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
0answers
20 views

Rules for multiplying exponents by scalars

If I have the formula $A * B^2$ where I know $B= 2*C$, can I change the original formula to $A * (2*C)^2$ ?
2
votes
2answers
21 views

Solving for a three dimensional vector.

Let $a = \begin{pmatrix} 2 \\ 5 \\ -1 \end{pmatrix}$ and $b = \begin{pmatrix} -6 \\ 4 \\ -3 \end{pmatrix}$ There exists two nonzero three-dimensional vectors ${v} = \begin{pmatrix} x \\ y \\ z ...
-2
votes
1answer
32 views

Write an algebraic expression in terms of $x$ (where $x > 0$) for “$\tan(2 \tan^{-1} x)$”

Write the following as an algebraic expression in terms of $x$ (where $x > 0$): $\tan(2 \tan^{-1} x)$ This is what I got so far: Let $\tan^{-1} x = \theta$. So, $\tan(\theta) = x$. So, opposite ...
0
votes
1answer
25 views

formula in Spivak calculus, ch 2-6ii f

Without going to much into details about the question itself I would like to draw attention to the fact that Spivak assumes knowledge of a formula (I got it from the solutions in the back of the book) ...
0
votes
0answers
8 views

System of Equations & Approximations

I am trying to derive Eq. (3.6) in the following thesis: http://drum.lib.umd.edu/bitstream/1903/14898/1/Khalil_umd_0117E_14726.pdf This is the equation I am trying to show: \begin{equation} ...
0
votes
1answer
24 views

Expansion for Partial Fractions for $(3-2x)/(x^2+6x+9)$

I'm trying to expand $(3-2x)/(x^2+6x+9)$ into partial fractions to integrate. I'm doing $$(3-2x)/((x+3)^2)=A/(x+3)+B(x+3)^2$$ $$(A(x+3)+B)/((x+3)^2)=3-2x$$ for x=0:$$(3A+B)/9=3$$ for x=1: ...
1
vote
3answers
43 views

How to simplify $\frac{\sqrt{x^3}}{\sqrt[3]{x^4}}$? [on hold]

Please, could someone help simplify and show the steps on how to simplify $$\frac{\sqrt{x^3}}{\sqrt[3]{x^4}}?$$ Thank you.
-1
votes
0answers
16 views

If P(4, -3) is a point on angle A, find the exact value of tan(2s) [on hold]

*I need to brush up on this. Could someone explain?
5
votes
1answer
100 views

Prove by combinatorial method that $ \frac{(2m)! \cdot (2n)!}{(m)! \cdot (n)! \cdot (m+n)!} $ is an integer [duplicate]

Prove that $$ \dfrac{(2m)! \cdot (2n)!}{(m)! \cdot (n)! \cdot (m+n)!} $$ is a positive integer, where $(m,n) \in \mathbb{Z^{+}}$ I have already solved it using Legendre's Formula ...
4
votes
3answers
62 views

Solving $7[x]+23\{x\}=191$

For every real number $x$, $[x]$ denotes the largest integer less than or equal to $x$ and $\{x\}=x-[x]$. The number of real solutions of $$7[x]+23\{x\}=191$$ is (a) 0 $\quad$ ...
6
votes
2answers
62 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
12 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
41 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
39 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
37 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
36 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
25 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
58 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
32 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
45 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
31 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
37 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
83 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
54 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
52 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
27 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
55 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
51 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
51 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
42 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
17 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 ...
-3
votes
1answer
27 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
42 views

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

$2x^2 + 32$ $\Rightarrow$ $2(x + 4)^2$ Is this correct?
4
votes
7answers
89 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
279 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, ...