Maple Exam 1 

Answer Key 

The solutions shown here are representative. There may be other methods to solve the problems. 

Question 1 

>	restart:with(plots):

 

>	p1:=polarplot(1,theta=0..2*Pi):

 

>	p2:=polarplot(cos(3*theta),theta=0..2*Pi):

 

>	display(p1,p2);

 

Question 2 

>	restart:f:=taylor(exp(I*x),x=0);

 

(1)

 

>	g:=taylor(cos(x)+I*sin(x),x=0);

 

(2)

 

>	f:=convert(f,polynom);

 

(3)

 

>	g:=convert(g,polynom);

 

(4)

 

>

f-g;

 

(5)

 

Question 3 

>	restart:a:=m->(((-1)^m)/(factorial(m)*factorial(m+1)))*(x/2)^(2*m+1);

 

(6)

 

a) Apply ratio test 

>	limit(a(m+1)/a(m),m=infinity);

 

(7)

 

The radius of convergence is the inverse of the above, hence it is infinity. This shows the series converges for all x. 

>

 

b) Plot the first ten terms for x=1. 

>

x:=1:

 

>

a(m);

 

(8)

 

>	plot( [ seq([n,a(n)],n=1..10) ] , style=point, symbol=diamond);

 

>