Math 696 -- Exercises

The moral of the presence of integration bugs and miscellaneous bugs in Maple is that you should always ask if the answer you get from a computer program is sensible. The computer is only a tool: it does not substitute for thinking. This is a lesson that ought to be reinforced in the high schools.

The Swamped Snake

Try to get Maple to make a decent plot of the function x + x² sin(1/x) when x is in a neighborhood of the origin.
Would a calculus student be able to deduce from looking at Maple plots that this function has an infinite number of local maxima and local minima in every neighborhood of the origin?

Exercise

Use Maple to "prove" that e is an algebraic number.
Hint: read the online help about the command PolynomialTools[MinimalPolynomial].

Exercise

Consider the following initial value problem for a separable, first-order differential equation.

> diffeq:=diff(y(x),x)= -y(x)*tan(x+1);
  init:=y(0)=cos(1);
                    d
          diffeq := -- y(x) = -y(x) tan(x + 1)
                    dx
                  init := y(0) = cos(1)

It is easy to verify that the solution is y=cos(x+1). Ask Maple to solve the differential equation via the command dsolve({diffeq,init}, y(x)); and watch what happens. You may be surprised at the answer Maple returns.

Is Maple's answer correct? Try comparing Maple's answer with the known answer at a dozen integral values of x. Explain what you observe.

Exercise

Use Maple to "prove" that 1=0.
Extra credit if your instructor cannot find the flaw in the proof.

Exercise

Write an essay addressing the following question.

Given that computer programs such as Maple make mistakes, do they have a legitimate role in mathematical education and research?