0
votes
2answers
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. $S_1=[1,2,4,4,8]$ $S_2=[8,4,4,2,1]$ ...
1
vote
2answers
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.
0
votes
2answers
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 ...
0
votes
1answer
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 $(2+6i)^6$
5
votes
7answers
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 ...
1
vote
2answers
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 ...
0
votes
1answer
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 $2^3+2^2a+2b+c=4^3+4^2a+4b+c$. $8+4a+2b+c=64+16a+4b+c$. ...
1
vote
0answers
24 views

Predictor-Corrector for Adams-Moulton

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 ...
0
votes
1answer
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 ...
5
votes
3answers
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 ...
0
votes
4answers
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 ...
1
vote
1answer
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 ...
0
votes
2answers
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 ...
0
votes
0answers
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 \begin{equation} ...
4
votes
2answers
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 ...
0
votes
1answer
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 ...
1
vote
2answers
43 views

Bounding $\sum_{n=n_1}^\infty x^n (n+1)^2$

I need to upperbound the sum $$\sum_{n=n_1}^\infty x^n (n+1)^2$$ where $0<x<1$ is a parameter. I know it can be done starting from $$\sum_{n=n_1}^\infty x^n (n+1)^2\le \sum_{n=0}^\infty x^n ...
0
votes
0answers
30 views

Is this factorization true for all $n$ in the natural numbers

I need to know if $x-a=(x^{\frac{n}3}-a^{\frac{n}3})(x^{\frac{n+1}3}+a^{\frac{n}3} x^{\frac{n}3}+a^{\frac{n+1}3})$ Is true. I know its true for $n=1$, is it true for all natural numbers though?
0
votes
1answer
23 views

Fraction Simplification

I am working on some calc2 power series problem and Im not sure why I can not simplify the following: $(\frac{(-2)^{n+1}(x-4)^{n+1}}{(n+1)!})(\frac{(n!)}{(-2)^n(x-4)^n})$. If someone could kindly help ...
3
votes
3answers
49 views

Why does $f(x)=ax^2 + bx + c \ge 0\ \forall x \in \mathbb R$ imply $f$ has at most one real distinct root and discriminant $D \le 0$?

Why does $f(x)=ax^2 + bx + c \ge 0 \ \forall x \in \mathbb R$ imply $f$ has at most one real distinct root and discriminant $D \le 0$? I've been wondering why the following result is true. ...
2
votes
1answer
23 views

Flash question about expanding the derivative definition

I'm trying to do a physics estimation and I need your help with an assumption that I'm making: My flash question is: Is it true that $$(f(x) \cdot x^2)'=\lim_{h \to 0}\ ...
1
vote
1answer
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 ...
1
vote
1answer
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 ...
1
vote
2answers
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 ...
0
votes
1answer
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 ...
0
votes
1answer
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 ...
0
votes
1answer
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 ...
0
votes
1answer
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? ...
1
vote
2answers
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
0
votes
3answers
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: $$\lim_{x\rightarrow1}\dfrac{\dfrac{1-\sqrt{x}}{\sqrt{x}}}{x-1}$$ ...
0
votes
1answer
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, ...
0
votes
0answers
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 ...
0
votes
1answer
49 views

How do I properly set up this optimization equation?

So I've been the given the task to fully optimize any packaging. I chose a DS game box. So first I took the measurements of the cartridge itself ($3.5 \text{ cm} \times 3.3 \text{ cm} \times 0.38 ...
2
votes
1answer
42 views

Find a unit vector and the rate of change

Could anyone help me answer this question? Or point me in the right direction? Find a unit vector in the direction in which f increases most rapidly at P and find the rate of change of f at p in that ...
1
vote
1answer
38 views

Let f (x,y) = (xy)/(x+y)

How would I go about answering this question? Let f (x,y) = (xy)/(x+y) Find a vector u for which Du f (1,1) = 0.
1
vote
4answers
52 views

Find the point on a plane $3x + 4y + z = 1$ that is closest to $(1,0,1)$

Is anyone able to help me with regards to this question? Find the point on a plane $3x + 4y + z = 1$ that is closest to $(1,0,1)$ http://i.imgur.com/ywdsJi7.png
2
votes
2answers
127 views

Simplifying Second Derivatives

I can't seem to figure out how my professor simplified this second derivative. Any help is much appreciated. I'm having trouble simplifying the second derivatives of most problems so step by step ...
1
vote
1answer
45 views

Prove that $4x(x-5)^3 + (x-5)^4=5(x-5)^3(x-1)$

Given the expression $x(x-5)^4$ I need to differentiate it. Upon using the product rule followed by the chain rule I get this answer, $4x(x-5)^3 + (x-5)^4$. The answer in the back of the book is ...
0
votes
1answer
30 views

proof of this equation

How can I show the gradient of trace $(W^{T}MW)$ with respect to $W$ is equal to $MW+M^{T}W$. where W is an $m\times n$ matrix and $M$ is an $m\times m$ matrix. Can anyone help me in this case?
1
vote
2answers
37 views

Simplifying ratio test with exponents $k+1$

Question: Find the interval and radius of convergence. $$\sum_{k=1}^\infty\frac{(x-1)^k(k^k)}{(k+1)^k} .$$ I applied the ratio test. ...
1
vote
0answers
78 views

Solving an 8th degree polynomial

I know that through the Abel Ruffini Theorem the general solution to a polynomial of degree five or more cannot be found explicitly. But are there are any other ways to find the roots of such a ...
0
votes
2answers
17 views

How do you get calculations when you reverse the the values of a scale?

I created a questionnaire with values of 1 (very easy) to 5 (very difficult) and did the calculations of the average based on the results. However, now my client wants the values to be 5 (very easy) ...
1
vote
1answer
22 views

calculus difference question

Find the difference between the maximum and minimum values of the function $y = |sin x - 0.75|$ attempt to solve: I began all multiplications on the function without absolute value; $y' = cos x$ ; ...
0
votes
1answer
45 views

Fiding a derivative

I need to find the derivative of $\sqrt{x^2+3x}$ using the definition of derivative. e.g. $\frac{f(x)-f(a)}{x-a}$ as x->a. Normally I get these but the $x^2$ is messing me up. I am at $$\lim ...
1
vote
1answer
63 views

Fitting a Circle Arc to a Parabola

Reading this paper for a project. In section 2.1 it says an approximate formula for the smooth curve described by the edge of the ski is y ≈ $x^2/2R_{SC} − d$. Why is the $x^2$ value divided by ...
0
votes
5answers
149 views

Application of Composition of Functions: Real world examples?

Do you know of a real world example where you'd combine two functions into a composite function? I see this topic in Algebra 2 textbooks, but rarely see actual applications of it. It's usually plug ...
1
vote
0answers
56 views

Physics of Skiing

I am conducting a research paper on the physics of skiing, specifically how ski parameters affect the ski's ideal carve. I have come across this paper, which is incredibly relevant but am having ...
1
vote
3answers
50 views

Problem about an Inequality

I need a hint to solve this inequality If $x_i >0$ for all $i$ then $(x_1 x_2 ... x_n)^{1/n}$ $<$ $(x_1+...x_n)\over n$ I tried a little by induction over n, but i dont go anywhere with that
0
votes
0answers
21 views

Sum of a Sequence (Relating to natural log)

Okay guys, so this is the first time I've posted, so I may not know the shorthand you guys will. I'm looking for help for a calc 2 problem I'm working on, specifically trying to figure out if this ...
1
vote
2answers
42 views

What kind of functions can be Riemann integrable?

I have learned that every continuous, or piecewise continuous function can be Riemann integrated. But then, are there uncontinuous functions that are Riemann integrable? And if there is, can I still ...