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Higher-order Derivatives

The first derivative is the slope of a curve, and the second derivative is the slope of the slope, like acceleration. So what's the third derivative? (Fun fact: it's actually called jerk.)

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 \(x=0\).

Give your answer to 3 decimal places.

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 \(f.\)


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