lecture 15 - users.sussex.ac.uktn41/pdfs/mathematicalconcepts/lectur… · mathematical concepts...
TRANSCRIPT
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Mathematical Concepts (G6012)
Lecture 15 Thomas Nowotny Chichester I, Room CI-105 Office hours: Tuesdays 15:00-16:45 [email protected]
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LIMITS
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The idea
• Problem: Sequence of numbers, e.g.
and
What happens to if we make n ever larger?
Mathematicians write
Two typical cases: The numbers go ever larger (the series diverges), or (more interesting) it converges to a number (here 0).
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Formal Definition
A sequence converges to a number
if for all (small) there is a number
such that for all .
Intuition: BB
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BB Intuition: Convergence
Blue dots: up to some
Green dots: for all larger
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Examples
The limit for the sequence for to infinity does not exist, it grows without bound, people write:
BB
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BB
Claim:
Proof: Let .
We choose .
Then, for all , we know
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Practical rules for limits
• If all limits involved exist,
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“Thumb” rules for calculating limits
(Please don’t show this slide to proper mathematicians)
If
then
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Example
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Some more (Please don’t show this slide to proper mathematicians)
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Another example (Please don’t show this slide to proper mathematicians)
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Warning: Not everything converges or diverges (goes to )
What about ? It does not exist at all!
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Big O notation: Speed of divergence
• Is used especially in Computer Science
• A function is , where is another function, if for , and some .
• In other words, if is on the order of for large (or less than …)
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Examples
It is also
is
is
Note: if is and converges to 0, then converges to 0 as well!
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DERIVATIVES
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Slope, Derivative
The slope or derivative is the ratio of the change of f(x) and the change of x.
First for linear functions:
BB
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BB
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BB Calculating the linear example
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Linear functions are easy …
• Because the slope is the same everywhere
• We can make any size we want and get the same value
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Generalisation for any smooth function
• Locally any smooth function looks more and more linear the further we zoom in:
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Derivative of a smooth function
• The derivative of a smooth function is the value the ratio converges to for smaller and smaller
, mathematicians write
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In the form as for the limits before
Sequence
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Alternative notations
If is a smooth function
Then the derivative of f is denoted as
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Note …
The derivative f’(x) of a function is again a function because we can calculate it for any point x.
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BB Example - Derivative of
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Applications
• If f(x) is your distance from home as a function of the time x. Then f’(x) is the speed you are driving towards (or away from) home.
• If you take the derivative of the derivative f’’(x), that would be your acceleration.
(These are important for animating things!)
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More Applications
• If f(x) describes the height of a hill, then f’(x) is the steepness.
• f(x) is your total money as a function of time, f’(x) is your instantaneous spending rate.
• (your example here)
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Derivative of a polynomial
• For we saw just now
• Generally, for the derivative is
then
• Last time:
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For those interested: General case
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Derivatives: Basic rules
Rule name Function Derivative
Polynomials
Constant factor
Sum and Difference
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Examples: Polynomial rule Example 1
Example 2
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Special functions Function Derivative