Tagged Questions

18 views

How to measure monotonicity of a list of values

I need to compare monotonicity of lists of values. I have $S=(n_1,n_2,...n_n)$, I need a function $\mathrm f(S)$ to return the monotonicity of the S. $S_1=[1,2,4,4,8]$ $S_2=[8,4,4,2,1]$ ...
55 views

How to Simplify Sin/tan problem.

I am trying to simplify $\displaystyle\frac{\sin^2}{\tan^2}$ but I don't know how to go about it. Any help is appreciated.
56 views

Logarithm of a negative number

We know this identity: $\ln(\frac{a}{b}) = \ln(a) - \ln(b)$ Suppose both $a$ and $b$ are negative. Then the left-hand size evaluates to something, it is a logarithm of a positive number (minus signs ...
20 views

How to calculate Polar coordinates for Complex Polynomials of Higher Degree?

When such I have a complex number such as $3 - 4i$, I can calculate the $r$ with $r=\sqrt{X^2+Y^2} = \sqrt{3^2+4^2}$. But how do I solve this when I have a complex number such as $(2+6i)^6$
113 views

Find value range of $2^x+2^y$

Assume $x,y \in \Bbb{R}$ satisfy $$4^x+4^y = 2^{x+1} + 2^{y+1}$$, Find the value range of $$2^x+2^y$$ I know $x=y=1$ is a solution of $4^x+4^y = 2^{x+1} + 2^{y+1}$ , but I can't go further more. I ...
37 views

the geometric explain of $t = x-\frac{a}{3}$ in the simplify of cubic equation $x^3+ax^2+bx+c=0$

Assume $$f(x) = x^3+ax^2+bx+c$$ we have $$f''(x)=2a+6x$$. we get $x = -\frac{a}{3}$ Magically, If we take the transformation: $$t = x -\left(-\frac{a}{3}\right)$$. we can transform the above ...
22 views

What are a,b, and c in $(3)/(x^3+ax^2+bx+c)$?

What are a,b, and c for the function $(3)/(x^3+ax^2+bx+c)$ with asymptotes at x=0, y=0, x=2 and x=4? I got stuck when I tried to calculate $2^3+2^2a+2b+c=4^3+4^2a+4b+c$. $8+4a+2b+c=64+16a+4b+c$. ...
24 views

What is the order of the corrector of Adams-Moulton type required in order to apply Milne's method for estimating the error in PECE mode? Find the coefficient of the leading term in the truncation ...
45 views

Sequence of learning mathematics from basic algebra to calculus.

What would be a step by step sequence of learning mathematics from basic algebra to basic calculus? I pose this question because I am in the process of self-learning mathematics as a preparation for a ...
660 views

Preventing “proof by homework”?

I am doing problem 3d in the Prologue of Spivak: Prove $(a+b)^n = a^n + {n\choose1}a^{n-1}b + {n\choose2}a^{n-2}b^2 + ... + {n\choose n-1}ab^{n-1} + b^n$ I feel like my proof could pass off as ...
119 views

Does $xy\geq x+y$?

I just see the GM-AM inequality. But I would like to compare $xy$ with $x+y$ for any $(x, y)\in\mathbb{R}^2$. It looks like $xy>x+y$ since the first one is multiplication and the second one is ...
16 views

Iterating the chain rule in multiple variables

$$f:\mathbb{R^3}\rightarrow\mathbb{R},\quad g:\mathbb{R^2}\rightarrow\mathbb{R},\quad h:\mathbb{R}\rightarrow\mathbb{R}$$ $f,g,h$ are differentiable along their domain. I'm asked to find the total ...
46 views

What is a derivative of a function?

What does it mean by getting a derivative of a function?And how is it different from getting a limit? I know how to find the limit of a function but really do not understand what it is all ...
9 views

Analytic solution to a maximization problem - Solve for $R$

I'm trying to use a CARA utility function $U(x)=e^{-\theta x}$ in the context of the Schumpeterian growth model to solve for the R&D spending. I set up a maximization problem ...
337 views

How can I prove this trigonometric statement true?

$${1+\sin^{2}\left(x\right) \over \cos^{2}\left(x\right)} = 1 + 2\tan^{2}\left(x\right)$$ This statement is part of a larger problem, but I need to prove that this is true before moving on. I'm ...
27 views

Let f(x) = x+(2)/x. Rewrite f(x+h)-f(x)/h as a single reduced fraction

Let $f(x) = x+(2)/x$. Rewrite $(f(x+h)-f(x))/h$ as a single reduced fraction. My answer is $x^2 + xh - 2 / x (x+h)$ Am I right? Also, what is the best way to think of problem such as this? Should ...
43 views

41 views

I need to do math from ground up, so what is a good workbook?

Can you guys recommend me a workbook that begins with arithmetic and ends with calculus. Or from pre-algebra to calculus. Like all "Master Math Series" books but in one complete book. It would really ...
57 views

prove that $\sqrt{4-a^2}+\sqrt{4-b^2}+\sqrt{4-c^2}+(\sqrt{3}-1)(|a-1|+|b-1|+|c-1|)\ge 3\sqrt{3}$ if $a+b+c=3$

$a,b,c\in[0,2]$ observation by triangle inequality $|a|+|b|\ge |a+b|$ $|a-1|+|b-1|+|c-1|\ge |a+b+c-3|$ but $a+b+c=3$ hence ...
43 views

Real world situation with System of Equation with 3 variables?

Where do you run into a real world situation involving 3 variables and 3 equations? Can someone think of a specific example from business, etc? I recall taking an operations research course that ...
29 views

Prove this logarithm equation

I keep getting the wrong answer. Can someone please correct my working out a^x=b^(1-x) In(a)^x=In(b)^(1-x) xIn(a)=(1-x)In(b) xIn(a)=In(e)-xIn(b) xIn(a)+xIn(b)=In(e) x[In(a)+In(b)]=Ine ...
26 views

Rewriting integrals over a symmetrical set

I have $\int_\mathbb{R}f(x)\cdot \mathbf{1}_{A}(x)\,dx$, with $f(x)$ integrable and $A=[-a,-b]\cup[b,a]$. I rewrote the integral as: $\int_{-a}^{-b}f(x)\,dx+\int_{b}^{a}f(x)\,dx$ but that is ...
39 views

Inverse Function of Logarithm

The answer is A but I don't understand why! $-2 \log_e (x^2)$ can be re-written as $-4 \log_e(x)$ right? but why do these two graphs look different? the graph $-2 \log_e (x^2)$ is one to ...
15 views

Maximum value of constant in logarithm problem

The first thing I did was: make: (x-1)^2 - k > 0 (x-1)^2 > k don't know what to do after this point... the maximum value of k is 9 i dont really understand what the maximum value of k is? ...
34 views

Describing asymptotic behaviour of a function

For question B! x^2+x+1/x^2 = 1+ [x+1/x^2] shouldnt the answer be asymptote at x=0 and y=1 ?? i dont understand the textbook solution
72 views

Finding the Limit in: $\lim\limits_{x\rightarrow1}\frac{\frac{1}{\sqrt{x}}-1}{x-1}$

Need some help finding this limit: $$\lim_{x\rightarrow1}\frac{\frac{1}{\sqrt{x}}-1}{x-1}$$ Here is what I have so far: $$\lim_{x\rightarrow1}\dfrac{\dfrac{1-\sqrt{x}}{\sqrt{x}}}{x-1}$$ ...
48 views

Where is the Pinching/Squeeze theorem in Spivak Calculus?

So I got Spivak Calculus 3. Edition. I'm starting it now but I want to know if there is the squeeze theorem clearly explained in the book, as I can't find it in the Appendix(pinching theorem, ...
33 views

“Tessellate” $e^{-x}$

Given an exponentially decaying function $f(x) = e^{-kx}$ that passes through the points $\vec a_0= (x_0, y_0)$ and $\vec a_2 = (x_2, y_2)$ such that $x_0 < x_2$ and $y_0 > y_2$, what is the ...
49 views