Extrema And Average Rate of Change
-Increasing
-decreasing
-constant
-maximum
-minimum
-extrema
-average rate of change
-secant line
Vocabulary
What is Extrema
Singular extremum; an extremum is a maximum or minimum. An extremum may be a relative extremum which is in a given region which is not the overall maximum or minimum or absolute.
Applications of Extrema
Rocket testing; how high will a rocket go?
Finding the maximal profit you can earn with a certain production method.
Finding the maximal and minimal volume capacity in a kettle, boiler,etc.
Average Rate of Change
limit is the value that a function or sequence "approaches" as the input or index approaches some value.
What is Avg. Rate of Change?
The average rate of change is finding how much something changes over time.
What is Limit?
Formula
lim f{n}=L
Limit of a function
Applications
Mainly used in statistics and comparisons.
For instance, comparing the new specs of the iphone X to the specs of the iphone 7 in a graph.
Augustin-Louis Cauchy in 1821, followed by Karl Weierstrass, formalized the definition of the limit of a function which became known as the (ε, δ)-definition of limit. The definition uses ε (the lowercase Greek letter epsilon) to represent any small positive number, so that "f(x) becomes arbitrarily close to L" means that f(x) eventually lies in the interval (L − ε, L + ε), which can also be written using the absolute value sign as |f(x) − L| < ε
Applications of limit
If I keep tossing a coin as long as it takes, how likely am I to never toss a head?
If I toss a coin NN times, what is the probability p(N)p(N) that I have not yet tossed a head? Now what is the limit as N→∞N→∞ of p(N)p(N)?
The mathematical answer to this is p(N)=(12)Np(N)=(12)N. Then
A derivative is a limit.
Speed is the derivative of position
We use the concept of speed everyday
