Overview of Calculus Book
Overview
Chapter 1 overview
Intuitive Definition of Limit
In this applet, we see a function [math]f[/math] graphed in the [math]xy[/math]plane. 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. 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]. You can use the preloaded examples chosen with the slider or type in your own functions with option 10. 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] [list] [*]Start by dragging the blue point on the [math]x[/math]axis. What is the relationship between the red segment on the [math]x[/math]axis and the green segment(s) on the [math]y[/math]axis? [*]What does the [math]\delta[/math] slider do? Notice that [math]\delta[/math] does not ever take on the value of zero. You can "fine tune" [math]\delta[/math] by clicking on the slider button then using the left and right keyboard arrows. [*]As [math]\delta[/math] shrinks to [math]0[/math], does the green area always get smaller? Does it ever get larger? Does the green area always shrink down to a single point? [*]Try the various examples in the applet to get a good feeling for your answers in the previous problem. [*]Example 5 shows a function that is not defined at [math]x=1[/math]. Even though [math]f(1)[/math] has no value, we can make a good estimate of [math]\lim_{x\to 1} f(x)[/math]. In this case, [math]\lim_{x\to 1} f(x)[/math] tells us what [math]f(1)[/math] "should" be. Use zooming to estimate this limit. [*]In Examples 6 and 7, the function is undefined at [math]x=2[/math]. (The function truly is undefined, even though the applet shows [math]f(2) = \infty[/math]. Check this yourself by plugging in [math]2[/math] for [math]x[/math] in the function). What is the value of [math]\lim_{x\to 2} f(x)[/math]? [*]Example 8 is a function that gets "infinitely wiggly" around [math]x=1[/math]. 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]. What will happen as you move [math]c[/math] slowly toward [math]1[/math]? Make a guess before you do it. [/list] [b]Project idea[/b] Let [math]f(x)[/math] be a function and define [math]g(x) = \lim_{t \to x} f(t)[/math]. Be careful to distinguish between [math]t[/math] and [math]x[/math] You may have to read the definition of [math]g(x)[/math] several times and think carefully about the situation. (This mixture of variables [math]x[/math] and [math]t[/math] comes up again later when we discuss integrals.) [list] [*]What is [math]g(c)[/math] when [math]f[/math] is continuous at [math]x = c[/math]? [*]What is [math]g(c)[/math] when [math]f[/math] has a removable discontinuity at [math]x = c[/math]? [*]What is [math]g(c)[/math] when [math]f[/math] has a jump discontinuity at [math]x = c[/math]? Does it depend on whether or not [math]f(c)[/math] is defined? [*]What is [math]g(c)[/math] when [math]f[/math] has an infinite discontinuity at [math]x = c[/math]? [*]Give an example where the domain of [math]g(x)[/math] is bigger than[math] f(x)[/math]. [*]Give an example where the domain of [math]g(x)[/math] is smaller than [math]f(x)[/math]. [*]Give an example where [math]g[/math] and [math]f[/math] have the same domain. [*]Is [math]g(x)[/math] always a continuous function? [*]Is it possible for [math]g(c)[/math] and [math]f(c)[/math] to be defined but not equal? [/list] This is a modification of an applet designed by Marc Renault. 
Numeric Derivative at a Point
Derivative Plot
As A moves slope of tangent line is plotted. Move A along xaxis or use Animate button. 

Newton's Method
This applet shows some of the features of Newton's method. When Newton's method works, it converges quickly, 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. Each illustrates some features: [list] [*][math]f(x)=2x \cos(x).84[/math] This is a nice function with lots of roots. [*][math]f(x)=x^42x^3x^22x+2[/math] This is a nice polynomial with two roots. [*][math]f(x)=x^43x^3+x^2+2x2[/math] Although this looks similar to the previous problem, starting at some points causes an endless loop that never converges. [*][math]f(x)x^2+1[/math] This problem clearly has no roots. It shows what Newton tries to do in such a case. [*][math]f(x)=cbrt(x)[/math] This is a classical problem where Newton's method does not work. [*][math]f(x)=\cos(x)x/5[/math] This function has an obvious root. It also has a lot or relative extrema so there are i lots of bad starting points. [*]Finally, you can enter your own function. [/list] With each functions there are some interesting questions to ask:[list] [*] How big is the region around a root where the function will converge quickly, say within 15 steps? [*] 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? [*] Are there regions where we don't find a root, even with lots of iterations? [/list] 
Riemann Sum from a to b.
Shows the value of an approximating Riemann sum and Trapepzoid sum also. 

Change a and b by sliding points along Xaxis OR by typing in values in the input boxes. Change n, the number of intervals, and p, the position in the interval where the function is evaluated by using the sliders. 