All Tasks
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
Values of inverse trigonometric functions
Compute the value of $x$ that satifies the equation
\[6\arcsin\left(x+7\right)-4\;\text{arccot}(-1)=0\]
# Functions of one variable
Training
13