulus and argument of $\frac{1}{z}$ Show on an Argand diagram the points A,B and C representing the ...
3
votes
1answer
30 views

Proof by induction from Spivak's calculus ch 2- 3b

I was cracking my head over the following proof (by induction) from Spivak's calculus. Givens: $ \binom{n+1}{k}=\binom{n}{k-1}+\binom{n}{k} $ and $ n \ge k $ Task: Proof by induction that $ ...
0
votes
1answer
28 views

Complex Numbers (Find p and q)

The complex numbers z1 and z2 are given by $$z_1=5+i,z_2=2-3i$$ Determine the values of the real constants $p$ and $q$ such that $$\frac{p+iq+3z_1}{p-iq+3z_2}=2i$$ My attempt, I substitute $z_1$ and ...
0
votes
3answers
108 views

Complex Numbers (Modulus)

The complex numbers $z_1$ and $z_2$ are given by $$z_1=5+i,z_2=2-3i$$ Find the modulus of $z_1-z_2$ My attempt, modulus of $z_1-z_2=\sqrt{5^2+1^2}-\sqrt{2^2+3^2}$ $=\sqrt{26}-\sqrt{13}$ ...
0
votes
1answer
32 views

vertices of a hyperbola the silliest question ever

I'm given that the center of the hyperbola is $(2,1)$ and $a=3$ and asked to find the vertices. Since vertices are on the same line with the axis of symmetry I thought the coordinates should be $(2,1 ...
2
votes
1answer
58 views

Solutions of $\sqrt{x+4+2\sqrt{x+3}}-(x^2+4x+3)^{1/3}=1$

$\sqrt{x+4+2\sqrt{x+3}}-(x^2+4x+3)^{1/3}=1$ I get that $-3$ as a solution, but apparently 1 is as well a solution, and I don't see a mechanism through which I could find it. Any help would be ...
0
votes
3answers
62 views

Prove that A is invertible if $A^2 - 4A -7I = 0$. [duplicate]

The $2 \times 2$ matrix $A$ satisfies $$A^2 - 4A -7I = 0,$$ where $I$ is the identity matrix. Prove that $A$ is invertible. I'm not sure how to do this. Help would be appreciated.
0
votes
1answer
20 views

Let ${A}$ be a $2 \times 2$ matrix. For every two-dimensional vector ${v}$, there… [duplicate]

Let ${A}$ be a $2 \times 2$ matrix. For every two-dimensional vector ${v}$, there exists a two-dimensional vector ${w}$ such that Aw = v. Show that ${A}$ is invertible. I'm not sure how to do this.
1
vote
1answer
95 views

Proving that matrix in equation is invertible

The $2 \times 2$ matrix ${A}$ satisfies ${A}^2 - 4 {A} - 7 {I} = {0}$ where ${I}$ is the $2 \times 2$ identity matrix. Prove that ${A}$ is invertible. I have tried to solve it like a quadratic, but ...
0
votes
2answers
30 views

Solving $f(x) = x^5 +x + 1 = 0$ with halving the interval / bisection method

Question: Use halving the interval / bisection method to approximately solve: $$f(x) = x^5+ x + 1 = 0$$ with a precision of $\pm 0.1$ Attempted solution: The general idea, as I understand it, is ...
0
votes
2answers
18 views

Complex number (Rhombus)

Given that $z_1=1+2i$ and $z_2=\frac{3}{5}+\frac{4}{5}i$, write $z_1z_2$ and $\frac{z_1}{z_2}$ in the form $p+iq$, where $p$ and $q \in R$. In an Argand diagram, the origin O and the points ...
-1
votes
1answer
22 views

Argument of Complex Number (Am I wrong?)

I'm given $z=-2+\sqrt{3}i$. So I worked out the argument of $arg(z)=\tan^{-1}(\frac{\sqrt{3}}{-2})$. I got the answer $2.256$rad. But the given answer is $2.45$rad. Am I wrong?
-1
votes
1answer
28 views

Doubt with Intervals and Inequalities

This doubt has been bothering me for ages. I would be truly grateful for any help. Problem 1: $\dfrac{2}{|x-4|}>1$ Express the solutions using intervals Solution: $x\in(2,4)\cup(4,6)$ ...
2
votes
3answers
28 views

Absolute Value Inequality Problem

Problem: $\dfrac{2}{|x-4|}>1$ Express the solutions using intervals My attempt using the Definition of Modulus: $$\dfrac{2-|x-4|}{|x-4|}>0$$ $$CASE A:x-4\ge 0\Rightarrow x\ge4\Rightarrow ...
1
vote
1answer
34 views

Simplify the expression and find the minimum value

I want to simplify the expression \begin{equation} A = \frac{\sqrt{1-\sqrt{1-x^2}}\Big[\sqrt{(1+x)^3} + \sqrt{(1-x)^3} \Big]}{2-\sqrt{1-\sqrt{1-x^2}}} \end{equation} and find the minimum value of ...
1
vote
1answer
27 views

On finite sums and products

I'd like to get a good book on finite summations and products before I study infinite series more in depth next year. The book should cover geometric/ harmonic sums and prove different formulas for ...
1
vote
0answers
28 views

Calculating a formula for variables with multiple values equaling the same total

I'm having a bit of trouble puzzling a formula for some code I'm using to develop a piece of software. I'm not very savvy with what the technical terms for all of what I'm describing are, but I'll try ...
0
votes
3answers
39 views

Complex number $\tan \alpha+i$

Given that $z=\tan \alpha+i$, where $0<\alpha<\frac{1}{2}\pi$ Find $\left |z \right |$. I've never seen this kind of example in my book. Can anyone guide me? Thanks a lot. How to find $arg ...
0
votes
1answer
25 views

Quick Factoring/Multiplying with recursion question

I am wondering if anyone can help to shed some light on something that I think should be very easy but I dont quite understand. In my textbook, how does author make this conclusion, From $$ ...
3
votes
2answers
176 views

Square roots of Complex Number. [duplicate]

Calculate, in the form $a+ib$, where $a,b\in \Bbb R$, the square roots of $16-30i$. My attempt with $(a+ib)^2 =16-30i$ makes me get $a^2+b^2=16$ and $2ab=−30$. Is this correct?
0
votes
2answers
39 views

Find the number of possible values of $a$

Positive integers $a, b, c$, and $d$ satisfy $a > b > c > d, a + b + c + d = 2010$, and $a^2 − b^2 + c^2 − d^2 = 2010$. Find the number of possible values of $a.$ Obviously, factoring, ...
2
votes
4answers
62 views

Trigonometric equation $\sec(3\theta/2) = -2$ - brain dead

Find $\theta$ with $\sec(3\theta/2)=-2$ on the interval $[0, 2\pi]$. I started off with $\cos(3 \theta/2)=-1/2$, thus $3\theta/2 = 2\pi/3$, but I don't know what to do afterwards, the answer should be ...
2
votes
2answers
25 views

Ratios and percents

Mrs. Smith has 80 birds: geese, hens and ducks. The ratio of geese to hens is 1:3. 60% of the birds are ducks. How many geese does Mrs. Smith have? a) 16 b) 8 c) 12 d) 11 I know that if 60% if ...
0
votes
5answers
44 views

How does this seemingly-trivial simplification work?

In a section on inductive proofs in the book Modelling Computing Systems: Mathematics for Computer Science (Muller, Struth) there is a simplification that is assumed to be trivial, but that I can't ...
3
votes
1answer
48 views

weird trig problem $\tan(\theta)=-\sqrt{2}\sin(\theta)$ on the interval $0 \leq \theta \leq 2\pi$

$\tan(\theta)=-\sqrt{2}\sin(\theta)$ on the interval $0 \leq \theta \lt 2\pi$ I started off with $[(\sin(\theta)/\cos(\theta)] \times (1/\sin(\theta) )= - \sqrt 2$, then after simplification i got ...
2
votes
2answers
99 views

Understanding why $a+b\sqrt {2}\neq \sqrt {3} $

I want to intuitively understand why $a+b\sqrt {2}\neq \sqrt {3} $ for $a, b \in \mathbb Q $ I really have no intuition regarding this matter, and have to deal with similar concepts regularly while ...
0
votes
1answer
34 views

Question regarding a loan from a bank for my friend

I have a question please, I'm having a difficult times calculating the right way this one... I took a loan on July 3 2014 of \$50,000 from my bank for my friend, split it on 12 installments (about ...
4
votes
1answer
40 views

Find the range of a $4$th-degree function

For the function $y=(x-1)(x-2)(x-3)(x-4)$, I see graphically that the range is $\ge-1$. But I cannot find a way to determine the range algebraically?
2
votes
2answers
53 views

$x^n + y^n = z^n$, $n>1$ To show that $x,y,z$ is greater than $n$

Problem: If $x$,$y$,$z$ and $n>1$ are natural numbers with $$x^n+y^n = z^n$$ then show that x,y and z are all greater then $n$. My approach, from Fermat's Theorem we know that $x^n + y^n = z^n$ ...
0
votes
0answers
38 views

Sum of zeros polynomial

There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x)={x}^{3}-a{x}^{2}+bx-65$. For each possible combination of $a$ and $b$, let ...
25
votes
11answers
1k views

Computing irrational numbers

I am genuinely curious, how do people compute decimal digits of irrational numbers in general, and $\pi$ or nth roots of integers in particular? How do they reach arbitrary accuracy?
1
vote
1answer
20 views

No. f ordered pair $(a,r)$ in Logarithmic equation.

If $a_{1},a_{2},a_{3},.............$ be a Geometric Progression, Where $a_{1} = a$ and common ratio $r$ are positive integers. If $\displaystyle ...
1
vote
3answers
35 views

Why $x=u+v$ substitution works?

I have the solution for the follwoing example : $$x^4+y^4=82$$ $$x-y=2$$ The author substitutes $x=u+v$ and $y=u-v$ My question is: If we have two numbers ($x, y$), can we always find ...