# AP Calculus AB : Antiderivatives following directly from derivatives of basic functions

## Example Questions

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### Example Question #1 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Consider the function

Find the minimum of the function on the interval .

Explanation:

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

Explanation:

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 #2 : Antiderivatives Following Directly From Derivatives Of Basic Functions

What is the local minimum of  when ?

The y-value is constant throughout that range.

There is no local minimum in that range.

Explanation:

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 #3 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Integrate,

Explanation:

Integrate

1) Apply the sum rule for integration,

2) Integrate each individual term and include a constant of integration,

Further Discussion

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.

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