All AP Calculus AB Resources
Example Questions
Example Question #2 : 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?
It represents another way to write the car's speed.
It has no relation to the previous function.
It represents the change in distance over a given time .
It represents the rate at which the speed of the car is changing.
It represents the total distance the car has traveled at time .
It represents the rate at which the speed of the car is changing.
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 #2 : Derivative As A Function
Find the critical numbers of the function,
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 #3 : 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 ?
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 #4 : Derivative As A Function
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.
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 . ^{}