Calculus

# Higher-order Derivatives: Level 2 Challenges

Given the graph of $$y=f(x)$$ above, which of the following is a possible graph of $$y=f''(x)?$$

Find the 2016-th derivative of $$\sin ^{ -1 }{ (x) }$$ at $$x=0$$.

###### Image Credit: Flickr Richard Stocker.

If a function $$f(x)$$ that is differentiable over $$(-\infty,\infty)$$ is monotonically decreasing and $$\displaystyle\lim_{x\rightarrow\infty}f(x)\neq-\infty,$$ then as $$x$$ approaches infinity, $$f(x)$$ is

$y = \tan^{-1}(x) , k! = \left. \dfrac { { d }^{ 21 }y }{ d{ x }^{ 21 } } \right|_{x=0}, \ \ \ \ \ k = \ ?$

Suppose $$f$$ is a function defined on the closed interval $$-3 \le x \le 4$$ with $$f(0)=42$$ such that the graph of $$f',$$ the derivative of $$f,$$ on the interval is as shown in the above diagram. Find the $$x$$-coordinates of the points of inflection of $$f.$$

×