Found problems: 913
Solve the equation $x^2+1 = log_2(x+2)- 2x$.
For a positive real number $ a > 1$, prove the following inequality.
$ \frac {1}{a \minus{} 1}\left(1 \minus{} \frac {\ln a}{a\minus{}1}\right) < \int_0^1 \frac {x}{a^x}\ dx < \frac {1}{\ln a}\left\{1 \minus{} \frac {\ln (\ln a \plus{} 1)}{\ln a}\right\}$
Solve the equation:
\[2^{\lg x}+8=(x-8)^{\frac{1}{\lg 2}}\]
Note: $\lg x=\log_{10}x$.
[i]Daniel Jinga [/i]
For a function $ f(x)\equal{}(\ln x)^2\plus{}2\ln x$, let $ C$ be the curve $ y\equal{}f(x)$. Denote $ A(a,\ f(a)),\ B(b,\ f(b))\ (a<b)$ the points of tangency of two tangents drawn from the origin $ O$ to $ C$ and the curve $ C$. Answer the following questions.
(1) Examine the increase and decrease, extremal value and inflection point , then draw the approximate garph of the curve $ C$.
(2) Find the values of $ a,\ b$.
(3) Find the volume by a rotation of the figure bounded by the part from the point $ A$ to the point $ B$ and line segments $ OA,\ OB$ around the $ y$-axis.
Let $L$ be a positive constant. For a point $P(t,\ 0)$ on the positive part of the $x$ axis on the coordinate plane, denote $Q(u(t),\ v(t))$ the point at which the point reach starting from $P$ proceeds by distance $L$ in counter-clockwise on the perimeter of a circle passing the point $P$ with center $O$.
(1) Find $u(t),\ v(t)$.
(2) For real number $a$ with $0<a<1$, find $f(a)=\int_a^1 \sqrt{\{u'(t)\}^2+\{v'(t)\}^2}\ dt$.
(3) Find $\lim_{a\rightarrow +0} \frac{f(a)}{\ln a}$.
[i]2011 Tokyo University entrance exam/Science, Problem 3[/i]
Without tables and such, prove that $\frac{1}{\log_2 \pi}+\frac{1}{\log_5 \pi} >2$
Define $a_{n}$ as satisfying: $\left(1+\frac{1}{n}\right)^{n+a_{n}}=e$.
Find $\lim_{n\rightarrow\infty}a_{n}$.
What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?
Given that $\log_{10}2 \approx 0.30103$, find the smallest positive integer $n$ such that the decimal representation of $2^{10n}$ does not begin with the digit $1$.
Suppose $ M(y)$ is a polynomial of degree $ n$ such that $ M(y) \equal{} 2^y$ for $ y \equal{} 1, 2, \ldots, n \plus{} 1$. Compute $ M(n \plus{} 2)$.
Let $a_n$ be the local maximum of $f_n(x)=\frac{x^ne^{-x+n\pi}}{n!}\ (n=1,\ 2,\ \cdots)$ for $x>0$.
Find $\lim_{n\to\infty} \ln \left(\frac{a_{2n}}{a_n}\right)^{\frac{1}{n}}$.
Prove that if $1 < a < b < c$, then $$\log_a(\log_a b) + \log_b(\log_b c) + \log_c(\log_c a) > 0.$$
Calculate the following indefinite integrals.
[1] $\int \frac{dx}{e^x-e^{-x}}$
[2] $\int e^{-ax}\cos 2x dx\ (a\neq 0)$
[3] $\int (3^x-2)^2 dx$
[4] $\int \frac{x^4+2x^3+3x^2+1}{(x+2)^5}dx$
[5] $\int \frac{dx}{1-\cos x}dx$