In short, by introducing the modulus of continuity, we reveal the practical, computational and quantitative aspect of continuity, and are able to do this because computer graphics step over the fact that the computations are quite ugly in all but the simplest examples. They also reveal an important connection between slopes of tangent lines and epsilon-delta ratios that is explored in the project.
f(x) = x
or they can be rewritten that way. One procedure for solving such equations is to take an approximate solution -- a starting guess -- x0 -- and to define x1 = f(x0) and then inductively, xn+1 = f(xn). Under certain circumstances, the sequence xn will have a limiting value z as n tends to infinity, and this limiting value will be a solution of our equation; i.e., f(z) = z. The goal of this project is help the student understand the the key to whether or not this works is the slope of the tangent line to the graph of f at the solution z: If this slope is less than one in absolute value, the iteration quickly converges to the solution. but otherwise, it does not. Moreover, they reveal that the smaller the absolute value of the slope, the more rapid the convergence is. Hence the best possible case is if this slope is zero. These experiments pave the way for understanding why Newton's method works, and more, why it provides such rapid convergence.
f(x) = 0
and of course any equation can be rewritten in this form. The key is replace f(x) with the equation for its tangent line at some point near a solution -- an initial guess x0. One then solves for the point x1 where the tangent line crosses the x-axis, which is easy. If we are in a region where the tangent line is a good fit to the function's graph, then this point x1 will be a better approximation. Now, take the tangent line at x1, and keep going... The goal of this project is help the student understand how astonishingly quickly the method works, to understand the geometry behind it -- and why, once again, slopes of tangent lines are the key, and to get a feel for how close the initial guess has to be for the method to work -- a matter of great practical concern. It should also help the student understand Newton's method in the general context of iteration of functions explored in the previous project. In fact, at the level of formulas, The method is to iterate the function
F(x) = x - f(x)/f'(x)
Notice that when F(x) = x it must be the case that f(x)/f'(x)=0, so that if f'(x) is always finite, f(x) = 0 which is what we want. Moreover F'(x) = f(x)f''(x)/(f'(x))2 which vanishes where f(x) = 0, at least provided (f'(x))2>0. Thus, with Newton's method, we are in the "best possible" situation with the function we are iterating: its slope is zero at the solutions of F(x) = x. The project will also reveal that the sizes of f''(x) and f'(x) play the key role in determining how close an initial guess has to be, and reveal some very surprising phenomena when there are more than two solutions to f(x) = 0.
p(x) = anxn + an-1xn-1 + ... + a1x + a0
there are n complex numbers z1, z2, ..., zn so that p(x) can be completely factored as
p(x) = an(x-z1)(x - z2) ... (x - zn)
for all x. Although the result is algebraic, it can best be understood in geometric terms. The goal of this project is to help the student understand the validity of the theorem in geometric terms. A secondary goal is to make the point that even though a problem may not appear to be geometric at first sight, geometric reasoning may still provide the key. It is important to fully understand this point to successfully apply mathematics in many instances.