Found problems: 348
The function $f: \mathbb{R}\rightarrow\mathbb{R}$ is twice differentiable and satisfies $f(0)=2,f'(0)=-2,f(1)=1$.
Prove that there is a $\xi \in ]0,1[$ for which we have $f(\xi)\cdot f'(\xi)+f''(\xi)=0$.
Suppose the function $f(x)-f(2x)$ has derivative $5$ at $x=1$ and derivative $7$ at $x=2$. Find the derivative of $f(x)-f(4x)$ at $x=1$.
Let $a$, $b$, $c$ be positive real numbers. Prove that \[ \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge\frac52\] and determine when equality holds.
Define mapping $F : \mathbb{R}^4\rightarrow \mathbb{R}^4$ as $F(x,\ y,\ z,\ w)=(xy,\ y,\ z,\ w)$ and let mapping $f : S^3\rightarrow \mathbb{R}^4$ be restriction of $F$ to 3 dimensional ball $S^3=\{(x,\ y,\ z,\ w)\in{\mathbb{R}^4} | x^2+y^2+z^2+w^2=1\}$.
Find the rank of $df_p$, or the differentiation of $f$ at every point $p$ in $S^3$.
The formula which expresses the relationship between $x$ and $y$ as shown in the accompanying table is:
\[ \begin{tabular}[t]{|c|c|c|c|c|c|}\hline
x&0&1&2&3&4\\\hline
y&100&90&70&40&0\\\hline
\end{tabular}\]
$\textbf{(A)}\ y=100-10x \qquad
\textbf{(B)}\ y=100-5x^2 \qquad
\textbf{(C)}\ y=100-5x-5x^2 \qquad\\
\textbf{(D)}\ y=20-x-x^2 \qquad
\textbf{(E)}\ \text{None of these}$
Calculate
\[T = \sum \frac{1}{n_1! \cdot n_2! \cdot \cdots n_{1994}! \cdot (n_2 + 2 \cdot n_3 + 3 \cdot n_4 + \ldots + 1993 \cdot n_{1994})!}\]
where the sum is taken over all 1994-tuples of the numbers $n_1, n_2, \ldots, n_{1994} \in \mathbb{N} \cup \{0\}$ satisfying $n_1 + 2 \cdot n_2 + 3 \cdot n_3 + \ldots + 1994 \cdot n_{1994} = 1994.$
The elliptic curve $y^2=x^3+1$ is tangent to a circle centered at $(4,0)$ at the point $(x_0,y_0)$. Determine the sum of all possible values of $x_0$.
Let $f(x)=\int_0^{x} e^{t} (\cos t+\sin t)\ dt,\ g(x)=\int_0^{x} e^{t} (\cos t-\sin t)\ dt.$
For a real number $a$, find $\sum_{n=1}^{\infty} \frac{e^{2a}}{\{f^{(n)}(a)\}^2+\{g^{(n)}(a)\}^2}.$
Find the positive value of $ a$ such that the curve $ C_1: x \equal{} \sqrt {2y^2 \plus{} \frac {25}{2}}$ tangent to the parabola $ C_2: y \equal{} ax^2$, then find the equation of the tangent line of $ C_1$ at the point of tangency.
Calculate the following indefinite integrals.
[1] $\int \sin x\cos ^ 3 x dx$
[2] $\int \frac{dx}{(1+\sqrt{x})\sqrt{x}}dx$
[3] $\int x^2 \sqrt{x^3+1}dx$
[4] $\int \frac{e^{2x}-3e^{x}}{e^x}dx$
[5] $\int (1-x^2)e^x dx$
Suppose that the $n$ times differentiable real function $f(x)$ has at least $n+1$ distinct zeros in the closed interval $[a,b]$ and that the polynomial $P(z)=z^n +c_{n-1}z^{n-1}+\ldots+c_1 x +c_0$ has only real zeroes. Show that
$f^{(n)}(x)+ c_{n-1} f^{(n-1)}(x) +\ldots +c_1 f'(x)+ c_0 f(x)$ has at least one zero in $[a,b]$, where $f^{(n)}$ denotes the $n$-th derivative of $f.$
Differentiable functions $f_1,\ldots,f_n:\mathbb R\to\mathbb R$ are linearly independent. Prove that there exist at least $n-1$ linearly independent functions among $f_1',\ldots,f_n'$.
A real function $f$ is defined for $0\le x\le 1$, with its first derivative $f'$ defined for $0\le x\le 1$ and its second derivative $f''$ defined for $0<x<1$. Prove that if $f(0)=f'(0)=f'(1)=f(1)-1 =0$, then there exists a number $0<y<1$ such that $|f''(y)|\ge 4$.
Suppose that $p(x)$ is a polynomial and that $p(x)-p^\prime (x)=x^2+2x+1$. Compute $p(5)$.
Let $ f:[0,2]\longrightarrow \mathbb{R} $ a differentiable function having a continuous derivative and satisfying $ f(0)=f(2)=1 $ and $ |f’|\le 1. $ Show that
$$ \left| \int_0^2 f(t) dt\right| >1. $$
If $r$ is positive and the line whose equation is $x + y = r$ is tangen to the circle whose equation is $x^2 + y ^2 = r$, then $r$ equals
$\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt{2}\qquad \textbf{(E) }2\sqrt{2}$
The equation $x^3+px+q=0$ has three distinct real roots. Show that $p<0$
A cat is trying to catch a mouse in the non-negative quadrant \[N=\{(x_1,x_2)\in \mathbb{R}^2: x_1,x_2\geq 0\}.\]
At time $t=0$ the cat is at $(1,1)$ and the mouse is at $(0,0)$. The cat moves with speed $\sqrt{2}$ such that the position $c(t)=(c_1(t),c_2(t))$ is continuous, and differentiable except at finitely many points; while the mouse moves with speed $1$ such that its position $m(t)=(m_1(t),m_2(t))$ is also continuous, and differentiable except at finitely many points. Thus $c(0)=(1,1)$ and $m(0)=(0,0)$;
$c(t)$ and $m(t)$ are continuous functions of $t$ such that $c(t),m(t)\in N$ for all $t\geq 0$; the derivatives $c'(t)=(c'_1(t),c'_2(t))$ and $m'(t)=(m'_1(t),m'_2(t))$ each exist for all but finitely many $t$ and \[(c'_1(t)^2+(c'_2(t))^2=2 \qquad (m'_1(t)^2+(m'_2(t))^2=1,\] whenever the respective derivative exists.
At each time $t$ the cat knows both the mouse's position $m(t)$ and velocity $m'(t)$.
Show that, no matter how the mouse moves, the cat can catch it by time $t=1$; that is, show that the cat can move such that $c(\tau)=m(\tau)$ for some $\tau\in[0,1]$.
The zeroes of a fourth degree polynomial $f(x)$ form an arithmetic progression. Prove that the three zeroes of the polynomial $f'(x)$ also form an arithmetic progression.
Let $f:[0,1] \to [0,1]$ a increasing continuous function, diferentiable in $(0,1)$ and with derivative smaller than 1 in every point. The sequence of sets $A_1,A_2,A_3,\dots$ is define as: $A_1 = f([0,1])$, and for $n \geq 2, A_n = f(A_{n-1}).$ Prove that $\displaystyle \lim_{n\to+\infty} d(A_n) = 0$, where $d(A)$ is the diameter of the set $A$.
Note: The diameter of a set $X$ is define as $d(X) = \sup_{x,y\in X} |x-y|.$
Let $a$ and $b$ be real numbers satisfying $a>b>0$. Evaluate \[\int_0^{2\pi}\dfrac{1}{a+b\cos(\theta)}d\theta.\] Express your answer in terms of $a$ and $b$.
([b]5[/b]) Find $ p$ so that $ \lim_{x\rightarrow\infty}x^p\left(\sqrt[3]{x\plus{}1}\plus{}\sqrt[3]{x\minus{}1}\minus{}2\sqrt[3]{x}\right)$ is some non-zero real number.
Let $r\geqslant 0$ be a real number and define $f(x)=1/(1+x^2)^r$. Prove that \[|f^{(k)}(x)|\leqslant\frac{2r\cdot(2r+1)\cdots(2r+k-1)}{(1+x^2)^{r+k/2}},\]for every natural number $k{}$. Here, $f^{(k)}(x)$ denotes the $k^{\text{th}}$ derivative of $f$.
Find the derivatives of the lengths of the semiaxes of the ellipsoid $x^2 + y^2 + z^2 + xy + yz + zx = 1 + \epsilon xy$ with respect to $\epsilon$ at $\epsilon = 0$.
The function $f(x)$ has the property that, for some real positive constant $C$, the expression \[\frac{f^{(n)}(x)}{n+x+C}\] is independent of $n$ for all nonnegative integers $n$, provided that $n+x+C\neq 0$. Given that $f'(0)=1$ and $\int_{0}^{1}f(x) \, dx = C+(e-2)$, determine the value of $C$.
Note: $f^{(n)}(x)$ is the $n$-th derivative of $f(x)$, and $f^{(0)}(x)$ is defined to be $f(x)$.