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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5
votes
2answers
38 views

A simple way to find $\lim_{n\rightarrow\infty}{\frac{1}{n^2}\sum_{k=1}^n{\sqrt{n^2-k^2}}}$

I was reading an exam paper used to identify gifted high-school students, and I encountered the following problem: $$\lim_{n\rightarrow\infty}{\frac{1}{n^2}\sum_{k=1}^n{\sqrt{n^2-k^2}}}$$ Using ...
1
vote
1answer
11 views

To find, wether '1' lies in the range of f, where $f(x)=[ln(\frac{7x-x^2}{12})]^\frac{3}{2}$?

$f(x)=[ln(\frac{7x-x^2}{12})]^\frac{3}{2}$, For the given function, the question is whether, f(x) can equal 1 for some real value of x?
1
vote
1answer
37 views

Is this a correct way to use triangle inequality

If I have: $$|g_1(x) - g_2(x) - (g_1(a) - g_2(a))| \leq f(x^*)$$ Can I proceed to say: $$|g_1(x) - g_2(x) - (g_1(a) - g_2(a))| \leq |g_1(x) - g_2(x)| - |(g_1(a) - g_2(a))|$$ $$ \implies |g_1(x) - ...
0
votes
2answers
22 views

Rewriting a particular sequence in respect to inverses

I'm having a large amount of difficulty on piecing together the intermediate algebra between the following formulas. $$ \frac{n^2 + 1}{2n^2 - 3} = \cdots = \frac {1 + \frac{1}{n ^ 2}}{2 - ...
0
votes
0answers
38 views

Show that the expansion of $(1+x)^n$ by Binomial Theorem is convergent when $x<1$

To show that the expansion of $(1+x)^n$ by Binomial Theorem is convergent when $x<1$ Let $u_r, u_{r+1}$ represent the $r^{th}$ and $(r + 1)^{th}$ terms of the expansion; then ...
2
votes
1answer
35 views

If $ x=\frac{\sin^3 t}{\sqrt{\cos 2t}}$ and $y = \frac{\cos^3 t}{\sqrt{\cos 2t}}\;,$ Then $\displaystyle \frac{dy}{dx}$ in terms of $t$

If $\displaystyle x=\frac{\sin^3 t}{\sqrt{\cos 2t}}$ and $\displaystyle y = \frac{\cos^3 t}{\sqrt{\cos 2t}}\;,$ Then $\displaystyle \frac{dy}{dx}$ in terms of $t$ $\bf{My\; Try::}$ Using The Formula ...
1
vote
2answers
31 views

partial fraction decomposition braindead

decompose $\frac{x^2-2x+3}{(x-1)^2(x^2+4)}$ the way my teacher wants us to solve is by substitution values for x, I set it up like this: (after setting the variables to the common denominator and ...
1
vote
0answers
21 views

Complex vector identity

Let $f=(f_1,f_2,f_3)$ be a complex vector. Can we see that $$G:=\frac{2\Im(f_2\bar{f_3})+i2\Im(f_3\bar{f_1})}{|f|^2-2\Im(f_1\bar{f_2})}=\frac{f_3}{f_1-if_2}$$ I tried using $f_j=x_{j,u}-ix_{j,v}$ ...
1
vote
2answers
53 views

Graph of the function $y = 2 + (x + 1)^3$

I know that this function will have the behavior of $Y = X^3$ but as I will translate for this function $(Y = X^3)$? I do this: $$(x + 1)^3 = x^3 + 3x^2 + 3x + 3 \quad y = x^3 + 3x^2 + 3x + 5$$ But ...
2
votes
1answer
31 views

Function with Multiple Periods

Basically I'm trying to fit some data with seasonal effects to a periodic function, and the problem I'm running into is that the local minima usually occur around April, and the local maxima usually ...
2
votes
2answers
43 views

Find h in terms of r

A sphere and a cylinder have equal volumes. The sphere has a radius 3r. The cylinder has radius 2r and height h. Find h in terms of r. I'm only 15, someone walk me through this as simple as ...
2
votes
2answers
28 views

Simplifying Surds, or square root fractions

So I have to write $\sqrt{2\over 18}$ in its simplest form. How would I work this out?
1
vote
3answers
63 views

Finding the formulae in terms

The cost, $£C$, of building a circular pond is proportional to the square of its diameter, $d$ meters. A pond with diameter $2$ meters costs $£52$. Find the formulae for $C$ in terms of $d$. Okay ...
3
votes
2answers
36 views

Proving this two equations are same and true

If $\sqrt{a} - \frac{1}{\sqrt{a}} = 1$, then $a + \frac{1}{a} = 3$. Why this statement is true? I tried to square the first equation, but it didn't work. I can't understand why there is a 3 in the ...
-2
votes
1answer
46 views

Polynomial and squares

Let f be the polynomial $f\in\mathbb{Z[x]}$ defined by $f(x)=x^4-22x^3+135x^2-154x-34$. How many times f(n) is a perfect square when $n\in\mathbb{Z}$ ? This problem I solved another way than the ...
-2
votes
2answers
42 views

Proving the equation has no root. [on hold]

How to show that for $a\in \mathbb R$, the equation $x^2+12a^2+4ax-8a+8=0$ has no root?
5
votes
2answers
80 views

$3^x + 4^y = 5^z$ [duplicate]

This is an advanced high-school problem. Find all natural $x,y$, and $z$ such that $3^x + 4^y = 5^z$. The only obvious solution I can see is $x=y=z=2$. Are there any other solutions?
2
votes
4answers
52 views

Sum of Series as $1,(2),1,(2,2),1,(2,2,2),1,(2,2,2,2),1…$

The Sum of First $2015$ terms of the Series... $1,2,1,2,2,1,2,2,2,1,2,2,2,2,1,.......................$ $\bf{My\; Try::}$ We Can Write the Given Series as ...
6
votes
3answers
50 views

“Rationalizing the denominator” of $1/(a + b\sqrt[3]{2} + c\sqrt[3]{4})$?

If $(a, b, c) \in \mathbb{Q}^3 \setminus \{(0, 0, 0)\}$, so that $a + b\sqrt[3]{2} + c\sqrt[3]{4}$ is a nonzero element of $\mathbb{Q}(\sqrt[3]{2})$, is there a formula for $${1\over{a + b\sqrt[3]{2} ...
-3
votes
0answers
30 views

Change the subject of a formula [on hold]

$150 \cdot 10^6 = \dfrac{3pR^2}{4t^2}$ How do I find out what $t$ is, hence make it the subject of the equation. I think I know what the answer should be: $p=1.5 \cdot 10^6$ $R= 0.075$ ...
1
vote
1answer
23 views

Question in regard to solving for inverse laplace transform

I am having some confusion when it comes to solving for the inverse laplace transform. ( We are allowed the tables with the common values by the way). Il give an example. Take, ...
-1
votes
0answers
21 views

Excel Exponential Line Values [on hold]

I am trying to use Excel to graph an Exponential Trendline, but would also like to extract the values to be used in a spreadsheet I am developing. The trendline formula is showing as: $y = ...
-4
votes
2answers
39 views

Logarithm with nth root [on hold]

I made it but the result is very strange. I want every step to the result $$ \large 6\log_{10}\frac{\sqrt2}{\sqrt[3]{3+\sqrt5}} $$
1
vote
1answer
26 views

Expanding a term with a sum

We have the following quantity: $$E\left[\left(\sum^n_{j=1} (X(t_j) - X(t_{j-1}))^2-t\right)^2\right]$$ My textbook says this can be expanded in the following way (colors are my touch) ...
-4
votes
0answers
26 views

solve the equation for the maximum positive integral value [on hold]

$$\large\displaystyle \sum_{x=1}^{\infty} \displaystyle \log_{n}\left(\frac{(x+a-1)(x+a+1)}{(x+a-3)(x+a+3)}\right)=1$$ How do I solve the above equation for the maximum positive integral value of $n$ ...
1
vote
0answers
27 views

Simplifying cyclometric function

How does one simplify this function? $$ f(x) = \arccos(\frac{\pi}{2} - \sin(x)) $$ A plot in GeoGebra showed a graph that looked like semicircle, so can one expect something in this form: ...
-4
votes
2answers
54 views

Simplifying the expression $\frac{x + y}{x - y} + \frac{1}{x + y} - \frac{x^2 + y^2}{y^2 - x^2}$

Can you tell me why my answer is wrong? $$\frac {x+y} {x-y} + \frac 1 {x+y} - \frac {x^2+y^2} {y^2-x^2} = \frac {x^2 + y^2} {x^2-y^2} + \frac {x-y} {x^2-y^2} + \frac {x^2+y^2} {x^2-y^2} = 2x^2 + 2y^2 ...
-8
votes
1answer
50 views

How many? I need help, please help me. [on hold]

I need help with this right now. How many $\$$ is $100\%$, if $25\%$ is $15\$$.
-4
votes
1answer
50 views

How to simplify $ \frac{x^2-9x+14}{x^2+7x+12} \div \frac{3x^2-21x}{4x^3+16x^2} $? [on hold]

I'm having trouble simplifying a fraction: $$ \frac{x^2-9x+14}{x^2+7x+12} \div \frac{3x^2-21x}{4x^3+16x^2} $$ I tried it but I think my factoring is wrong keep coming out wrong answers.
0
votes
3answers
38 views

How do I factorise the following expression?

How do I go from the left expression to the right one? $$ (2-x)^2 \cdot (-2-x) - (-2-x) = - (x+2)(x-3)(x-1) $$ I'm guessing that I have to solve the third degree equation. What are the steps for ...
0
votes
0answers
16 views

Simplifying $\theta u^2 + (1-\theta)v^2 - [\theta u + (1-\theta)v]^2$

I've been working through a problem in Chiang's Fundamental Methods of Mathematical Economics and I ran into a little bit of trouble. So the problem is to check whether a function is concave or convex ...
-1
votes
1answer
75 views

Solve this equation for x [on hold]

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
41 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
27 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 ...
-4
votes
1answer
26 views

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

Find the domain and range of $f$ and $f^{−1}$ $f(x) = x^2 − 9, \ \ \ x \le 0$ $f^{−1}(x) = -\sqrt{x+9}$
-7
votes
1answer
55 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
41 views

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

$2x^2 + 32$ $\Rightarrow$ $2(x + 4)^2$ Is this correct?
4
votes
7answers
83 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
18 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
278 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
18 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
34 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
30 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: $\dfrac{k^3 - 1}{k - 1}$ ...
3
votes
4answers
364 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
88 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
42 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
21 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
31 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 $ ...