Deriving Derivatives Using a Derived Derivative

Today, a monday, a tiring monday, a monday that wasn't the same as last monday. Sorry this has nothing to do with Calculus. Okay back to class...

Remember when we were working with derivatives last week? We found the derivative of f(x) = x² at the point where x=1. that could be found using the derivative definition found on previous slides. There are many ways to use this definition, some are more tedious than others but they should all arrive at the same answer. It is like Mr. K's analogy for using more force than necessary.

When you are going to hang up a painting. You need a nail in the wall, should you use the little hammer or should you use the sledge hammer. Common sense would tell us to use the little hammer, because the sledge hammer would blast a hole through the wall. So use the way that would save you time and effort for more tougher questions.

The easier way to find this is to plug the function and x = 1 into the definition and you get:

Picture courtesy of Sitmo

Now all you have to do is to simply the function. What you have to do is to get rid of the h in the denominator or else when you apply the limit, the equation will explode or be undefined. The answer will be 2 + h. so when you apply the limit, h will be so small towards zero that it will be seen as negligible and so the derivative will be 2.

After that little review we learned about locally linear points. Points can be called locally linear when you zoom into a point on your calculator and eventually it appears as a straight line. Take the graph of f(x)=|x²+1| for example:

Picture courtesy of Fooplot.com

If you zoom into the point (2, 3) you will eventually find a straight line meaning that the point is locally linear. But if you zoom into any point where this function has a zero you will never find a line. You will only see a corner or a spiky spot. It will never be a line so it is not locally linear.

So that is what we did today in class. Homework is exercise 2.2 question numbers: 1, 2, 3, 5, 10a, 11, 12, 14, 15, 17. But remember if you do not feel comfortable with the subject just practice more. Everybody is more than welcome to do the entire exercise.

So the next scibe will be Kristina.
Good night...