Use Euler\'s method with change t = 1 to estimate the position at t = 4 starting

Question

                    Use Euler's method with change t = 1 to estimate the position at t = 4 starting from the initial condition p(0) = 10.0. Then find the exact position t = 4.                 

                    The velocities of the object measure at discrete times are time 0 = 1.0, time 1 = 3.0, time 2 = 6.0, time 3 = 10.0, and time 4 = 15.0.                 

                    The velocities follow a quadratic equation of the form v(t) = t^2/2 + at + b for some values of a and b.                 

Explanation / Answer

velocity is derivative of position.

here v(t)=t^2/2 + a*t+b

at t=0,

velocity=1

so b=1

at t=1,velocity=3

so a=3/2

so v(t)=dx/dt=t^2/2 + 1.5*t+1

so now position at t=1: 10+1*v(t=0)=11

position at t=2 : 11+1*v(t=1)=11+1*3=14

position at t=3: 14+1*v(t=2)=20

position at t=4: 20+1*v(t=3)=30

to get the exact position ,we need to solve dx/dt = t^2/2 +1.5*t+1

x=t^3/6 + 1.5*t^2/2 + t+c

c=a constant

at t=0,x=10

so c=10

so at t=4,

x=36.66

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