Brief Calculous Review.Doc

Brief Calculus Review

Functions

A function is a relationship between 2 or more variables. For example, we might say that the variable Y is related to the variable X as follows:

We call X the independent variable and Y the dependent variable.

Thus, when X takes the value of 1 Y is 1, when X is 2 Y is 4 and so on.

By taking the square root in both sides we can invert this function as follows.

Thus, when Y takes a value of 4, X equals 2, when Y takes a value of 1 X equals____

Rules of differentiation

Broadly speaking, the rules of differentiation help us answer this question: if the independent variable changes by one unit, how much will be the change in the dependent variable.

Suppose we have the following function,

If we increase X by one unit, how much will Y change? Let’s pick some numbers. Suppose first that X is 10. This implies that Y is 10x2 or 20. Now we increase X by one unit to 11. Then Y equals 11x2 or 22. Thus the change is 2. If we increase X from 11 to 12? Y equals 24, which means that the change in Y equals 2.

We have the following result,

Suppose we have,

where A is any positive or negative number (in the previous example A=2), the derivative of Y with respect to X (the change in Y for a one unit change in X), which we write as is

Constants

Suppose instead that the function is

where B and A are any positive or negative constants (numbers). How does Y change? Since B is a constant, when X changes the result will be the same as before (try by picking some numbers for X and Y). That is, the derivative of a constant is zero.

Power rule

Now suppose we have a function that looks like this

where A and b can be any positive or negative number.

How does Y change if X increases by one unit?

The rule is

For example let the function be,

, then

Now let’s combine two things (a constant multiplying a power variable)

the derivative of Y with respect to X is,  

Product rule

Suppose we have two...