Articles of calculus

Strictly convex function: how often can its second derivative be zero?

It’s a basic fact that a twice-differentiable function from $\mathbb{R}$ to $\mathbb{R}$ is strictly convex if its derivative is positive everywhere. The converse is not true: consider, e.g., $f(x) = x^4$, which is strictly converse, with $f ”(0)=0$. Is there a partial converse, however? Is it true, e.g., that a strictly convex twice-differentiable function from […]

$\int {e^{3x} – e^x \over e^{4x} + e^{2x} + 1} dx$

$$I = \int {e^{3x} – e^x \over e^{4x} + e^{2x} + 1} dx$$ Substituting for $e^x$, $$I = \int {u^2 – 1 \over u^4 + u^2 + 1} du = \int { u^4 + u^2 + 1 + – 2 – u^4 \over u^4 + u^2 + 1} du = u – \int {u^4 + […]

Proving inequalities about integral approximation

We can state that, with $n$ integer, $$\int_1^n \log x \ \mathrm{dx} \leq \sum_{m = 1}^n \log m$$ because the second is the area of $n$ rectangles with unity base, while the first is “just” the area under the function. 1) How can it analitically or geometrically be proved? 2) Can this be stated in […]

Help with a limit of an integral: $\lim_{h\to \infty}h\int_{0}^\infty{{ {e}^{-hx}f(x)} dx}=f(0)$

I’m not sure how to handle limits and integral and I would like some help with the following one: let $f:[0,\infty)\rightarrow \Bbb{R}$ be a continuous and bounded function, show that $$\lim_{h\to \infty}h\int_{0}^\infty{{ {e}^{-hx}f(x)} dx}=f(0)$$ I tried many things from The fundamental theorem of calculus and define $F$ such that $F’=f$ and use integration by parts […]

the approximation of $\log(266)$?

Consider the following exercise: Of the following, which is the best approximation of $\sqrt{1.5}(266)^{1.5}$? A 1,000 B 2,700 C 3,200 D 4,100 E 5,300 The direct idea is using the “differential approximation”: $$f(x)\approx f(x_0)+f'(x_0)(x-x_0)$$ where $f(x)=\sqrt{x}266^x$ and $x_0=1$, $x=1.5$. Finally, one may have to approximate $\log 266$. So here are my questions: How to approximate […]

Integral without using Euler substitution

Help me please with integral: $$\int \frac{2x-\sqrt{4x^{2}-x+1}}{x-1}\;dx$$ I must solve it without using Euler substitution. Thanks!

How calculate the indefinite integral $\int\frac{1}{x^3+x+1}dx$

How do I calculate the following indefinite integral? $$\int\frac{1}{x^3+x+1}dx$$ Approach: $x^3+x+1=(x-a)(x^2+ax+c)$ where $a:$ real solution of the equation $a^3+a+1=0$ $c:$ real solution of the equation $c^3-c^2+1=0$ Then $$\int\frac{1}{x^3+x+1}dx=\int\frac{1}{(x-a)(x^2+ax+c)}dx=\int\frac{A}{(x-a)}dx+\int\frac{Bx+C}{(x^2+ax+c)}dx$$

A contest math integral: $\int_1^\infty \frac{\text{d}x}{\pi^{nx}-1}$

My school holds a math contest that has problems that vary level to level. Nobody managed to solve this particular one: $$\int_1^\infty \frac{\text{d}x}{\pi^{nx}-1}$$ In terms of $n$ I was wondering if there is a solution to this integral?

Find the value of : $\lim_{x \to \infty}( \sqrt{4x^2+5x} – \sqrt{4x^2+x})$

$$\lim_{x \to \infty} \left(\sqrt{4x^2+5x} – \sqrt{4x^2+x}\ \right)$$ I have a lot of approaches, but it seems that I get stuck in all of those unfortunately. So for example I have tried to multiply both numerator and denominator by the conjugate $\left(\sqrt{4x^2+5x} + \sqrt{4x^2+x}\right)$, then I get $\displaystyle \frac{4x}{\sqrt{4x^2+5x} + \sqrt{4x^2+x}}$, but I can conclude nothing […]

Proving roots with Mean Value Theorem

Stewart wants me to prove stuff but I have no idea how to. a) Show that a polynomial of degree 3 has at most three real roots. b) Show that a polynomial of degree n has at most n real roots.