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. 

Chapter working notes
Working notes on appplications
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. 
Area Between Curves
Taylor Polynomial of f(x) centered at point a
Slope Fields
This worksheet gives a demonstration of the use of new features in release 4.0 to find solution curves of a slope field. The user can create named points as the starting point of new solution curves. The point A can be moved around to show the value of the vector field at a point. 

The second graphics window is used as a control panel. With each vector field you should see if the solution curves fit into one pattern or if there are several different kinds. You should check to see if there are limit or boundary curves. The default field Dy=x, Dx=y+.02x spirals out. You should find the variation that makes it spiral in or form circles The field Dx=x, Dy=y diverges out. What happens with a field like Dx=sin(x*y), Dy=cos(x*y) Mike May, S.J. maymk@slu.edu 