AB Notes

The main AB topics are limits, derivatives, and integrals, all of which are interrelated:

A limit is the value a function approaches as you approach a specified x-value.
The definition of a derivative is the limit as h approaches 0 of [f(x+h) - f(x)]/h.
Integrals are antiderivatives.

Some basic notes on these topics are shown below:

LIMITS:
To find limits algebraically, simply plug in c - the value x approaches - directly into the function and solve. There are 3 possibilities for an answer:

 

DERIVATIVES:

The formal definition of a derivative of a function is stated above.

 



If a function is differentiable, it is continuous. If it is continuous, it is not necessarily differentiable:

 

Continuity VS Differentiability

 

There are many rules for finding derivatives, including:



Rules for Derivatives

Once you can find derivatives, you can use them to understand the original function better:



Relationships Between Derivatives

 

Using these relationships, you can accurately graph the original function by hand:

 

 

INTEGRALS:

There are 2 types of integrals, definite and indefinite. Both are related to functions of area:

 

Integrals

In order to find them mathematically, basically use the reverse of the deriving rules. If there is a more complicated integral, you can use u-substitution:

 

U-Substitution

 

FUNDAMENTAL THEOREM OF CALCULUS:

So:

 

From this, we can find:

 

 

 

OTHER THEOREMS:

Mean Value Theorem:

 

Rolle's Theorem:

 

APPLICATIONS:

 

Every topic in Calculus has common, real life applications, including volumes of irregularly shaped objects, related rates, exponential growth and decay, and more.

 

Exponential Growth and Decay, using differential equations: