# Tagged Questions

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

38 views

### Express g's Fourier coefficients using f's ones, if $g(x)=f(x+c)$.

The Fourier coefficients are defined (in our course) as: $$\hat{f(n)}={1\over 2\pi}\int_{0}^{2\pi}{f(t)e^{-int}dt}$$ I am asked to express g's coefficients as a combination of f's ones, given ...
44 views

### Prove that for any positive integer $n$ and $d$, $\sum_{k=0}^d 2^k\log_2(\frac{n}{2^k})=2^{d+1}\log_2(\frac{n}{2^{d-1}})-2-\log_2{n}$

I could prove it by induction, but I need to see how I might have discovered it for myself (cause that's what's gonna be on exam).
29 views

### finding the supremum

Let $A=\{x:\frac{[b\cdot n]}{n}\}$ when $n\in \mathbb{N}$ find the supremum we know that $b\cdot n-1<[b\cdot n]<b \cdot n$ therefore $b- \frac{1}{n}<\frac{[b\cdot n]}{n}<b$ so b is a ...
157 views

51 views

### There are two periodic functions $f(x)$ and $g(x)$, provide an example when $f(x)*g(x)$ is unbounded, and $f(x)+g(x)=0$ has infinitely many solutions

There are two periodic functions $f(x)$ and $g(x)$ which are defined on $\mathbb{R}$, provide an example when $f(x)\cdot g(x)$ is unbounded, and $f(x)+g(x)=0$ has infinitely many solutions ?
107 views

### Limit $(e^x+x)^{1/x}$, when $x\to 0$

Can I expand e^x in the limit $$\large{\lim _{x\to 0}(e^x+x)^{1/x}},$$ just as $1+x$ according to the Taylor expansion? I mean is it normal to think about limits of a form $(1+x+o(x))^{1/x}$ just as ...
55 views

### what is the value of $\int \sin(x)\cos(x)dx$? $\frac{\sin^2(x)}{2}$ or $\frac{-\cos^2(x)}{2}$ or $\frac{-\cos(2x)}{4}$

$\int \sin(x)\cos(x)dx = \frac{\sin^2(x)}{2}$ because $$\frac{d}{dx}\frac{\sin^2(x)}{2}=\sin(x)\frac{\sin(x)}{dx}=\sin(x)\cos(x)$$ but also ...
62 views

### To check the convergence of an integral

I tried to find out if this integral is convergent or divergent, $$\int _0^{\frac{\pi}{2}}\left(\frac{\ln\left(\sin x\right)}{\sqrt{x}\:}\right)\:dx$$ I know that the problematic point is near ...
223 views

### Why doesn't using the approximation $\sin x\approx x$ near $0$ work for computing this limit?

The limit is $$\lim_{x\to0}\left(\frac{1}{\sin^2x}-\frac{1}{x^2}\right)$$ which I'm aware can be rearranged to obtain the indeterminate $\dfrac{0}{0}$, but in an attempt to avoid L'Hopital's rule ...
7k views

### The shortest distance between any two distinct points is the line segment joining them.How can I see why this is true?

On a euclidean plane, the shortest distance between any two distinct points is the line segment joining them. How can I see why this is true?
65 views

### Problem with understanding a Differential in Multivariable Calculus

I have just started with Partial Differentiation and the book from where I'm learning (Mathematical Methods in the Physical Sciences) had the following problem on approximations using differentials ...
29 views

### Differential Equation with Cross Products [without separating into system of equations]

I need to solve the following equation: $$\frac{d m}{d t}=-m\wedge b-\alpha m\wedge (m\wedge b),$$ where $b$ is constant However, I was instructed specifically not to separate the calculation into ...
33 views

### Finding the volume of a cone with and oblique base.

The base of $S$ is an elliptical region with boundary curve $9x^2+4y^2=36$. Cross-sections perpendicular to the $x$-axis are isosceles right triangles with hypotenuse in the base. The base of $S$ is ...
25 views

### How proceeded author from $(3)$ to get $a_n<f(n)-f(0)<1$

We have $f:(-1,\infty)\rightarrow\mathbb{R}, f(x)=\frac{x}{x+1}$ and $$a_n=\sum_{k=1}^n f(k)-\int_0^1 f(x)dx$$ We need to prove that $a_n$ is bounded. I don't need another method, I want to help ...
52 views

### $f$ is integrable & continuous over $[a,b]$ , $\int_{a}^{b}f(x)dx \geq 0$ for any subinterval $(\alpha,\beta)$ of $(a,b)$, then $f \geq 0$ in $[a,b]$

Some known things about this problem are: if $f(c) < 0$, $a < c < b$, then $f(x) < f(c)/2$ in some neighborhood of $c$, but I am not exactly sure how to use this to get to my goal of ...
74 views

### If $k$ is a non-zero constant, determine by inspection the indefinite integral of $\int e^{kx} dx$.

I have to solve this exercise: If $k$ is a non-zero constant, determine by inspection the indefinite integral of $\int e^{kx} dx$. By inspection, I guess it means that it should be solved by ...
69 views

### prove continuity

Let $f:\Bbb R \to \Bbb R$ satisfy the property $f(x+y)=f(x)+f(y)$ for all $x,y$ in $\Bbb R$ I have to show that 1)$f(0)=0 , f(-x)=-f(x),$ for all $x$ in $\Bbb R$, and $f(x-y)=f(x)-f(y)$ $y$ in ...
41 views

### Derive Inverse Laplace transform by differentiating F(s) and Integrating f(t) (5.5-21)

Request: Please check my work. I cannot duplicate the answer in text although it is very close. I believe the problem lies in how I take the derivative. Is there a better way to calculate the inverse ...
121 views

### find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges
Find examples of two series $\sum a_n$ and $\sum b_n$ both of which diverge but for which $\sum \min(a_n, b_n)$ converges. To make it more challenging, produce examples where $a_n$ and $b_n$ are ...