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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.)

Level 1

         

What is the fewest number of derivatives one needs to compute so that \(f(x) = 568{ x }^{ 499 }\) becomes zero?

In the above diagram, the red curve is the graph of a function \(f,\) and the blue curve is the graph of its first derivative \(f'.\) What is the relationship between \(f'(-1)\) and \(f''(1) ?\)

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

SecondDerivatives

SecondDerivatives

Let \(P\) be a point moving in the \(xy\)-plane whose coordinates at time \(t\) are given by \[x(t)=2e^t\sin t, y(t)=6\cos t.\] What is the minimum value of the magnitude of the acceleration of \(P?\)

Find \(f^{(100)}(0)\). If

\[\large{ f(x) = \sin x + x^{100000}}\]

Note: \(f^n(x)\) denotes the \(n^{th}\) derivative of \(f(x)\).

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