MAT120 Asst. Prof. Dr. Ferhat PAKDAMAR (Civil Engineer) M Blok - M106 Gebze Technical University Department of Architecture Spring – 2014/2015 Week 8

Subjects WeekSubjectsMethods Introduction Set Theory and Fuzzy Logic.Term Paper Real Numbers, Complex numbers, Coordinate Systems Functions, Linear equations Matrices Matrice operations MIDTERM EXAM MT limits. Derivatives, Basic derivative rules Term Paper presentationsDead line for TP Integration by parts, Area and volume Integrals Introduction to Numeric Analysis Introduction to Statistics Review 15 Review 16 FINAL EXAM FINAL

LIMITS

Example Now 0/0 is a difficulty! We don't really know the value of 0/0 (it is "indeterminate"), so we need another way of answering this. limits

So instead of trying to work it out for x=1 let's try approaching it closer and closer: x (x 2 − 1)(x − 1) 0,5 1, ,9 1, ,99 1, ,999 1, ,9999 1, , , limits

We are now faced with an interesting situation:  When x=1 we don't know the answer (it is indeterminate)  But we can see that it is going to be 2 We want to give the answer "2" but can't, so instead mathematicians say exactly what is going on by using the special word "limit" And it is written in symbols as: limits

So Limit is…..

The right-hand limit of f (x), as x approaches a, equals L written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the right of a. One-Sided Limits

The left-hand limit of f (x), as x approaches a, equals M written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the left of a. One-Sided Limits

1. Given Find Examples of One-Sided Limit Find

Find the limits: More Examples

Limit Theorems

Examples Using Limit Rule Ex.

More Examples

Indeterminate forms occur when substitution in the limit results in 0/0. In such cases either factor or rationalize the expressions. Ex. Indeterminate Forms Notice form Factor and cancel common factors

More Examples

A function f is continuous at the point x = a if the following are true: Continuity

At which value(s) of x is the given function discontinuous? Example: Continuous everywhere except x=-3 g(x)=-3 is undefined

Limits at Infinity For all n > 0, provided that is defined.

For all n > 0, Infinite Limits

DERIVATIVES

Derivatives

We will use the slope formula: to find the derivative of a function y = f(x) x changes from x to x+Δx y changes from f(x) to f(x+Δx) Follow these steps: Derivatives

In Turkish: (Türev=Eğimin Değişimi)

Example: the function f(x) = x 2 We know f(x) = x 2, and can calculate f(x+Δx) : Start with the slope formula: Put in f(x+Δx) and f(x): Simplify (x 2 and −x 2 cancel): Simplify more (divide through by Δx): And then as Δx heads towards 0 we get: Result: the derivative of x 2 is 2x Start with: ExpandExpand (x + Δx) 2 : We write dx instead of "Δx heads towards 0", so "the derivative of" is commonly written Derivatives

x 2 = 2x "The derivative of x 2 equals 2x" or simply "d dx of x 2 equals 2x" What does x 2 = 2x mean? It means that, for the function x 2, the slope or "rate of change" at any point is 2x. So when x=2 the slope is 2x = 4, as shown here: Or when x=5 the slope is 2x = 10, and so on. Derivatives

Common Functions FunctionDerivative Constantc0 x1 Squarex2x2 2x Square Root √x(½)x -½ Exponentia l exex exex axax a x (ln a) Logarithmsln(x)1/x log a (x)1 / (x ln(a)) Trigonomet ry (x is in radians)radians sin(x)cos(x) −sin(x) tan(x)sec 2 (x) sin -1 (x)1/√(1−x 2 ) cos -1 (x)−1/√(1−x 2 ) tan -1 (x)1/(1+x 2 ) Derivatives Rules

Rules FunctionDerivative Multiplication by constant cfcf’ Power Rulexnxn nx n−1 Sum Rulef + gf’ + g’ Difference Rulef - gf’ − g’ Product Rulefgf g’ + f’ g Quotient Rulef/g(f’ g − g’ f )/g 2 Reciprocal Rule1/f−f’/f 2 Chain Rule (as "Composition of Functions") f º g(f’ º g) × g’ Chain Rule (in a different form) f(g(x))f’(g(x))g’(x Derivatives Rules

Example: What is (5z 2 + z 3 − 7z 4 ) ? Using the Power Rule: z 2 = 2z z 3 = 3z 2 z 4 = 4z 3 And so: (5z 2 + z 3 − 7z 4 ) = 5 × 2z + 3z 2 − 7 × 4z 3 = 10z + 3z 2 − 28z 3 More Examples

Example: What is (5x−2) 3 ? The Chain Rule says: the derivative of f(g(x)) = f’(g(x))g’(x) (5x-2) 3 is made up of g 3 and 5x-2: f(g) = g 3 g(x) = 5x−2 The individual derivatives are: f'(g) = 3g 2 (by the Power Rule) g'(x) = 5 So: (5x−2) 3 = 3g(x) 2 × 5 = 15(5x−2) 2 More Examples

Example: What is (1/sin(x)) ? 1/sin(x) is made up of 1/g and sin(): f(g) = 1/g g(x) = sin(x) The Chain Rule says: the derivative of f(g(x)) = f’(g(x))g’(x) The individual derivatives are: f'(g) = −1/(g 2 ) g'(x) = cos(x) So: (1/sin(x))’ = −1/(g(x)) 2 × cos(x) = −cos(x)/sin 2 (x) More Examples

Second Derivative A derivative basically gives you the slope of a function at any point. The "Second Derivative" is the derivative of the derivative of a function. So: Find the derivative of a function Then take the derivative of that A derivative is often shown with a little tick mark: f'(x) The second derivative is shown with two tick marks like this: f''(x)

Example: f(x) = x 3 Its derivative is f'(x) = 3x 2 The derivative of 3x 2 is 6x, so the second derivative of f(x) is: f''(x) = 6x

You are cruising along in a bike race, going a steady 10 m every second. Distance, Speed and Acceleration Distance: is how far you have moved along your path. It is common to use s for distance (from the Latin "spatium"). So let us use: distance (in meters): s time (in seconds): t

Speed: is how much your distance s changes over time t and is actually the first derivative of distance with respect to time: And we know you are doing 10 m per second, so

Acceleration: Now you start cycling faster! You increase your speed to 14 m every second over the next 2 seconds. When you are accelerating your speed is changing over time. So is changing over time We could write it like this: But it is usually written Your speed changes by 2 meters per second per second. And yes, "per second" is used twice! It can be thought of as (m/s)/s but is usually written m/s 2

Finding Maxima and Minima Using Derivatives In a smoothly changing function a low point (a minimum) or high point (a maximum) are where the function flattens out : Where does it flatten out? Where is the slope zero? Where the slope is zero. The Derivative tells us!

Example: A ball is thrown in the air. Its height at any time t is given by: h = t − 5t 2 What is its maximum height? Using derivatives we can find the slope of that function: Now find when the slope is zero: 14 − 10t = 0 10t = 14 t = 14 / 10 = 1,4 The slope is zero at t = 1,4 seconds

And the height at that time is: h = ×1,4 − 5×1,4 2 h = ,6 − 9,8 = 12,8 And so: The maximum height is 12,8 m (at t = 1,4 s)

When a function's slope is zero at x, and the second derivative at x is: less than 0, it is a local maximum greater than 0, it is a local minimum equal to 0, then the test fails (there may be other ways of finding out though) "Second Derivative: less than 0 is a maximum, greater than 0 is a minimum" Second Derivatives

Example: Find the maxima and minima for: y = 5x 3 + 2x 2 − 3x The derivative (slope) is: Which is quadratic with zeros at: x = −3/5 x = +1/3 The second derivative is y'' = 30x + 4

At x = +1/3: y'' = 30(+1/3) + 4 = +14 it is greater than 0, so +1/3 is a local minimum At x = −3/5: y'' = 30(−3/5) + 4 = −14 it is less than 0, so −3/5 is a local maximum The second derivative is y'' = 30x + 4

Chaos Theory and The Butterfly Effect

Chaos Theory and Butterfly Effect  When a tiny variation changes the results of a system dramatically (over a period of time), this sensitivity is what we call the butterfly effect.  Why was it called like that? Answer in the history section.

Chaos Theory and Butterfly Effect  Linear (Normal) System  We can guess very precisely its behavior.  Random Systems  We cannot guess at all! Ex: Throwing the dice.  Chaotic Systems  Deterministic with no random, but unpredictable on long term because they are very complex and sensitive.  Seems to be illogical and paradoxal?

Chaos Theory and Butterfly Effect InputOutput of a normal system Output of a random system Output of a chaotic system Can you predict? System Input Output

Chaos Theory and Butterfly Effect InputOutput of a normal system Output of a random system Output of a chaotic system (??)37 (??) 10Can you predict? System Input Output

Chaos Theory and Butterfly Effect InputOutput of a normal system Output of a random system Output of a chaotic system (value doubled)9 (??)29(??) 5Can you predict? System Input Output

Chaos Theory and Butterfly Effect InputOutput of a normal system Output of a random system Output of a chaotic system (??) (??)10 System Input Output So that’s how the chaotic system is: 1.Deterministic 2.Highly sensitive (butterfly effect) 3.Unpredictable on the long term

Chaos Theory and Butterfly Effect  Edward Lorenz in 1961 used a numerical computer model to run a weather prediction.  Lorenz was modeling the atmosphere with a set of three simple partial differential equations.  One day he wanted to restart his computations where he ended the day before. The previous day’s last output was He entered expecting to continue on.  The result was a completely different weather scenario!!!

Chaos Theory and Butterfly Effect This is what he observed: Small difference of the input: instead of Extreme Differences of the output After a period of time

Chaos Theory and Butterfly Effect  In 1972, Philip Merilees quoted Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?  This is why we call it butterfly effect….

Chaos Theory and Butterfly Effect  1- The weather: Can you predict weather of the next year? 1- The weather:  2- The stock market: Can you predict exchange rates? 2- The stock market:  3- Biology: Can you predict how a virus is going to spread? 3- Biology:  4- Physics: Can you predict the motion of gas in vacuum? 4- Physics:  5- Evolution of life: small changes in the chemistry of the early Earth gives rise to life. 5- Evolution of life:  6- Fractals: Even art has been touched with the chaos theory: A fractal is generally a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole, a property called "self-similarity. Fractals are often considered to be infinitely complex. 6- Fractals:  7- Aviation safety: The Swiss cheese model. 7- Aviation safety:  8- Highway traffic jams: Unpredictable and very complex system. 8- Highway traffic jams:  9- Psychology: A small psychological fact in childhood can lead to problems or suicide in adolescence. 9- Psychology:  10- Time travel: The butterfly effect theory presents scenarios involving time travel with "what if " scenarios. 10- Time travel:

Chaos Theory and Butterfly Effect  You can stay at home and be happily introspective or you can make the choice, step out, and be the Butterfly that begins the tempest that changes the world. (John Sanford)  Sometimes it's the smallest decisions that can change your life forever. (Keri Russell)  Everything we do affects other people.

Have a nice week!