We take the vocabulary of mathematics for granted and, more fundamentally, we accept from experience the need for a well defined vocabulary. Students frequently refuse to believe in either the words or their necessity. Problems that explore and reinforce the meaning of mathematical vocabulary develop mathematical literacy, a fundamental requirement of successful analysis, problem solving, and communication.
Like a new language, the words must be heard and spoken. We frequently ask:
Which of the following equations are linear (or homogeneous or ...)?Mastery of the vocabulary demands the converse, though:
Give three examples of linear (or homogeneous or ...) equations.Give an example of a linear homogeneous equation.
Give an example of a linear equation that has the trivial solution.
Give an example of a nonlinear homogeneous equation.
Give an example of an equation that is nonhomogeneous and has the solution y = 0.
Ask for connections among ideas upon which the vocabulary is based:
What is the amplitude of the trivial solution?What is the amplitude of the solution of a linear homogeneous second order equation subject to zero initial conditions?
Find the amplitude and the maximum displacement of the mass in the spring-mass system governed by x'' + x = 0, x(0) = 1, x'(0) = 0.
Find the period of the mass in the preceding exercise. Would doubling the initial displacement change the period? Would it change the amplitude?
The Laplace transform of a function is
. Is it oscillatory, periodic or neither?