a) Obtain the Taylor polynomial P5(x), of degree 5, about x0 = 0, of the function f(x) = 0.5e2x.

1. Complicated functions such as combinations of sin(x), ln(x) or ex often crop up in physics engines in games, usually as solutions to differential equations.  These complicated functions can often be approximated very accurately using much simpler forms like Taylor polynomials, which are very much quicker to evaluate and so help speed up game play.  However, when making approximations, it is important to know what limits on accuracy (if any) such approximations might have. This question allows you to form a polynomial approximation to a known exponential function and to explore any limits on accuracy.  
a) Obtain the Taylor polynomial P5(x), of degree 5, about  x0 = 0, of the function f(x) = 0.5e2x.  
b) Use this polynomial to estimate the value of f(0.2) to 8 decimal places.  
c) Construct an upper bound ε, for the error obtained in b) and hence obtain an interval in which the true value of 0.5e0.4 is guaranteed to lie.  
d) Find the actual error in your estimate obtained in b) and compare the magnitude of the actual error with the magnitude of the upper bound ε.


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