# High School Math : Conceptualizing Derivatives

## Example Questions

### Example Question #1 : Introduction To Derivatives

The speed of a car traveling on the highway is given by the following function of time:

Note that

What does this mean?

Possible Answers:

The car is not moving at time .

The car is not decelerating at time .

The car's speed is constantly changing at time .

The car takes  seconds to reach its maximum speed.

The car is not accelerating at time .

Correct answer:

The car is not moving at time .

Explanation:

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.

### Example Question #1 : Conceptualizing Derivatives

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?

Possible Answers:

It represents another way to write the car's speed.

It has no relation to the previous function.

It represents the total distance the car has traveled at time .

It represents the change in distance over a given time .

It represents the rate at which the speed of the car is changing.

Correct answer:

It represents the rate at which the speed of the car is changing.

Explanation:

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.