Mathematics with Maple

S.Duzhin

Solutions to problems of Lesson 3

> with(plots):

Problem #1

> plot({sin(x),sin(x+2/3*Pi),sin(x+4/3*Pi)}, x=0..2*Pi);

Problem #2

> solve(16*x^5-20*x^3+5*x=1);

Problem #3

> solve(32*x^6-48*x^4+18*x^2=1);

This equation has 4 different roots, but two of them have multiplicity 2, thus the total number of roots is 6.

Problem #4

> solve (x-x^3/6+x^5/120=0);

Here I:=sqrt(-1) is the imaginary unit

Problem #5

The equation has only one root x=0. This fact can be easily proved, although the Maple command

> solve(sin(x)=x-x^3/6);

cannot solve it.

Problem #6

> fsolve(x-x^3/6+x^5/120-x^7/5040+x^9/9!-x^11/11!+x^13/13!, x=3..3.5);

This number is very close, but not equal, to Pi.

Problem #7

abs(a)=sqrt(2) => only one solution

abs(a)>sqrt(2) => no solutions

abs(a)<sqrt(2) => two solutions

Problem #8

Check that the expression q is a constant 1.