All AP Calculus AB Resources
Example Question #2 : Calculus I — Derivatives
Consider the function
Find the minimum of the function on the interval .
To find potential minima of the function, take the first derivative of using the power rule.
Set the derivative to 0:
We solve for to obtain and then plug in 0.5 into the original function to obtain the answer of
We can double check that is indeed a minimum by using the second derivative test
which means the function is concave up, so that the point we found is a minimum.
Example Question #1 : Antiderivatives Following Directly From Derivatives Of Basic Functions
Find the x coordinate of the minimum of on the interval .
First, find the derivative, which is:
Set that equal to 0 to find your critical points:
Evaluate f at each critical point and endpoint over [-2, 2]:
The least of those numbers is the minimum. Because f(-2)=-32, that is the minimum.
Example Question #7 : Identifying Absolute And Local Extrema
What is the local minimum of when ?
There is no local minimum in that range.
The y-value is constant throughout that range.
To find the maximum, we need to look at the first derivative.
To find the first derivative, we can use the power rule. To do that, we lower the exponent on the variables by one and multiply by the original exponent.
We're going to treat as since anything to the zero power is one.
Notice that since anything times zero is zero.
When looking at the first derivative, remember that if the output of this equation is positive, the original function is increasing. If the derivative is negative, then the function is decreasing.
Because we want the MINIMUM, we want to see where the derivative changes from negative to positive.
Notice that has a root when . In fact, it changes from negative to positive at that particular point. This is the local minimum in the interval .
Example Question #2 : Antiderivatives Following Directly From Derivatives Of Basic Functions
1) Apply the sum rule for integration,
2) Integrate each individual term and include a constant of integration,
Since indefinite integration is essentially a reverse process of differentiation, check your result by computing its' derivative.
This is the same function we integrated, which confirms our result. Also, because the derivative of a constant is always zero, we must include "C" in our result since any constant added to any function will produce the same derivative.
Example Question #3 : Antiderivatives Following Directly From Derivatives Of Basic Functions
Example Question #4 : Antiderivatives Following Directly From Derivatives Of Basic Functions
Example Question #5 : Antiderivatives Following Directly From Derivatives Of Basic Functions
Example Question #6 : Antiderivatives Following Directly From Derivatives Of Basic Functions
Example Question #81 : Integrals
Example Question #82 : Integrals