Articles of calculus

Limit with Integral and Sigma

Problem: $f'(x)$ is a continuous function for $[0, 1]$. Show that $$\lim_{n \to \infty}n \left( \frac{1}{n}\sum_{i=1}^n {f\left(\frac{i}{n}\right)-\int_0^1f(x)dx} \right)=\frac{f(1)-f(0)}{2}$$ I tried to use the definition of the definite integral to change it to a limit but it doesn’t seem to work. And I also wonder why a continuous condition for $f'(x)$ was given. I thought it […]

Computing limit of $(1+1/n^2)^n$

This question already has an answer here: Prove that $( 1 + n^{-2}) ^n \to 1$. 8 answers

How to evaluate this limit? Riemann Integral

$$\large f(x)= \lim_{n\rightarrow \infty}\left( \dfrac{n^n(x+n)\left( x+\dfrac{n}{2}\right)\left( x+\dfrac{n}{3}\right)… \left( x+\dfrac{n}{n}\right)}{n!(x^2+n^2)\left( x^2+\dfrac{n^2}{4}\right)\left( x^2+\dfrac{n^2}{9}\right)…\left( x^2+\dfrac{n^2}{n^2}\right)}\right)$$ $x\in R^+$ Find the coordinates of the maxima of $f(x)$. My Work: Is the method correct? Is there an easier way?

Show that $|\sin{a}-\sin{b}| \le |a-b| $ for all $a$ and $b$

I’ve recently been going over the mean value and intermediate value theorems, however I’m not sure where to start on this.

How to find the line that splits the area into two equal parts?

Let $R$ be the region bounded by the graphs of $y = \cos \left(\frac{\pi x}{2}\right)$ and $y=x^2 -1$. The line $y=k$ splits the region $R$ into two equal parts. Find the value of $k$. First find the area. $$A = \int\limits_{-1}^1 \left[\cos \left(\frac{\pi x}{2}\right) – x^2 +1\right]\, \mathrm{d}x = \frac{4}{\pi} + \frac{4}{3}$$ Not exactly sure […]

Calculating/Estimating difference between Harmonic numbers

The Harmonic number $H_n$ is defined as $H_n=\sum_{t=1}^n\frac{1}{t}$. I wish to compute $\lim_{n\to\infty}\frac{k}{n}(H_n-H_k)$ where $k$ is a function of $n$ (which can be a constant function, e.g. $k=3$). Moreover, I wish the computation to be as simple as possible and self-sufficient (which might already be too much to ask). If a general solution is not […]

Implicit Differentiation Quesrion

$$x^y = y^x$$ is the equation My solution is :$$\frac { dy }{ dx } =\left( \ln { y-\frac { y }{ x } } \right) /\left( \ln { x-\frac { x }{ y } } \right) $$ Just wondering if this is correct!

$\lim_{(x,y)\rightarrow (0,0)}\dfrac{xy^2}{x^2-y^2}$

I want to solve the following question. $$\lim_{(x,y)\rightarrow (0,0)}\dfrac{xy^2}{x^2-y^2}$$ I am going to use polar coordinates. $x=r\cos\theta$ and $y=r\sin\theta$ $$\lim_{(x,y)\rightarrow (0,0)}\dfrac{xy^2}{x^2-y^2}=\lim_{r\rightarrow 0^+}\frac{r^3\cos\theta\sin\theta}{r^2(\cos^2\theta-\sin^2\theta)}=\lim_{r\rightarrow 0^+}\dfrac{r\cos\theta\sin^2\theta}{\cos^2\theta-\sin^2\theta}=0$$ if $\theta\neq \dfrac{\pi}{4},\dfrac{3\pi}{4},\dfrac{5\pi}{4}, \dfrac{7\pi}{4}.$ For values of $\theta$ we have the lines $y=x$ and $y=-x$, but they are bot in the domain of the function. Wolfram says it does not exist. Why?

The following series is an irrational number

This question already has an answer here: Prove the series $ \sum_{n=1}^\infty \frac{1}{(n!)^2}$ converges to an irrational number 4 answers The sum of the series $\sum_{n=0}^{\infty}\frac{\epsilon_n}{n!}$ is an irrational number 2 answers

Graphing and differentiation

I am having trouble making sense of how to figure out what a graph looks like just by knowing the critical numbers and the intervals the function is increasing and when it is decreasing. For example I have $(-\infty, -1), (0,1)$ as negative and $(-1,0), (1,+\infty)$ as positive. I know that the it can only […]