Mike May Copy of Calculus
This is a collection of applets for single variable calculus. It should grow as time goes on. Let me know if you can think of demonstrations that should be added.
Table of Contents
- Overview
- Overview of Calculus Book
- AP Calculus AB: Average Value of a Function
- Derivative drill without trig functions
- Chapter 1 Preliminary material
- Chapter 1 overview
- Exponential Functions
- Translations and Compressions
- Sin Curve Fitting
- Translation Fitting Drill
- Family of Functions
- Unit Circle - exact values
- Polar Grapher
- Chapter 2 Limits
- Intuitive Definition of Limit
- Delta-Epsilon Limit demonstration
- Chapter 3 Derivatives conceptually
- Numeric Derivative at a Point
- Zoom to tangent line
- Secant to Tangent Line
- Derivative Builder
- Second Derivative Plot
- Product and Quotient Rules Derivatives Drill
- chain rule demonstration
- Chapter 4 Derivatives symbolically
- Derivative Plot
- Simple Derivative Drill Applet
- Chain Rule Derivatives Drill
- Chain Rule With One Inner Function
- Chain Rule with Two Inner Functions
- parametric derivative
- Calculus - The Derivative of an Inverse Function
- Chapter 4 Applications of the derivative
- Newton's Method
- Linearization and Linear Approximation Demonstration
- Linear (Tangent Line) Approximation
- Optimize area of side-by-side rectangular plots
- Section 2.8-Related Rates-Ladder Falling
- Related Rates - Square
- Slope Fields
- Chapter 5 Integrals
- Riemann Sum from a to b.
- Comparing Numeric Integration Methods
- U-Substitution demonstration
- Integration Drill Applet
- Euler's Method
- Chapter 6 Applications of integration
- Area Between Curves
- Solids Of Revolution: Disk Method
- Volumes: Disk Method
- Solids Of Revolution Shell Method
- Visualizing solids with regular cross sections
- Arc Length Exploration
- Solids with Known Cross-Sections Project
- Chapter 7 Series
- Taylor Polynomial of f(x) centered at point a
- Customizable Solids of Revolution (y-axis)
- Radius of convergence
Overview of Calculus Book
Overview
This GeoGebraBook is intended as a collection of material for single variable calculus.[br][br]It will be organized following the normal syllabus of a standard course.[br][br]This is also something of a proof of concept project. It will have a number of additions that [br]can be described as me checking to see if I can do things with features of Geogebra or GeoGebraBooks[br]or to show off constructions others have made that I think of as cool.[br][br]This page is my first attempt at looking at the new page structure in GeoGebraBooks, allowing richer content on a page.[br][br][br]*************************************************************[br][br]As mentioned above this is also a proof of concept project.[br]Some of the applets are my own work. Some are modifications of applets created by others. Some are simply the work of others.[br][br]The applets included span a variety of models for what an applet can do.[br][br]A number of the applets are visual demonstrations of some concept. [br]The applets I choose tend to give you the option of using your favorite function in the demonstration.[br][br]I think of this as "classic GeoGebra"
As I modification of the first model, I have moved toward having a number of preset examples ready for selection, typically with a slider bar. This makes it easy to work through the examples that show features of interest.
I have also started playing with applets that are designed as drill tools. This uses the CAS engine. It lets students do unlimited drill and get immediate feedback.
I also have found interesting applets that do a fixed example, but use several sheets for explanation.
In the review material I use an applet that simply reviews trig functions of standard angles.
Chapter 1 overview
Whenever we teach calculus we find a need to do some review of previous material.[br][br]This chapter provides applets for review of material where calculus depends on previous material.[br]1) Family of Functions is designed to look at functions with parameters. We want students to have a good intuitive grasp of the generic graph of classes of functions,[br]2) Trig review[br]3) Compression and translation
Intuitive Definition of Limit
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.
Numeric Derivative at a Point
Finding the derivative at a point turns into a viewing window problem.[br]I want to find a delta-epsilon window so that the curve looks like a line.[br](I should not be able to tell the difference between the curve and the secant line from [br](c-delta,f(c-delta)) to (c+delta, f(c+delta)).)[br][br]I then approximate the derivative by finding rise/run for the line.
For most functions, it is easiest to use the calculator definition of derivative. Set epsilon to 0.001, and set delta to something that keeps the curve in the viewing window. [br][br]Use this method to find the derivative of the given function at three points. then try with another function.[br][br]If you look at a badly behaved function, like [math]f\left(x\right)=sin\left(\frac{1}{x^2}\right)[/math] at x=0.1, you need a smaller epsilon.[br][br]This applet is meant as an illustration of the definition of a derivative at a point.[br][br]It has the advantage of showing that numeric differentiation is quite robust.[br]The default curve is a parabola, where students will be able to find the derivative symbolically.[br][br]
Derivative Plot
As A moves slope of tangent line is plotted. Move A along x-axis or use Animate button.
Derivative Plot
Newton's Method
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]
Riemann Sum from a to b.
Shows the value of an approximating Riemann sum and Trapepzoid sum also.
Riemann Sum from a to b.
Area Between Curves
A standard application of integration is to find the area between two curves.[br][br]The integration unit is the top function minus the bottom function.[br][br]The basic integral is [br][br][math]\int_{LeftEnd}^{RightEnd}\left(TopCurve-BottomCurve\right)dx[/math] [br][br]It should be noted that if top and bottom, or left and right, are reversed, the area is negative.
It is always good to start with a problem where we can find the answer without using integration.[br][list][*]A rectangle, Top(x)=4, Bottom(x)=1, Left = -2, Right =3[/*][*]A triangle, Top(x)=x, Bottom(x)=-1, Left = -1, Right =3[/*][*]A circle, Top(x)=sqrt(4-x^2), Bottom(x)=-sqrt(4-x^2), Left=-2, Right=2[/*][/list][br]One can then look are areas we cannot compute with simple geometry.[br][list][*]Bottom(x)=x^2, Top(x)=4, Left=-2, Right=2[/*][*]Bottom(x)=x^2, Top(x)=x+2, Left=-1, Right=2[/*][/list][br]One should also look at cases where the "top" and "bottom" curves cross.
Taylor Polynomial of f(x) centered at point a
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.