Essential Knowledge 2.1D1

Essential Knowledge 2.1D1 Students will know that differentiating f ' produces the second derivative f '', provided the derivative of f ' exists; repeating this process produces higher order derivatives of f.

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Perhaps I'm using this question too often, but this is no accident. I think questions about objects falling are a great way to learn calculus.

Example

If I drop a ball off of a 1000 foot tall building, what is the acceleration of the ball at 2 seconds? At 5 seconds? Remember the equation for the position of the ball is y = -16t² + 1000

This time the question asks you to find acceleration. What is acceleration? Acceleration is the rate of change of velocity. In other words, velocity measures how quickly the ball changes its position, while acceleration measures how quickly the ball changes its velocity. A car increases its speed by accelerating

With this language, you might have figured out how to find acceleration. Let's go through the logic

Velocity: Rate of change of position. To find this we took the derivative of position with respect to time.

Therefore,

Acceleration: Rate of change of velocity. To find this we must take the derivative of velocity with respect to time.

So let's do this. First I'll need to find the velocity.

and therefore

Now that we have the velocity, we can take the derivative of velocity to find acceleration

and therefore

This means that the ball accelerates downwards at a rate of 32 ft per second, per second. Also known as 32ft/s²

Because we took the derivative of a derivative, acceleration is known as the "second derivative" of position with respect to time. In the next Essential Knowledge, we'll learn all of the proper terminology and symbols for higher order derivatives.