*Monotonicity*

We know what happens when the derivative is zero. It's a peak or trough.

(I secretly wish that I could apply this to the stock market!)

Now, suppose we say that the derivative is bigger than or smaller than 0. what is going on here?

If the derivative is bigger than zero, the tangent line is pointing upwards. (slope>0) Now, using the same logic, we conclude that when it's smaller than 0, the tangent line points down.

It takes no Einstein to notice than when the tangent line goes up, so does the function. if it goes down, the function follows. (sort of like the economy and the stock market. the market is a good representation of the trend of an economy.)

Remember, the monotonicity is a powerful weapon in analyzing functions.

(I secretly wish that I could apply this to the stock market!)

Now, suppose we say that the derivative is bigger than or smaller than 0. what is going on here?

If the derivative is bigger than zero, the tangent line is pointing upwards. (slope>0) Now, using the same logic, we conclude that when it's smaller than 0, the tangent line points down.

It takes no Einstein to notice than when the tangent line goes up, so does the function. if it goes down, the function follows. (sort of like the economy and the stock market. the market is a good representation of the trend of an economy.)

__To summarize:__**1. When derivative>0, we have an increasing function.****2. When derivative<0, we have a decreasing function.****3. At derivative=0, we have an extreme value.**

Remember, the monotonicity is a powerful weapon in analyzing functions.

*Concavity*

Up there, we learned about monotonicity. We still have one big question to ask ourselves-- is the monotonicity increasing or decreasing? I know this is technically incorrect but think of it as this-- we only know from monotonicity whether a function is increasing or decreasing. How do we know if it's increasing at an increasing pace, or increasing at a decreasing pace?

There, we invented something called concavity. Analogical to monotonicity, concavity is based around the second derivative of a function. (Recall higher order derivatives). So, basically, it's "the derivative of the monotonicity". Now you might see some sense in the question I asked up there. It is defined as follows:

1. A function is concave up at x if f''(x)>0.

2. A function is concave down at x if f''(x)<0.

3. x is called an

The concavity measures the speed the function is increasing or decreasing. For example, an exponential function y=2^x would be concave up for x>0, while a logarithmic function y=log(x) would be concave down for x>0.

It is really as simple as that. Now that we have extreme values, monotonicity, and concavity, we can draw the graph of a function even without a TI-84! Amazing?

There, we invented something called concavity. Analogical to monotonicity, concavity is based around the second derivative of a function. (Recall higher order derivatives). So, basically, it's "the derivative of the monotonicity". Now you might see some sense in the question I asked up there. It is defined as follows:

1. A function is concave up at x if f''(x)>0.

2. A function is concave down at x if f''(x)<0.

3. x is called an

*inflection point*if f''(x)=0.The concavity measures the speed the function is increasing or decreasing. For example, an exponential function y=2^x would be concave up for x>0, while a logarithmic function y=log(x) would be concave down for x>0.

It is really as simple as that. Now that we have extreme values, monotonicity, and concavity, we can draw the graph of a function even without a TI-84! Amazing?