Scribe Post: Day 55

OKAAAY! Well, once again, I can't upload my graphs/pics. So I'll go early tomorrow morning at school, as usual, and complete this post, or go on again later and see if it works. Sorry for the inconvenience.

Today’s class was about Global Extrema and the Extreme Value Theorem.

TERMS

Maxima – plural for maximum
Minima – plural for minimum
Extreme – either a minimum or a maximum
Extrema – minima and maxima
Local – just a vicinity on a graph
Global – considers the entire interval
Critical Numbers – where f’ = 0 or undefined

In previous lessons, we learned that local extrema are the lowest or highest possible points in some vicinity on a graph. For example, given this graph (figure A), we’ll see that the points highlighted with green appear to be the maximum and minimum. If we consider the entire graph, however, we know that these points aren’t the extrema.



Global extrema are the maxima and minima over the entire interval. Notice that I said “over the entire interval”. The interval must be set to determine the global extrema.



To find the global extrema, do the following steps:
1) Find f’
2) Find the critical numbers
3) Evaluate f at the critical numbers and end points**
4) Determine the global extrema (the lowest and highest values on the interval)

**Always check where the function is undefined. This is the most overlooked place where the function has a min/max

This leads to a theorem in calculus called the Extreme Value Theorem (EVT), which states that if a function is continuous on a closed interval, [a, b], then it has a global max and global min. This is an existence theorem, meaning that if you can draw the graph of a function without lifting your pen (continuous), then global extrema must exist. It doesn’t say where these exist; just says they should exist.

Let’s do an example, following the steps above:



What we can take away from this is that from the interval [-1, 3], the global extrema are also the local extrema.

Due for Wednesday are Exercises 5.1, odds.
Also, due on Monday is the derivative photo assignment.