#### Welcome to ASYMPTOTE

ASYMPTOTE is a free learning environment that enables adaptive and synchronous online teaching. We explain how to use the app in this learning graph. Enjoy!

# Unassigned

Learning
5

#### Values of inverse trigonometric functions

Find the values of $\alpha$ and $\beta$ given by $\alpha=\arccos\left(-\frac{1}{2}\right)$ and $\beta=\arctan\left(-\frac{\sqrt 3}{3}\right)$

# Functions of one variable

Training
13

#### Derivative arccot (I)

Let $f(x)=\text{arccot}(\alpha-5x)-\pi,\quad \alpha\in\mathbb{R}$. If $f'(1)=1$ then $\alpha$ is equal to

# Functions of one variable

Training
13

#### Matrix multiplication (B+T)

Let $A=\begin{bmatrix} 4&-2&5\\2&6&0\\3&3&3\end{bmatrix}$ and $B=\begin{bmatrix} 1&0&0\\0&2&-3\\6&-6&2\end{bmatrix}$ be real matrices $M_{3\times 3}$. The matrix $AB$ is:

# Operation

Training
13

#### Tangent and normal lines

Let $f(x)= \dfrac{\pi}{2} -2 \arcsin(1-2x)$ with domain $D_f = \left[0, 1\right]$. The reduced equation of the line normal to the graph of $f$ at the point with abscissa $\dfrac{1}{4}$ is:

# Functions of one variable

Training
13

#### Chain rule arccos

Let $y=y(x)=\arccos(1-5x)$ and $x=x(t)=\frac{1}{5}+ln(3t^2-2)$. Using the chain rule formula to express the derivative $\left.\frac{dy}{dt}\right|_{t=1}$.

# Functions of one variable

Training
13