In this applet, we see a function [math]f[/math] graphed in the [math]xy[/math]-plane. [br]You can move the blue point on the [math]x[/math]-axis and you can change [math]\delta[/math], the "radius" of an interval centered about that point. [br]The point has [math]x[/math]-value [math]c[/math], and you can see the values of [math]c[/math] and [math]f(c)[/math]. [br]You can use the pre-loaded examples chosen with the slider or type in your own functions with option 10.[br][br]We say [math]\lim_{x\to c} f(x)[/math] exists if all the values of [math]f(x)[/math] are "really close" to some number whenever [math]x[/math] is "really close" to [math]c[/math].

[b]Explore[/b][br][br][list][br][*]Start by dragging the blue point on the [math]x[/math]-axis. What is the relationship between[br]the red segment on the [math]x[/math]-axis and the green segment(s) on the [math]y[/math]-axis?[br][*]What does the [math]\delta[/math] slider do? Notice that [math]\delta[/math] does not ever take on the[br]value of zero. You can "fine tune" [math]\delta[/math] by clicking on the slider button[br]then using the left and right keyboard arrows.[br][*]As [math]\delta[/math] shrinks to [math]0[/math], does the green area[br]always get smaller? Does it ever get larger? Does the green area always shrink down[br]to a single point?[br][*]Try the various examples in the applet to get a good feeling for your answers[br]in the previous problem.[br][*]Example 5 shows a function that is not defined at [math]x=1[/math]. Even though [math]f(1)[/math] has[br]no value, we can make a good estimate of [math]\lim_{x\to 1} f(x)[/math]. In this case,[br][math]\lim_{x\to 1} f(x)[/math] tells us what [math]f(1)[/math] "should" be. Use zooming to estimate this limit. [br][*]In Examples 6 and 7, the function is undefined at [math]x=2[/math]. (The function truly is undefined,[br]even though the applet shows [math]f(2) = \infty[/math]. Check this yourself by plugging[br]in [math]2[/math] for [math]x[/math] in the function). What is the value of [math]\lim_{x\to 2} f(x)[/math]?[br][*]Example 8 is a function that gets "infinitely wiggly" around [math]x=1[/math].[br]What happens if [math]c=1[/math] and you shrink [math]\delta[/math]? Try this: make [math]c=1[/math] and [math]\delta=0.001[/math].[br]What will happen as you move [math]c[/math] slowly toward [math]1[/math]? Make a guess before you do it.[br][/list][br][br][b]Project idea[/b][br]Let [math]f(x)[/math] be a function and define [math]g(x) = \lim_{t \to x} f(t)[/math].[br]Be careful to distinguish between [math]t[/math] and [math]x[/math] You may have to read the definition[br]of [math]g(x)[/math] several times and think carefully about the situation. (This mixture of[br]variables [math]x[/math] and [math]t[/math] comes up again later when we discuss integrals.)[br][list][br][*]What is [math]g(c)[/math] when [math]f[/math] is continuous at [math]x = c[/math]?[br][*]What is [math]g(c)[/math] when [math]f[/math] has a removable discontinuity at [math]x = c[/math]?[br][*]What is [math]g(c)[/math] when [math]f[/math] has a jump discontinuity at [math]x = c[/math]?[br]Does it depend on whether or not [math]f(c)[/math] is defined?[br][*]What is [math]g(c)[/math] when [math]f[/math] has an infinite discontinuity at [math]x = c[/math]?[br][*]Give an example where the domain of [math]g(x)[/math] is bigger than[math] f(x)[/math].[br][*]Give an example where the domain of [math]g(x)[/math] is smaller than [math]f(x)[/math].[br][*]Give an example where [math]g[/math] and [math]f[/math] have the same domain.[br][*]Is [math]g(x)[/math] always a continuous function?[br][*]Is it possible for [math]g(c)[/math] and [math]f(c)[/math] to be defined but not equal?[br][/list][br][br]This is a modification of an applet designed by Marc Renault.

The metatheorem of calculus is that almost any function looks like a straight line when we zoom in far enough. [br]When we have zoomed in far enough we cannot tell the difference between the graph and the tangent line.[br]If we have zoomed in that far we can approximate the tangent line with a secant line.[br]The slope of that line is the derivative at that point.

This applet zooms in on y=f(x) around the point A.[br]A is a draggable point.[br]The box extends out from A by a distance of zoom up and down, right and left.[br]In the zoom window you can also see the secant line using A and a point zoom/2 to the right of A.

This applet shows some of the features of Newton's method.[br]When Newton's method works, it converges quickly,[br]The applet lets you focus on either the full sequence of points or following through step by step.

The Applet comes with number of functions preloaded.[br]Each illustrates some features:[br][list][br][*][math]f(x)=2x \cos(x)-.84[/math] [br]This is a nice function with lots of roots.[br][*][math]f(x)=x^4-2x^3-x^2-2x+2[/math] [br]This is a nice polynomial with two roots.[br][*][math]f(x)=x^4-3x^3+x^2+2x-2[/math] [br]Although this looks similar to the previous problem, starting at some points causes an endless loop that never converges.[br][*][math]f(x)-x^2+1[/math] [br]This problem clearly has no roots. It shows what Newton tries to do in such a case.[br][*][math]f(x)=cbrt(x)[/math] [br]This is a classical problem where Newton's method does not work.[br][*][math]f(x)=\cos(x)-x/5[/math] [br]This function has an obvious root. It also has a lot or relative extrema so there are i lots of bad starting points.[br][*]Finally, you can enter your own function.[br][/list][br][br]With each functions there are some interesting questions to ask:[list][br][*] How big is the region around a root where the function will converge quickly, say within 15 steps?[br][*] Are there places where the root found is not stable, that is where a small change in the starting point gives a big change in the root found?[br][*] Are there regions where we don't find a root, even with lots of iterations?[br][/list]

Shows the value of an approximating Riemann sum and Trapepzoid sum also.

Enter a function of [math]x [/math].[br]Choose the degree of the polynomial by sliding point [math]n[/math] on the slide bar.[br]Choose the center of the polynomial by sliding point [math]a [/math] on the slide bar.