## All Tasks

#### Graphs - Inverse trigonometric functions

Consider the following functions: $f_1(x)=\arccos\left(-\frac{x}{4}\right)-\frac{\pi}{4}$ $f_2(x)=\frac{\pi}{4}+\arctan(2x)$ $f_3(x)=\frac{\pi}{4}-\arcsin\left(\frac{x}{4}\right)$ $f_4(x)=\text{arccotan}(2x)-\frac{\pi}{4}$ Match each function with its respective graph shown in the figure.

# Functions of one variable

Reasoning
13

#### Chain rule arcsin

Let $f$ be a diferentiable function in $\mathbb{R}$ and $g(x)=f(\pi/2+\arcsin(2x-1))$. Find the derivative of $g$ at the point where the line $2x-2y=1$ intercepts The $x$ axis.

# Functions of one variable

Reasoning
13

#### Scale balance

How heavy is the boy?
# Terms with variables
# Systems of linear equations

Modeling
9

#### Side length of a rectangle

Which of the following equations fits the picture if the goal is to calculate the length of x?
# Linear equations
# Square & rectangle

Reasoning
9

#### Differential of the function arcsin

Find the differential of the function $y=\arcsin(\frac{u}{v})$ where $u$ and $v$ are differentiable functions of $x$.

# Functions of one variable

Training
13

#### Derivation rules arcsin

Let $\displaystyle f(x)= a -2 \arcsin(1-2x)$ with $a \in \mathbb{R}$ and let $f'(x)$ and $f''(x)$ be the first and the second derivative of a function $f$. Knowing that $\displaystyle \dfrac{f\left(\dfrac{1}{2}\right)}{f'\left(\dfrac{1}{2}\right)} + f''\left(\frac{1}{2}\right) = 2$, then the value of $a$ is:

# Functions of one variable

Training
13