All Tasks - Asymptote

Consider $f(x)= -3\pi +a \arccos(3x+1)$ with $a \in \mathbb R$ and domain $D_f=\left[-\dfrac{2}{3}, 0\right]$. Knowing that the value of the differential of $f(x)$ at the point of abscissa $x_0=-\dfrac{1}{3}$ with $\Delta x=0.1$, is $df\left(-\dfrac{1}{3}\right)=-9$, then the value of $a$ is:

Consider the function $f(x)= 5 \arccos\left(2x\right)$ with domain $D_f = \left[ -\dfrac{1}{2}, \dfrac{1}{2} \right]$. The differential of the function $y=f(x)$ at the point of abscissa $x=0$ is:

Consider the function $f(x)= 5 \arccos\left(2x\right)$ with domain $D_f = \left[ -\dfrac{1}{2}, \dfrac{1}{2} \right]$. Using the notion of differential, the approximate value of $5 \arccos\left(0.2\right)$ is:

In a gym there is an exercise machine that consists of an inclined plane (2 meters long) leaning against a wall (see the schematic of the machine in the figure). Initially, the base of the plane was placed at a distance $x = 1 \ m$ from the wall, and in this case, the plane is inclined at an angle of $\theta=\dfrac{\pi}{3} \ rad$ to the floor. The steepness is given by the function $$\theta (x)= \arccos \left( \dfrac{x}{2}\right).$$ Estimate, using the differential, what is the new steepness of the machine if the distance from the base of the plane to the wall is increased to $x = 1.05 \ m$.

Jon is an American guy who loves Big Macs. While on a trip around the world, he eats eight Big Macs: two in Germany, one in the United States, two in Switzerland, and three in India. Assuming that the prices of the Big Mac are: In Germany 3.59 €, in the United States 4.59 €, in Switzerland 5.99 € and in India 1.77 €. a) What was Jon's total expenditure on Big Macs during his vacation? b) If his flight was canceled, Jon would stay in the United States. In the United States, how much would he spend on the same number of Big Macs?

The population of a town is 20.000 citizens. From them 8.206 are men. Men are 426 fewer than women. How many children are living in the town?