Conceptualizing Derivatives - High School Math
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The speed of a car traveling on the highway is given by the following function of time:
Note that
What does this mean?
The speed of a car traveling on the highway is given by the following function of time:
Note that
What does this mean?
The function 
gives you the car's speed at time 
. Therefore, the fact that 
means that the car's speed is 
at time 
. This is equivalent to saying that the car is not moving at time 
. We have to take the derivative of 
to make claims about the acceleration.
The function
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The speed of a car traveling on the highway is given by the following function of time:
Consider a second function:
What can we conclude about this second function?
The speed of a car traveling on the highway is given by the following function of time:
Consider a second function:
What can we conclude about this second function?
Notice that the function 
is simply the derivative of 
with respect to time. To see this, simply use the power rule on each of the two terms.
Therefore, 
is the rate at which the car's speed changes, a quantity called acceleration.
Notice that the function
Therefore,
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