All Tasks

IFDT_arctan_B_T_1

Consider $y=f(x)= -2 \arctan\left(2x\right)$ with domain $D_f=\mathbb{R}$. Applying the inverse function derivative theorem, the expression of $\dfrac{dy}{dx}$ is:
 
# Complements of differential calculus in real numbers

Training
13

IFDT_arctan_I_T_1

Consider $y=f(x)= \dfrac{\pi}{2} -2 \arctan(1-2x)$ with domain $D_f=\mathbb{R}$. Applying the inverse function derivative theorem, the expression of $\dfrac{dy}{dx}$ is:
 
# Complements of differential calculus in real numbers

Training
13

(A) Derivation arctan

Let $ \displaystyle f(x)= \dfrac{\pi}{2} -b \arctan(1-2x)$ with $b \in \mathbb{R}$, and let $f'(x)$ and $f''(x)$ be the first and the second derivative of the function $f$. Knowing that $\displaystyle f\left(\dfrac{1}{2}\right) \times f'\left(\dfrac{1}{2}\right) + f''\left(\frac{1}{2}\right) = 4 \pi$, then the value of $b$ is:
 
# Complements of differential calculus in real numbers

Training
13

(B) Derivation arctan

Consider $f(x)= -2 \arctan\left(2x\right)$ with domain $D_f=\mathbb{R}$. Let $f'(x)$ be the first derivative of function $f(x)$, then $f'\left(-\dfrac{1}{2}\right)$ is:
 
# Complements of differential calculus in real numbers

Training
13

(I) Derivation arctan

Let $f$ and $g$ be two differentiable functions in their domain, such that: $f(x)= \dfrac{\pi}{2} -2 \arctan(1-2x)$ and $g(x)=\left( x^2+1\right) f(x)$. Considering that $g'(x)$ is the first derivative of function $g(x)$, then $g'\left(\dfrac{1}{2}\right)$ is:
 
# Complements of differential calculus in real numbers

Training
13

(B) Inverse arctan

Consider $f(x)= -2 \arctan\left(2x\right)$ with domain $D_f=\mathbb{R}$ and range $D'_f = \left] -\pi, \pi \right[$. The analytic expression of the inverse function of $f$, $f^{-1}(x)$, and $f^{-1}\left(-\dfrac{\pi}{3} \right) $ are, respectively,
 
# Complements of differential calculus in real numbers

Training
13