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### june23

Course: ECE 2409, Fall 2009
School: Villanova University
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Word Count: 1023

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23,'99 DERIVATIVES June AND DIFFERENTIAL EQUATIONS i+ v i= 1 5/30/99 1999BG Mobasseri WHAT IS A DERIVATIVE? Derivative of f(x) at x=a is given by f'(a) y=f(x) f(a) a 1999BG Mobasseri 2 f(a+h) ) f (a=lima+h- f(a ) h 0f( h ) a+h 5/30/99 SIGNIFICANCE OF DERIVATIVE In the limit, derivative at x=a is equal to the slope m of the tangent line at a. y=f(x) f(a) a 1999BG Mobasseri 3 a+h 5/30/99...

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23,'99 DERIVATIVES June AND DIFFERENTIAL EQUATIONS i+ v i= 1 5/30/99 1999BG Mobasseri WHAT IS A DERIVATIVE? Derivative of f(x) at x=a is given by f'(a) y=f(x) f(a) a 1999BG Mobasseri 2 f(a+h) ) f (a=lima+h- f(a ) h 0f( h ) a+h 5/30/99 SIGNIFICANCE OF DERIVATIVE In the limit, derivative at x=a is equal to the slope m of the tangent line at a. y=f(x) f(a) a 1999BG Mobasseri 3 a+h 5/30/99 APPLICATIONS OF DERIVATIVE The most widely used application of derivative is in finding the extremum (max or min) points of a function. If a function has a local extremum at a number c then either f'(c)=0 or f'(c) doesn't exist 1999BG Mobasseri 4 5/30/99 CRITICAL NUMBERS A number c in the domain of a function f is a critical number of f if either f'(c)=0 or f'(c) does not exist f'(c)=0 x=c 1999BG Mobasseri 5 5/30/99 DERIVATIVES IN MATLAB:diff MATLAB computes two differentials, dy and dx, using diff, to arrive at the derivative If x and y are input array of numbers dy=diff( y) dx=diff(x) Then, yprime=diff(y)./diff(x) 1999BG Mobasseri 6 5/30/99 How diff works Take the array y=[3 5 7 5 9]; diff(y) simply is the pairwise differences, i.e. diff(y)=[53 75 57 95] which is equal to diff(y)=[2 2 2 4] Naturally, there is one fewer term in diff(y) 7 5/30/99 1999BG Mobasseri than it is in y Using diff to find derivatives Let y=[10 25 30 50 10] for x=[1 2 3 4 5] Then One fewer component diff(y)=[15 5 20 40] diff(x)=[1 1 1 1 1] Dividing termbyterm yprime=[15 5 20 40] 1999BG Mobasseri 8 5/30/99 WORKING WITH humps humps is a builtin MATLAB function, like peaks and is given by 1 1 y= + -6 2 2 [( - .3) + 0.01] [( - .9) + .04 ] 1999BG Mobasseri 9 5/30/99 HOW IT LOOKS 100 90 80 70 60 50 40 30 20 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1999BG Mobasseri 10 5/30/99 Try it!:derivative of humps First, we must know how humps was created in order to know dx x=0:0.01:1;%dx in this case is 0.01 y=humps(x); Now use dy=diff(y); yprime=dy./dx; dx=diff(x); 1999BG Mobasseri 11 5/30/99 PLOT OF dy/dx 100 80 60 HUMPS Function peaks 40 20 0 0 1000 500 0 -500 -1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Deriv. goes to 0 FIRST DERIVATIVE OF HUMPS 1999BG Mobasseri 12 5/30/99 Finding critical points:theory vs. application To find max. or min.we should look for instances where y'=0. This is where the difference between textbook methods and the real world shows up. To see for yourself look for instances where yprime=0. 1999BG Mobasseri 13 5/30/99 Where are the zeros? If you did the search right, you will realize that nowhere in yprime you can actually see a zero Since there are no points where y' is exactly zero, we should look for points of transition + + from positive to negative, i.e. sign changes 1999BG Mobasseri 14 5/30/99 Zero crossing and sign transition In Matlab, all derivatives are discrete. Some are positive, some negative but no zeros crossing Zero Since derivative goes from + to , we can infer that it went to zero somewhere in between 1999BG Mobasseri 15 5/30/99 FINDING ZERO CROSSINGS There are two ways to find where a function crosses zero; 1. Numerical 2. Algebraic(polynomial fitting) 1999BG Mobasseri 16 5/30/99 ZC USING ARRAY OPERATION To find where a sign transition takes place, multiply two consecutive numbers in the derivative array. If there is a sign change, product the is negative [10 15 16 9 4 3 4 9] X X X X X X X Zero crossing + 1999BG Mobasseri + + 17 + + 5/30/99 MATLAB's WAY Let x be an n element array[x1,x2,...,xn]. To generate the zero crossing array, do y=x(1:n1).*x(2:n) Algebraically, this is what is happening y=[x x x x ,..., x x ] 1999BG Mobasseri 18 5/30/99 Try it! Evaluate humps in the range x=0 to 1 in increments of 0.01 This gives you a 1x101 array. Find yprime. This is now a 1x100 array Find where in this 101 positions sign changes occur 1999BG Mobasseri 19 5/30/99 LOCATING SIGN CHANGES Just look for instances of negative sign in the zero crossing array 800 600 400 200 0 -200 -400 -600 -800 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1st=0.29 2nd=0.63 3rd=0.88 1999BG Mobasseri 20 5/30/99 CLOSER LOOK 800 600 400 200 0 -200 -400 -600 -800 0.2 Sign change 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 1999BG Mobasseri 21 5/30/99 SECOND ZC 50 40 30 20 10 0 -10 -20 -30 -40 -50 0.55 0.6 0.65 0.7 0.75 1999BG Mobasseri 22 5/30/99 THIRD ZC 50 40 30 20 10 0 -10 -20 -30 -40 -50 0.85 0.86 0.87 0.88 0.89 0.9 0.91 0.92 0.93 0.94 0.95 1999BG Mobasseri 23 5/30/99 Differential Equations i+ v i= 1999BG Mobasseri 24 5/30/99 Simplest Diff.Eq A first order differential equation is of the form y '=dy/dx=g(x,y) We are looking for a function y such that its derivative equals g(x,y) 1999BG Mobasseri 25 5/30/99 FEW EXAMPLES Find y that such that y '=3x y `=y y y `=(2x)cos 1999BG Mobasseri 26 5/30/99 2 SOLVING ODE's USING MATLAB MATLAB solves ordinary DE in two ways: 1):numerical and 2):symbolic For numerical solution use [x,y]=ode23(`function',a,b,initial) See next slide for usage 1999BG Mobasseri 27 5/30/99 EXPLAINING ode23 Here are the elements of [x,y]=ode23(`function',a,b,initial) function this is g(x,y) in y '=g(x,y). Must be written as a separate MATLAB function a left point of the interval b right point of the interval initial initial condition, i.e. y(a) 1999BG Mobasseri 28 5/30/99 ...

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