For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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1
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2answers
32 views

How to mathematically prove the optimality conditions for a univariate function?

Consider a univariate function $f(x)$. I know the graphical intuition behind why $f'(x)=0$ at the extrema of $f$. But how do you prove it mathematically? I start with the assumption of $x^*$ being a ...
0
votes
0answers
31 views

Using Leibnitz Theory to find 2nd order differentiation of an equation.

Here is my solution but I am getting stuck at a later stage. Can someone point out my mistake. $ f(x)= x^2log x $ Let $u=log x$ and $v=x^2$ We know $u_n = (-)^{n-1} .{(n-1)!}. (x)^{-n}$ Using ...
2
votes
4answers
96 views

$(1, 1) \cdot (6, 0) = 6?$ Intuition?

$a = (1, 1)$ $b= (6, 0)$ $a \cdot b = (1, 1) \cdot (6, 0) = 6$ I have seen the dot product of $a$ and $b$ refered to as "What is the x-coordinate of $a$, assuming $b$ is the $x$-axis?". Well here ...
1
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1answer
25 views

Function Question - Main Features

Can anyone answer these 2 questions? Consider the function $f : \mathbb{R} \to \mathbb{R}$ given by $f(x) = x^2e^{-x}$. (a) Calculate local maxima/minima and points of inflection of f(x). (b) ...
1
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2answers
49 views

Showing the summation of numbers

Using each of the digits 1 through 9 once, form numbers whose sum is 100. If you think it can't be done, then prove it. My attempt: I say it can't be done because the sum of all numbers $1-9$ is ...
7
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4answers
196 views

Calculus Question: Improper integral $\int_{0}^{\infty}\frac{\cos(2x+1)}{\sqrt[3]{x}}dx$

How to evaluate integral $$\int_{0}^{\infty}\frac{\cos(2x+1)}{\sqrt[3]{x}}dx?$$ I tried substitution $x=u^3$ and I got $3\displaystyle\int_{0}^{\infty}u \cos(2u^3+1)du$. After that I tried to use ...
0
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1answer
28 views

Proving the existence of a harmonic function

Show that if $f(u,v)$ is harmonic, then $F(x,y)=f(x^2-y^2,2xy)$ is also harmonic. What i did For a harmonic function to exist, $F_{xx}$ +$F_{yy}$= 0 Hence i know that i must show this result. I ...
0
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0answers
126 views

Finding the equation for ellipse velocity

I am trying to figure out how to do a homework problem for my math class. The homework for the night is, given a set of parametric equations, has three parts, the first of which is to find the speed ...
2
votes
3answers
3k views

Computing limits which involve square roots

Is there a general strategy for this? For example I'm working on the limit $$\lim_{n\rightarrow\infty}\sqrt{n^2 + n} - n $$ I have a simple argument to show that this limit is less than or equal to ...
0
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2answers
42 views

HoPoMo in PHP: I don't know how to implement - “max{”

I am woefully undereducated in math - so I have to ask a question that I'm assuming is fairly basic. This is a formula in the HoPoMo Honeybee Population Model for modeling the effects of season on ...
1
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2answers
73 views

The limit of $f(x)= \sqrt{x^2+4x+3} +x$ as $x\to\infty$

The problem is to find $\lim_{x\to\infty} \sqrt{x^2+4x+3} +x$. Do I just divide everything by $x^2$ and get limit $= \sqrt{1}+0=1$?
3
votes
1answer
72 views

Function - Main Features?

I understand how to draw this function, but what does it mean by main features? any examples for the question below? Consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x) = ...
1
vote
1answer
70 views

Discontinuity of a piecewise defined function with a parameter

Let $$ f(x,y) = \left\{ \begin{array}{ll} cx+4, & \textrm{if $x<6,$}\\ cx^2-4, & \textrm{if $x\geq 6.$}\\ \end{array} \right. $$ respectively. For what value of $c$ is this function ...
0
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2answers
87 views

Solve $|x-2| \leq 2|x|$

This is an in-class example we were given in calculus class, I am having some difficulty understanding one of the instructor's steps. The following is my attempt of the question: Since this is an ...
1
vote
1answer
49 views

Infinite Product implies divergence or not?

If $\displaystyle\prod_{n=1}^{\infty} (1-a_{n}) = 0$ then is it always true that $\displaystyle\sum_{n=1}^{\infty} a_{n} $ diverges? ($0 \leq a_{n} < 1) $
0
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2answers
28 views

Maxima of sinusoid functions

We have the function $2\sin(x)\cos(x)$, and we want to know the distance between the maxima. How should one tackle this? I know that the derivative is $2\cos(2x)$.
1
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2answers
35 views

Lagrange Multipliers to determine min and max

I've got this question in a book of questions I'm doing. Can someone show me step by step how to solve this? Using Lagrange Multipliers for two constraints, determine the maximum and minimum of ...
7
votes
0answers
215 views

How to evaluate the integral $e^{-(c\ln(\frac{1}{x}))^s} dx$?

Can anyone help me evaluate $$\int_{\alpha}^1 \exp{\left\{-\left(c\ln\left(\frac{1}{x}\right)\right)^s\right\}} dx$$, Where $0 \leq \alpha \leq 1$ and $s \in \mathbb{R}$. I tried changing ...
5
votes
2answers
2k views

Integral of $\int \frac{du}{u \sqrt{5-u^2}}$

I am trying to find this integral and I can get the answer on wolfram of course but I do not know what is wrong with my method, having gone through it twice. $$\int \frac{du}{u \sqrt{5-u^2}}$$ $u = ...
1
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2answers
38 views

Replace x with -1 in $ y' = \frac 32({1+x^\frac 23})^\frac 12 \ ({\frac 23 x^\frac {-1}3}) $

Question: Find an equation of the tangent line to the given curve at the given point $ y = ({1+x^\frac 23})^\frac 32 \ $ at $ x = -1$ This gives the slope: $ y' = \frac ...
1
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1answer
65 views

Evaluate the limit, if it exists

In Exercises 5-34, evaluate the limit, if it exists. If not, determine whether the one side limit exists (finite or infinite). 26. $$\lim_{x\to ...
0
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1answer
40 views

Integer solutions of an equation that is set to a number

How many integer solutions for $a$ and $b$ in $(ab)/(a+b)=3600$? My attempt: $(ab)/(a+b)=3600$ = $ab=3600(a+b)$ = $ab=3600a+3600b$ =$ab=3600a=3600b$ Dividing $3600b$ on both sides ...
5
votes
1answer
144 views

Computing $\int_0^{\large1/1^2} \left\{ \frac{1}{t} \right\}\,\mathrm{d}t+\int_0^{\large 1/2^2} \left\{ \frac{1}{t} \right\}\,\mathrm{d}t+\cdots $

What tools would you recommend me for computing this series? $$\int_0^{\large1/1^2} \left\{ \frac{1}{t} \right\}\,\mathrm{d}t+\int_0^{\large 1/2^2} \left\{ \frac{1}{t} ...
1
vote
2answers
60 views

Limit approaching to negative infinity.

Q. Find $\lim _{x\to -\infty }\left(\frac{x^4\sin\frac{1}{x}+x^2}{1+|x|^3}\right)$ By inserting $x=-\frac{1}{y}$ and as $_{x\to \:-\infty \:}$ then $_{y\to \:0\:\:}$. By applying this my text arrive ...
2
votes
2answers
69 views

how to find iso-cost contours on a 2d plot efficiently

Consider a 2D plot in which dimension 1 and 2 represent quantity 1 and 2 respectively ranging over 0 to 100. Each point in the space corresponding to (x,y) represent cost of choosing quantity 1 as x ...
1
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0answers
108 views

Quaternion Calculus

I was reading a note on Quaternion(Link) and I am happened to read a section regarding a solution of quaternion differential equation. I am putting that segment as picture format here for more ...
0
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0answers
38 views

Integral $\int_1^{\infty}(1-f(x))\ dx$ convergent iff sum $\sum_{n=1}^{\infty} n(1-f(n^2))$ is convergent

$f:[1,\infty)\longrightarrow\mathbb{R}$ is nonincreasing and $\lim_{n\to\infty}f(n)=1$. One must prove that if $$\int_1^{\infty}(1-f(x))\ dx<\infty$$ then so is $$\sum_{n=1}^{\infty} n(1-f(n^2))$$ ...
2
votes
3answers
61 views

Verifying proof of $\lim_{x \to\sqrt{2}}\frac{x^2-2}{x^2+\sqrt{2}x-4} = \frac 2 3$

$$\lim_{x \to\sqrt{2}} \dfrac{x^2-2}{x^2+\sqrt{2}x-4} = \lim_{w \to2} \dfrac{w^2-4}{w^2+2w-8} =\lim_{w \to2} \dfrac{(w-2)(w+2)}{(w+4)(w-2)} = \frac 2 3$$ Change of variable: $$w=\sqrt{2}x ...
0
votes
5answers
78 views

Showing that $f(x) = \ln x - e^x$ has no real roots

Show that $f(x) = \ln x - e^x$ has no real roots Since $\displaystyle\lim_{x \to 0^+} f(x) = -\infty$ and $\displaystyle\lim_{x \to \infty} f(x) = \lim_{x\to \infty} e^x \left ( \frac{\ln x}{e^x} ...
0
votes
0answers
46 views

proving convergence and finding the limit of this sequence if it is convergent?

$a(n)=\sin(1+a(n-1))$ where $a(0)=0$ and $n$ belongs to integers , prove or disprove whether the sequence converges and find the limit . well i understood that this is bounded but i am not able to ...
0
votes
1answer
26 views

finding maximum of a function

Need help in finding maximum of the following function: $1-\sum_{i=1}^{n}p_i\times \frac{p_i-1}{log(p_i)}$ given that $\sum_{i=1}^{n}p_i=1$. Thank you for your time and help.
0
votes
1answer
20 views

How to find a function with two variables from two functions with one variable

I am trying to determine a function for an algorithm I wrote. The time $t$ it takes to run depends on two variables $w$ and $l$ (with $l > 0$ and $w > 0$) I measured $t$ with a fixed $w$ ...
2
votes
1answer
27 views

$\sum_{n=1}^{\infty} {(1-\cos(\sin 1/n))}^{w}$ with $w$ as parameter

Let $f(x)=(1-\cos(\sin x))$; $a_n=f(1/n)$ for $n\in\mathbb{N}$ For which $w>0$ series $$\sum_{n=1}^{\infty} {a_n}^{w}$$ converge? I haven't got a slicest idea how to check that, absolutely none ...
2
votes
0answers
45 views

How do I solve this tricky definite integral ?! [duplicate]

$$I=\int\limits_0^1\dfrac{x^2-1}{\ln x}\mathrm dx$$ I tried numerous substitutions but nothing seems to work.. any ideas ???!
0
votes
1answer
24 views

Optimisation question - rectangle under curve

I've done part a: i. $(\pi-x,0)$ ii.$(\pi-2x)\sin x$ Not sure how to do b though. Can someone explain what to do?
0
votes
1answer
33 views

Finding family of curve for given asymptotes

I need to find possible curves, with asymptotes given as $x=0 (x \to -\infty)$ and $y=mx \hspace{0.5cm} m>0$. it is easy to find curves for individual lines, $y= \exp(-\lambda_1 x) + mx$ for $y=mx$ ...
2
votes
1answer
79 views

Proving $\sum_{k=1}^{\infty}\frac{\sin kx}{x}=\frac{\pi-x}{2}$ for $0\le x\le 2\pi$

Refer to this OP: Sign of a series, we have the following equation \begin{equation} \sum_{k=1}^{\infty}\frac{\sin kx}{k}=\frac{\pi-x}{2} \end{equation} defined for $0\le x\le 2\pi$. Here is ...
2
votes
5answers
157 views

How to evaluate $\displaystyle \lim_{x \to \infty}\frac{8-\sqrt{x}}{8+\sqrt{x}}$

$$\displaystyle \lim_{x \to \infty}\dfrac{8-\sqrt{x}}{8+\sqrt{x}}$$ I tried rationalizing the numerator: $$\lim_{x \to \infty}\dfrac{8-\sqrt{x}}{8+\sqrt{x}} \times ...
0
votes
1answer
35 views

Fourier Transform and $f*g$ convolution

Given the 3 following: $$\mathfrak{F}(e^{-|t|})=\sqrt\frac{2}{\pi}\frac{1}{1+\omega^2}$$ $$\mathfrak{F}(r(t))=\sqrt\frac{2}{\pi}\frac{\sin \omega}{\omega}$$ where $$r(t)=\left\{\begin{matrix} 1, ...
1
vote
6answers
675 views

Why is this limit statement false?

Determine whether the statement is true or false. If $\displaystyle\lim_{ x→0} f(x) =\infty \text { and } \lim_{x→0} g(x) = \infty, \text{ then } \lim_{x→0}[f(x) − g(x)] = 0$. Can someone ...
2
votes
2answers
40 views

A Lagrange Multiplier Problem : How to deal with this case when $b< 8$

I was trying to solve the following problem of several variables calculus given in my class.I am stuck in a particular case of the problem.Please help me to solve the problem.Thnx in advance. Find ...
3
votes
1answer
62 views

Does such a function exist always?

Suppose that $f(x)$ is some smooth function on $[0,1]$ with $f(x) \geq c > 0$. Can we always find a function $g(x)$ smooth satisfying $g'(x) \not= 0$ for all $x \in [0,1]$ and $f'(x)g'(x) + ...
1
vote
1answer
49 views

How can I construct a specific sigmoid function?

The simple sigmoid function $$f(x)=1/(1+e^{−x})$$ approaches zero as x tends to negative infinity, and approaches $1$ as x tends to positive infinity. But I want to set $1$ and $20$ instead of $0$ and ...
0
votes
0answers
30 views

Partial derivatives using chain rule

If $u = \frac{1}{y}[\phi(ax + y) + \phi(ax - y)]$, and $\phi$ is twice differentiable, show that, $$\frac{\partial^2u}{\partial x^2} = \frac{a^2}{y^2}\frac{\partial}{\partial y}\left(y^2 ...
0
votes
0answers
46 views

Are these answers sufficient enough for this linear function question?

I am writing this to ask for some opinions as to whether or not my answer to this bonus question on my calculus assignment is sufficient enough. If I am missing something, please explain and show me ...
0
votes
0answers
51 views

limsup of a sup

I am stuck with the following limit: $$\overline{\lim}_{r\to+\infty}\frac{\sup_{n\in\mathbb N}\{(r+\frac32-n)2^n\}}{2^r}$$ I guess the result is $\frac{2^{\frac52}}{e\ln2}$, but I have no clue to ...
2
votes
2answers
49 views

How to prove that no constant can bound the function f(x) = x

I know this is a trivial question, but how would one mathematically demonstrate this using a proof?
0
votes
2answers
50 views

Basic question about limits and derivatives

the limit $\lim_{h \to 0}\dfrac{\sqrt{81+h}-9}{h}$ represents the derivative of some function $f(x)$ at some number a. Find $f$ and $a$. I don't quite understand what this question is asking. Is ...
8
votes
3answers
318 views

Proving that a definition of e is unique

We can define $e$ as the number such that $\lim_{h \to 0} \frac{e^h-1}{h}=1$. However, of course we can only define $e$ this way if it is unique, i.e., there is no other value $c$ for which that is a ...
0
votes
1answer
31 views

How to evaluate these expressions, notation

This is a very elementary question, but it has caused be a lot of confusion these last days when working on some problems. If I write: $f'(-x)$, does it mean that I have the function f, then i ...