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# Displacement, Velocity, Acceleration

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.

# Finding Acceleration Given Velocity

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 $$t=1?$$

At time $$t$$, the velocity of a motorcyclist is $$v(t) = 54 - 36e^{-t}$$ mph. What is his acceleration when his velocity is 11?

The position of a particle at time $$t$$ is given by $$f(t) = t^{6} + 3 t^{3}$$. What is the acceleration of the particle at time $$t=2?$$

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 $$2?$$

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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