Engineering, Calculus and Differential Equations

Differential Equation Model of a Spring

Legend:

• m ... mass attached to the spring

• c ... dampening coefficient

• k ... spring stiffness constant

• c1 and c2 are the initial condictions

Things to Explore.

Pause to think it through and hypothesize, then adjust the sliders to test your reasoning.

• What should reducing the dampening to zero do?

• What should makeing the spring stiffer do?

• What should adding to the mass on the end of the spring do?

Linear System of Differential Equations

A brief review of the characteristic polynomial's roots and related types of equilibrium solution at (0,0)

• purly imaginary roots => center/orbit

• complex roots => spiral sink or source depending on sign or the real part

• real roots of oposite sign => saddle

• real roots of the same sign => sink or source depending on sign

The Mean Value Theorem

When certain conditions are true, the derivative mean value theorem says that there must exist at least one point c between the two endpoints A and B such that the slope of f(x) at that point matches the secant slope between the endpoint value f(A) and f(B).

Again when certain conditions are true, the integral mean value theorem says that the average height of a function g(x) between the endpoints A and B must intersect the graph at some point(s) c.

Adjust the slider to explore the aplicability of these theorems for several different functions. See if you can figure out what the prerequisite properties are by finding the functions where the theorem doesn't apply.

Can you spot the relationship between f(x) and g(x)? Hint: it is this relationship that makes the two theorems equivalent.

Taylor Polynomial Approximation of a dampened Osillation