# AP Calculus AB : Corresponding characteristics of graphs of ƒ and ƒ'

## Example Questions

### Example Question #1 : Derivatives

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

Consider a second function:

It represents the change in distance over a given time .

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

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

It has no relation to the previous function.

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

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.

### Example Question #1 : Corresponding Characteristics Of Graphs Of ƒ And ƒ'

Find the critical numbers of the function,

Explanation:

1) Recall the definition of a critical point:

The critical points of a function  are defined as points , such that  is in the domain of , and at which the derivative  is either zero or does not exist. The number  is called a critical number of .

2) Differentiate

3) Set to zero and solve for

The critical numbers are,

We can also observe that the derivative does not exist at , since the term would be come infinite. However,  is not a critical number because the original function  is not defined at . The original function is infinite at , and therefore  is a vertical asymptote of  as can be seen in its' graph,

Further Discussion

In this problem we were asked to obtain the critical numbers. If were were asked to find the critical points, we would simply evaluate the function at the critical numbers to find the corresponding function values and then write them as a set of ordered pairs,

### Example Question #1 : Derivative As A Function

The function  is a continuous, twice-differentiable functuon defined for all real numbers.

If the following are true:

Which function could be ?

Explanation:

To answer this problem we must first interpret our given conditions:

•  Implies the function is strictly increasing.
•  Implies the function is strictly concave down.

We note the only function given which fufills both of these conditions is .

### Example Question #3 : Corresponding Characteristics Of Graphs Of ƒ And ƒ'

A jogger leaves City  at .  His subsequent position, in feet, is given by the function:

,

where  is the time in minutes.

Find the acceleration of the jogger at  minutes.

Explanation:

The accelaration is given by the second derivative of the position function:

For the given position function:

,

,

.

Therefore, the acceleration at  minutes is .  Again, note the units must be in .