COMP155Computer SimulationNovember 3, 2008Basic Concepts of Calculus Derivative – rate of change of a function Integral – area under the curve of a functionDerivative is the slopeImage from http://en.wikipedia.org/wiki/DerivativeExample: velocity and acceleration This curve represents velocity over time.What does acceleration look like?Acceleration is slope of velocity Velocity is blue, acceleration is redExample 2: Velocity Velocity(t) = -t2/2 + 10t + 70Example 2 Velocity is blue, acceleration is red70102)(2++−= tttv2tta−=10)(Example 3 Green curve is distance, what is velocity?Example 3 Green is position, blue is velocityExample 3 What is happening to green curve when blue curve crosses zero?The derivative is zero when function is at a local minimum or maximumExample 3 Position (green), velocity (blue), acceleration (red)Integral is the areaImage from http://en.wikipedia.org/wiki/IntegralExample: Integration Velocity is blue, acceleration is red+ area- areaExample: Integration Red line is acceleration, what is velocity?- area+ areaIntegration Requires Init Values Accumulated area gives shape, initial conditions locate shapeThese are both correct velocity curves, for given acceleration curveExample 2: Integration position at t=0 is 100,position at t=25 is 150What is position at t=50?Example 2: Integration position at t=0 is 100,position at t=25 is 150What is position at t=50?answer: 200This area must be 50.This area is also 50.Fundamental Theorem of Calculus f is a continuous real-valued function defined on a closed interval [a, b],  F is an anti-derivative of f:  F’ = f,the definite integral offover that interval is the definite integral offover that interval is given by differentiation is the reverse of integration∫−=baaFbFdxxf )()()(Fundamental Theorem of CalculusWhatever the shape of the velocity curve, we can determine its area in some interval from the position values at the ends of that interval.green = position∫−−−=5.25.2)5.2()5.2()( ppdxxvgreen = positionblue = velocityCalculus in Simulation We’ll commonly have a derivative of some function(or instantaneous values of the derivate)and need to construct values for the functionand need to construct values for the function Example: We know velocity, need to compute position Example: We know acceleration, need to compute velocity and positionTime Slicing Simulation is updated at regular interval in time dt = time step/sliceTo approximate integration:To approximate integration: compute derivative at time steps make assumption about derivative value at other times accumulate area to approximate integrationTime slice approximation dt = 1, assume current value held for previous time sliceblue: correct derivative functionover-estimate of positive functionred: approximated derivative functionunder-estimate of positive areaover-estimate of negative areaof positive areaunder-estimate of negative areaTime slice approximationdt = 1green: correct positionred: approximated positionTime slice approximationdt = 1green: correct positionred: approximated positionSmaller dt  better approximationdt= 1dt= 0.5dt= 1dt= 0.5dt = 0.1dt = 0.01Smaller dt  better approximationdt= 1This is similar to the limit in the definition of a derivative:hafhafafh)()(lim)('0−+=→dt= 1dt = 0.1The time slice is h.As it gets smaller, we get closer to the true value of the derivative.hh0→Time slice approximationdt= 1We’ve rewrittento getdttfdttftfdt)()(lim)('0−+=→dt= 1dt = 0.1to get(actually, we’re using))(')()( tfdttfdttf⋅+=+)(')()( dttfdttfdttf+⋅+=+Euler’s Method Taylor Series:as dt0, the higher order terms quickly ...)(''')('')(')()(32+⋅+⋅+⋅+=+ tfdttfdttfdttfdttfas dt0, the higher order terms quickly approach 0, giving our approximation The higher order terms are the error in the approximation)(')()( tfdttfdttf⋅+=+Instantaneous Derivative Values In interactive simulations, we often do not have a function for the derivative, rather we have instantaneous values for the derivativeExample: driving simulationExample: driving simulationacceleration value is set by driver’s joystick Euler’s method can still be applied (twice)to compute velocity and position assume that current value of acceleration was constant through previous time stepExample: Vehicle Dynamics Acceleration is set by user Function updates velocity, then positionvoid Vehicle::update(double elapsed_time){{double dt = elapsed_time/1000.0;velocity += acceleration*dt;double vx = velocity*sin(heading);double vy = velocity*cos(heading);double x = position.getX() + vx*dt;double y = position.getY() +