Articles of calculus

Right derivative of continuous function nonnegative implies increasing function?

Suppose $f$ is a continuous function on $(a,b)$ such that $$\lim_{h\to 0^+}\frac{f(x+h)-f(x)}{h}\geq 0$$ exists for all $x\in (a,b)$. Prove that $f$ is an increasing function on $(a,b)$, i.e. $f(x_1)\geq f(x_0)$ for all $x_1\geq x_0$, $x_0,x_1\in (a,b)$. What I tried is proof by contradiction: Suppose there exists $x_1> x_0$ but $f(x_1)<f(x_0)$. By Intermediate Value Theorem, there […]

What is $\frac{dx}{d}?$

The operator $\frac{d}{dx}$ is common in calculus to denote a derivative. However, this also begs the question, what is the operator $\frac{dx}{d}$? Is this operator used commonly? If so, what is it called/what does it do? I have played aroud with it before, and found a natural way to define it seems to be that […]

How to find the range of the given function?

Find the range of $$f(x)=\dfrac{x^2+14x+9}{x^2+2x+3}$$ where $x\in \mathbb R$ I thought of finding derivative but this will get too complicated so i am completely blank. Thanks in advance!

Evaluate this limit $\lim_{x\to – \infty}\frac{4^{x+3}-3^{x+2}-2^{x+1}}{4^{x+1}+3^{x+2}+2^{x+3}}$

Evaluate the limit $$\lim_{x\to – \infty}\frac{4^{x+3}-3^{x+2}-2^{x+1}}{4^{x+1}+3^{x+2}+2^{x+3}}.$$ I’ve tried and it always comes out $\frac{0}{0}$, and l’Hopital doesn’t seem to help me much here, what would you do?

integral of a product of sine function with a polynomial

What is the integral of this function : $$\int_{-\infty}^{+\infty}x^{\alpha} \sin{ax} \, \mathrm{d}x$$ with $\alpha, a$:real values?

Sum and average length of chords

From a specific point A on a circle’s circumference, am drawing various chords. a) what is the sum of length of such chords and b) what is the average length of such chords? Assuming a radius r, I’m trying to solve a) in the below manner: One specific chord length = 2rCos(Theta) To get the […]

Another way to express $\text{cis } 75^\circ + \text{cis } 83^\circ + \text{cis } 91^\circ + \dots + \text{cis } 147^\circ$

The number $$\text{cis } 75^\circ + \text{cis } 83^\circ + \text{cis } 91^\circ + \dots + \text{cis } 147^\circ$$ is expressed in the form $r \, \text{cis } \theta$, where $0 \le \theta < 360^\circ$. Find $\theta$ in degrees. Hint: $\text{cis}\ \theta=\cos \theta+i \sin \theta$ Edit: I simplified the expression down to$$\frac{\text{cis} \ 75^\circ \sin […]

How to find the point on the sphere that is closest to a plane?

Consider the plane $x+2y+2z=4$, how to find the point on the sphere $x^2+y^2+z^2=1$ that is closest to the plane? I could find the distance from the plane to the origin using the formula $D=\frac{|1\cdot 0+2\cdot 0+2\cdot 0-4|}{\sqrt{1^2+2^2+2^2}}=\frac43$, and then I can find the distance between the plane and sphere by subtracting the radius of sphere […]

Find the implicite and explicite set of this little cartesian product

$S$ is the set of all $(x_{1},x_{2}) \in \mathbb{R} \times \mathbb{R}$ with $x_{1}+x_{2} \geq 3$ and $-x_{1}+2x_{2}=6$ What’s its implicit and explicit set? Implicite: $S=\left\{(x_{1},x_{2})|x_{1},x_{2} \in \mathbb{R}\right\}$ For the explicite we somehow have to calculate the solutions for $x_{1}$ and $x_{2}$ (?) The problem is there is this inequality sign… $$-x_{1}+2x_{2}=6$$ $$x_{1}=2x_{2}-6$$ Now take the […]

Show that $\lim_{n \rightarrow \infty} \prod_{k=1}^{n} (1-e^{-ka})$ exists an is positive

If $a$ is a positive constant,then show that $\displaystyle \lim_{n \rightarrow \infty} \prod_{k=1}^{n} (1-e^{-ka})$ exists and is strictly positive.