## 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