How fast are you moving? How fast is how fast you are moving changing? Displacement, velocity, and acceleration form the art of understanding movement, Calculus-style.

The velocity of a particle at time \( t \) is given by \( f(t) = \sin ^{5} \left(2t \right).\) What is the acceleration of the particle at time \(t=1?\)

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The velocity of a particle at time \( t \) is given by a function \( f(t) \) which satisfies \[ \left( 2 + f(t) \right) e^{ (t-2) f(t)} = 9 .\] If \( f(t) \) is differentiable at time \(2,\) what is the particle's acceleration at time \(2?\)

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The velocity of a particle at time \( t \) is given by \( \displaystyle{ f(t) = \frac{t^{3}}{ t^{3} + 1} }. \) If the acceleration of the particle at time \( t=2 \) is \(A,\) what is \( 81 A ? \)

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