Calculus Year 11 maths methods

 Calculus Rhapsody   I Will Derive  Lets get starting with the horror’s of Calculus!!!!

 What would we refer to as a definition for a limit in every day life???  There are limits set for the amount of alcohol one given person can consume while driving.  The legal limit is for blood alcohol concentration is 0.05g/100ml which means as the number of drinks consumed in 1 hour approaches 2 the average blood alcohol levels approach Introduction to Limits

Example 1

 Therefore if we look back at our blood concentration example we can now say:  The limit of f(x) as x approaches 2 is equal to 0.05 Recap!

 By investigating the behavior of the function f(x)= x+3 in the vicinity of x=2 show that Example 2 x f(x)

 A continuous function is a graph that forms a continuous line: that means it has no breaks.  If a function is continuous at the point where a limit is being found, then the limit always exists and can easily be found by substitution.  f(x) is continuous when x=a : Limits and Continuous functions

 Find the  1 st let x = 2 -> = 6  Sketch the function to see if your correct  The function is continuous at x=2. Example 3

Theorems of Limits

 Questions 2, 3, 4, 5 in class  Questions 8 and 12 for homework Questions to be answered

Discontinuity

 If the function is discontinuous at the x-value where the limit is being investigated then the limit will exist only if the function is approaching the same value from the left as from the right. Definition of Discontinuity

 Point (where a hole exists at the point)  Hybrid functions (where this a clear jump in the graph from point to point)  Oscillating (never see this in any of your examples and will never reach a limit)  Infinite discontinuity x=0) There are 4 types of Discontinuities

Point Hole at the point 2 2

Hybrid Function 2 2 4

Oscillating

Infinite Discontinuity

Finding Discontinuities Find 1 1 The function is discontinuous at x=1

Example 2

Solution

Limit of hybrid function

Solution