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

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0
votes
4answers
2k views

How to find the derivative of the function $y=5^{-1/x}$?

$$y=5^{-1/x}$$ Help would be so greatly appreciated :] It's another homework problem...I unfortunately was not present during the lecture for these types of problems. I'm guessing from the $-1/x$ ...
-2
votes
4answers
9k views

How to find the derivative of the function $f(x) = (2x-3)^4 (x^2 + x + 1)^5$?

Can someone please tell me how to find the derivative of this function? I've been working on this one problem since yesterday and I still can't find the answer... $$f(x) = (2x-3)^4 (x^2 + x + 1)^5$$ ...
1
vote
2answers
65 views

Finding Concavity

I'm trying to find the concavity and inflection points to this equation: $f(x) = 3x^3 - x^2 + 2x - 7$ I've taken the derivative which is: $9x^2 - 2x + 2$, but it's not factorable so how would I find ...
1
vote
0answers
50 views

Trouble proving/disproving the existence of a particular derivative.

I am having trouble with the following question: Quesiton: Define $f$ such that $$ f(x) = \begin{cases} \cos\left(\frac{1}{x}\right) &\mbox{ if }x \ne 0 \\ 0 &\mbox{ if } x = 0 \end{cases} ...
4
votes
2answers
58 views

Finding when $\frac{1}{\pi}\int_0^{j\pi} \frac{\sin t}{t} - \frac{1}{2}$ is positive / negative

Could you help me with the following question? Show that the following numbers are positive for $j$ odd and negative otherwise: $$\frac{1}{\pi}\int_0^{j\pi} \frac{\sin t}{t} - \frac{1}{2}$$
1
vote
2answers
185 views

$\sum_{n=1}^{\infty } \left ( 1 - \cos(\frac{1}{n}) \right )$ converges or diverges? [duplicate]

$$\sum_{n=1}^{\infty } \left ( 1 - \cos\left(\frac{1}{n}\right) \right )$$ I tried all of the tests but I couldnt solve this.Does it converges or diverges?
0
votes
1answer
261 views

Maximization of sum of two functions

Is there any relationship similar to the following. Let $X$ be the maximum of functions $f_1(x)+f_2(x)$. Let $X_1$ be a maximum of $f_1(x)$ and let $X_2$ be a maximum of $f_2(x)$. Is there any ...
0
votes
2answers
56 views

How to find $\sum_{n=0}^{\infty } \left ( \frac{1}{n.(n+1).(n+2)))} \right )$?

$$\sum_{n=0}^{\infty } \left ( \frac{1}{n.(n+1).(n+2)))} \right )$$ I've tried all of the tests but I couldnt solve this. Does it converges? What is the sum if it converges?
0
votes
1answer
294 views

Discrete approximation - exponential function and integrals

Let $f$ be a complex-valued continuous function on $\mathbb{R}_+$ with compact support and let $g, h$ be two complex-valued continuous functions on $\mathbb{R}_+$ such that $g$ is bounded and ...
5
votes
3answers
137 views

Is there a simple geometrical description of $e$? [duplicate]

Of course I am not looking for a definition through $\int_1^e{1\over x} \, \mathrm{d}x=1$ or that slope of $a^x$ at $x=0$ is $1$ when $a=e$. I am looking for something understandable by a kid who has ...
3
votes
2answers
170 views

Please help me understand this. $\frac{dx}{dt} = S x (a-x)$. What does it mean for some constant $S$? How to find $x$ for fastest/slowest growth?

I am having some trouble understanding this problem. There is this function that calculates reaction rate of a substance for some constant positive $S$. $a$ = original amount of the first substance ...
4
votes
3answers
504 views

Improper Integral : $\int_{0}^{\infty } \frac{dx}{x\sin x}$?

$$\int\limits_{0}^{\infty}\frac{dx}{x\sin x}$$ How can I explain that this integral diverges?
5
votes
2answers
218 views

Find $\sum\limits_{k\, \text{ odd}} \frac{2(k^2-1)}{k^4+k^2+1}$

How to find $$\sum_{k \text{ odd}} \frac{2(k^2-1)}{k^4+k^2+1}$$ Here we find $\displaystyle\sum_{k=1}^{\infty} \frac{2(k^2-1)}{k^4+k^2+1}=1$ and we know that $\displaystyle\sum_{k \,\text{odd}} + ...
1
vote
1answer
71 views

Maximum value of the given expression

Assuming $\theta \in [\frac{-5\pi}{12},\frac{-\pi}{3}]$, find the maximum value of $$\frac{\tan(\theta+\frac{2\pi}{3})-\tan(\theta+\frac{\pi}{6})+\cos(\theta+\frac{\pi}{6})}{\sqrt{3}}$$ I know we can ...
3
votes
4answers
1k views

Limit of the geometric sequence

I have the following homework question: For what values of $r$ does $\lim_{n\rightarrow \infty} r^n$ exist? What does it converge to? Correct me if I am wrong, but if $r$ is greater than or equal ...
1
vote
3answers
255 views

Derivative of $x^2\sqrt{1+x}$

Given that $f(x)=x^2\sqrt{1+x}$, show that $f'(x)=\dfrac{x(ax+b)}{2\sqrt{1+x}}$ where $a$ and $b$ are constants to be found. I first tried using the product rule: ...
5
votes
6answers
324 views

Understanding calculus formulas intuitively

I am currently studying calculus in Russian and my course book is very rigorous.I used to think that I understand everything but I recently noticed that I only understand the logical steps in proofs ...
0
votes
2answers
52 views

Integral/Integrands identities

I need to know whether, under some assumptions about the functions behavior , and maybe some values of integration limits $a,b$, the following relation holds: $\int_a^bdx \ f(x) = \int_a^b dx \ g(x) ...
4
votes
3answers
190 views

Silly question: Why is $\sqrt{(9x^2)} $ not $3x$?

I had to find the derivative of $f(x) = \sqrt{(9x^2)}$. I applied chain rule with the following steps. Let $f(x)$ be $\sqrt{x}$ and $g(x)$ be $9x^2$ $$ \begin{align} &f'(g(x)) \times g'(x) \\ ...
0
votes
1answer
502 views

Find the shortest distance between $9x^2+9y^2-30y+16=0$ and $y^2=x^3$

Find the shortest distance between $9x^2+9y^2-30y+16=0$ and $y^2=x^3$. I know the shortest distance exists between the curves on the common normal line. Is there any other shorter way to attempt?
6
votes
3answers
276 views

Show this equation has at least one root in $(0,1)$

Let $ax^2+bx+c=0$ be a quadratic equation, where $a,b,c\in\mathbb{R}$. If $2a+3b+6c=0$, then show that this equation will have atleast one root in $(0,1)$. I think it involves either Rolle's Theorem ...
1
vote
1answer
61 views

Taylor polynomial

I need your help to solve this question. I tried something, but i can't finish my proof. Let $f(x)$ be a differentiable function in $(0, \infty )$, so that $|f'(x)|$ is bounded there. Prove that ...
0
votes
1answer
42 views

Interval for $\alpha$, so that the given line is normal to $xy=4$

Find the interval for $\alpha$ so that $(3-\alpha)x+{\alpha}y+({\alpha}^2-1)=0$ is normal to the curve $xy=4$. I don't understand why do we need an interval for $\alpha$? The curve is a hyperbola ...
0
votes
2answers
38 views

What does $\sum_{n=0}^{\infty} (-1)^n .\frac{x^{2n+1}}{2n+1}$ converge to at x= 1 and x = -1

This is what I did. $\sum_{n=0}^{\infty} (-1)^n .\frac{1^{2n+1}}{2n+1}=\sum_{n=0}^{\infty}(-1)^n .\frac{1}{2n+1}$ Now I broke it up to positive and negative. $\sum_{n=0}^{\infty}(-1)^n ...
29
votes
10answers
4k views

A really complicated calculus book

I've been studying math as a hobby, just for fun for years, and I had my goal to understand nearly every good undergraduate textbook and I think, I finally reached it. So now I need an another goal. ...
-4
votes
1answer
91 views

A complicate limit calculation

How to evaluate $$\lim_{n\to\infty+}\frac{\frac{\cos(n!)}{n!+2n}-\sin\left(\frac1{n^2}\right)(n^8+\ln n)^{\frac14}}{(n^3+n^2)^{\frac13}-n}$$
-2
votes
1answer
99 views

A non-homogeneous linear differential equation

Find the general solution of : $$y''+y=\cos x\cos 2x$$
16
votes
4answers
1k views

Wrong Wolfram|Alpha limit?

I have this function: $$ f(x,y) = \frac {xy}{|x|+|y|} $$ And I want to evaluate it's limit when $$ (x,y) \to (0,0)$$ My guess is that it tends to zero. So, by definition, if: $$ \forall \varepsilon ...
5
votes
3answers
339 views

Find the maximum area of the regular pentagon

Find the maximum area of the regular pentagon that inscribed a unit square.
1
vote
3answers
168 views

Formula to limit a number within a minimum and maximum value

I'm looking for a formula that can limit a specific input within a specific range. I want to limit a number within 0 and 100. Example: Input: -100 / Output: 0 Input: -1 / Output: 0 Input: 0 / ...
6
votes
1answer
93 views

Did I sum this correctly?

I was tutoring some students in calculus today and they were supposed to sum up all of the odd four digit numbers. They weren't really confident about their answers so I showed them what I would do. ...
4
votes
1answer
94 views

Determining the bigger of two numbers : $\left(\frac12\right)^e$ or $\left(\frac1e\right)^2$

The question says - use the function $f(x)=sin(x)^{sin(x)}$, where $0<x<\pi$, to determine the bigger of two numbers: $\left(\frac12\right)^e$ or $\left(\frac{1}{e}\right)^2$. Any tips on how to ...
2
votes
1answer
63 views

How to prove this inequality : $(x^a+y^a)^{\frac1{a}}>(x^b+y^b)^{\frac1{b}} \, ; x>0,\ y>0;\ 0<a<b$

Prove that when $\displaystyle x>0,\ y>0;\ 0<a<b$ $$\displaystyle(x^a+y^a)^{\frac1{a}}>(x^b+y^b)^{\frac1{b}}$$
-2
votes
1answer
483 views

Find $a$ and $b$ so that the function $f$ is differentiable everywhere

Find $a$ and $b$ so that the function $f$ is differentiable everywhere: $$f(x) = \begin{cases} 9\cos x & \mbox{if $x<0$} \\ ax + b & \mbox{if $x \geq0$} \end{cases}$$
2
votes
2answers
843 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 ...
1
vote
1answer
138 views

I got stuck on a formula in a proof on the mean value property of harmonic functions in a sphere

In looking over a proof on the mean value property for harmonic functions on spheres I came across the following formula: $$\frac{\partial u}{\partial n}=\textbf{n}\cdot \triangledown ...
1
vote
1answer
100 views

Solving for x with radicals and negative exponents

How do I go about solving for $x$ in this equation? $$\displaystyle -x^{-\large\frac{3}{4}} + \frac{15^{\large\frac{1}{4}}}{15} = 0$$
10
votes
3answers
697 views

A sine integral $\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$

The following question comes from Some integral with sine post $$\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$$ but now I'd be curious to know how to deal with it by methods of ...
2
votes
4answers
124 views

Algebraically Solve Limit

$$\lim_{x \to 0} \dfrac{2\sqrt{x+1}-x-2}{x^2}$$ I can solve it using l'Hôpital but just cannot find a way to do it algebraically.
0
votes
1answer
224 views

Integrating the step-wise integer function 1/[x]

I'm trying to find the integral, respective to $x$, of a function that utilizes the step-wise integer (or floor) function. $$\displaystyle z = \int {1 \over [[x]*1.1^{[y]}]+1}$$ It's for modelling a ...
4
votes
2answers
46 views

Is my logic correct?

The question says there is a function $f(x)$ which maps $R$ to $R$, and $f''(x)>0$ for all x. This means $f'(x)$ is always increasing. And it is given that ...
16
votes
4answers
428 views

Computing $\lim_{n\to\infty}n\sum_{k=1}^n\left( \frac{1}{(2k-1)^2} - \frac{3}{4k^2}\right)$

What ways would you propose for the limit below? $$\lim_{n\to\infty}n\sum_{k=1}^n\left( \frac{1}{(2k-1)^2} - \frac{3}{4k^2}\right)$$ Thanks in advance for your suggestions, hints! Sis.
3
votes
3answers
104 views

How to evaluate this rational limit

Evaluate: $$\lim_{x\to0^+}\frac{(-3\sqrt{x}+x^3+\sin(x^6))(\text{e}^{4\sqrt{x}}-1)}{4\ln (1+x)+x^4-x^6}$$
2
votes
1answer
116 views

Polynomial function, only 1 solution

I am given the following task: The graph of the polynomial function $f(x)$ is symmetric to the y-axis. It has exactly one local minimum on the x-axis and an inflection point at $x = 1$. Find the the ...
23
votes
4answers
450 views

$\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^{n} n^{1/k} $

What would you suggest here? $$\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^{n} n^{1/k} $$
2
votes
2answers
590 views

3D Fourier transform

I don't know how to evaluate an integral of the form $$\int d^3 r \exp(-i \vec r\cdot\vec q)\exp(-a^2 r^2)$$ where $a\in \mathbb R$. Could anyone please teach me how to do this integral? Many ...
1
vote
0answers
113 views

Proof of an inequality problem

Wise men or women over the world!! I badly ask you to help me. Let $N$ and $B$ be two positive integers such that $1\le B\le \frac{N}{2}$ and $N=ug$ (for convenience, assume that $N$ is even.) For ...
5
votes
2answers
123 views

Are there rigorous mathematical definitions for these waves?

My friend linked this .gif to me tonight, and asked me if I knew of any equations that might model these bottom two waves (the blue and green waves). Unfortunately, I am not far enough in my education ...
1
vote
2answers
1k views

Let $f$ and $g$ be differentiable functions with the following properties: [closed]

$g(x)>0$ for all $x$ $f(0)=1$ If $h(x)=f(x)g(x)$ and $h'(x)=f(x)g'(x)$, then $f(x)=$ a) $f'(x)$ b) $g(x)$ c) $e^x$ d) $0$ e) 1 Please give me an explanation as well. Thank you very much.
0
votes
1answer
96 views

An exercise from 'Giaquinta,Modica,Mathematical Analysis,2'

Here is a exercise from Giaquinta,Modica,Mathematical Analysis,2 It is a last exercise from chapter 2 Let $f: \Bbb{R}\rightarrow \Bbb{R}$. show that for any $y \in \Bbb{R}$ and $n \in \Bbb{N}$ the ...