Three friends, Sara, Cristina and Jimena, have a total of 15000 followers on a social network. If Jimena lost $25 \%$ of her followers she would still have three times as many followers as Sara. Furthermore, half of Sara's followers plus one-fifth of Cristina's followers equal one-quarter of Jimena's followers. Calculate how many followers each of the three friends has.
a) (1.25 points) Calculate, if they exist, the value of the following limits: a.1) $(0.5$ points) $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } ( 1 - 2 x ) } { x - 2 x ^ { 2 } - \operatorname { sen } x }$ a.2) (0.75 points) $\lim _ { x \rightarrow \infty } \frac { 1 } { x } \left( \frac { 3 } { x } - \frac { 2 } { \operatorname { sen } \frac { 1 } { x } } \right)$ (Hint: use the change of variable $t = 1 / x$ where necessary). b) (1.25 points) Calculate the following integrals: b.1) (0.5 points) $\int \frac { x } { x ^ { 2 } - 1 } d x$ b.2) (0.75 points) $\int _ { 0 } ^ { 1 } x ^ { 2 } e ^ { - x } d x$
Given the point $A ( 1,0 , - 1 )$, the line $r \equiv x - 1 = y + 1 = \frac { z - 2 } { 2 }$ and the plane $\pi \equiv x + y - z = 6$, find: a) ( 0.75 points) Find the angle formed by the plane $\pi$ and the plane perpendicular to the line $r$ that passes through point $A$. b) ( 0.75 points) Determine the distance between the line $r$ and the plane $\pi$. c) (1 point) Calculate an equation of the line that passes through A, forms a right angle with the line $r$ and does not intersect the plane $\pi$.
In an urn there are two white balls and four black balls. A ball is drawn at random. If the ball drawn is white, it is returned to the urn and another white ball is added; if it is black, it is not returned to the urn. Next, a ball is drawn at random from the urn again. a) (1 point) What is the probability that the two balls drawn are of different colors? b) (1.5 points) What is the probability that the first ball drawn was black, given that the second was white?
a) ( 0.75 points) Find a single system of two linear equations in the variables $x$ and $y$, which has as solutions $\{ x = 1 , y = 2 \}$ and $\{ x = 0 , y = 0 \}$. b) (1 point) Find a system of two linear equations in the variables $x , y$ and z whose solutions are, as a function of the parameter $\lambda \in \mathbb { R }$ : $$\left\{ \begin{array} { l }
x = \lambda \\
y = \lambda - 2 \\
z = \lambda - 1
\end{array} \right.$$ c) ( 0.75 points) Find a system of three linear equations with two unknowns, x and y, that has only the solution $x = 1$ and $y = 2$.
Let the function $$f ( x ) = x ^ { 3 } - | x | + 2$$ a) ( 0.75 points) Study the continuity and differentiability of $f$ at $x = 0$. b) (1 point) Determine the relative extrema of $f ( x )$ on the real line. c) ( 0.75 points) Calculate the area of the region bounded by the graph of $f$, the x-axis $y = 0$, and the lines $x = - 1$ and $x = 1$.
Given the lines $$r \equiv \frac { x - 2 } { 1 } = \frac { y + 1 } { 1 } = \frac { z + 4 } { - 3 } , \quad s \equiv \left\{ \begin{array} { l }
x + z = 2 \\
- 2 x + y - 2 z = 1
\end{array} . \right.$$ a) (1.5 points) Write an equation of the common perpendicular line to $r$ and $s$. b) (1 point) Calculate the distance between $r$ and $s$.
QB.4
2.5 marksApproximating Binomial to Normal DistributionView
According to meteorological statistics, in a Nordic city it rains on an average of $45 \%$ of the days. A climatologist analyzes rainfall records from 100 days chosen at random from the last 50 years. a) (1 point) Express how to calculate exactly the probability that it rained on 40 of them. b) (1.5 points) Calculate this probability by approximating it using a normal distribution.