Maple Tool 1

This tool is designed to be used with Activity 1 in Section 10.3.1.

>	with(plots): with(stats[statplots]): f:=x->exp(x);

Warning, the name changecoords has been redefined

Here are the first nine Taylor polynomials for the exponential function at .

>	P[0]:=x->1; P[1]:=x->P[0](x)+x; P[2]:=x->P[1](x)+x^2/2; P[3]:=x->P[2](x)+x^3/(3!); P[4]:=x->P[3](x)+x^4/(4!); P[5]:=x->P[4](x)+x^5/(5!); P[6]:=x->P[5](x)+x^6/(6!); P[7]:=x->P[6](x)+x^7/(7!); P[8]:=x->P[7](x)+x^8/(8!);

Now we examine the convergence of the polynomial approximations at a particular point .  The following code will display a graph of the function , graphs of the first nine Taylor polynomial approximations , and the first nine Taylor approximations at the particular pont .

Enter a value of  between  and .

>

x[0]:=3;

Below the plot we list the approximating values together with the value of  and the error in the last approximation.  The viewing window  may be altered by varying the definitions of the constants  and .

>	a:=4; b:=60; PlotA:=plot(f(x),x=-a..a, thickness=3, color=red, xtickmarks=[-a,-a/2,a/2,a], ytickmarks=[-b,-b/2,b/2,b], view=[-a..a,-b..b]):

>

xdata:=[x[0],x[0],x[0],x[0],x[0],x[0],x[0],x[0],x[0]]: ydata:=[P[0](x[0]), P[1](x[0]), P[2](x[0]), P[3](x[0]),P[4](x[0]), P[5](x[0]), P[6](x[0]), P[7](x[0]), P[8](x[0])]: PlotF:=scatterplot(xdata, ydata, color=navy): PlotG:=plot([P[0](x), P[1](x), P[2](x), P[3](x), P[4](x), P[5](x), P[6](x), P[7](x), P[8](x)], x=-a..a, color=green, xtickmarks=[-a,-a/2,a/2,a], ytickmarks=[-b,-b/2,b/2,b], view=[-a..a,-b..b]): display(PlotA,PlotF,PlotG); print(Approximations); evalf(ydata);

>	print(Actual); evalf(f(x[0]));

>	print(Actual-last_approximation); evalf(f(x[0])-ydata[9]);

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